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Vector Calculus
機械工程學系
白明憲
教授
MingsianMingsian R R BaiBai2
Vector Calculus (field theory)
Scalar field ( )Vector field ( ) in fluid mechanics (historical origin)
Key elements in this chapter + field integrals + curvilinedifferential opera ar coordinates
+ tors
integral theorem
p p tv v t==
xxr r
s potential t+ heory
MingsianMingsian R R BaiBai3
Divergence 散度
v velocityv
ˆ n outward normal
control volume piecewise smooth orientable closely connected surface
Net outflow (flux) per unit voluInterpr me per etat unio i in t t me
BS
rarr
0
ˆdiv ( ) ( ) lim the volume of
a spatial differential operator independent of coordinate system unique value
SV
n v dSv p v p V B
Vrarr
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
sdot= nabla
lowastrarr
intr
r r
MingsianMingsian R R BaiBai4
For the Cartesian coordinates
0 1 1
0
0 1 1
for some point ( ) on the - (by 2
mean valu
ˆ ˆ
ˆˆ ˆ ˆˆ ( ) ( )2
( )2
x face x front x back
x y z xx front x front
x
xx y z x front
n v dS n v dS
xn v dS i v i v j v k dS v x y z dydz
xv x y z y z
minus minus minus
minus minus
Δ+
sdot = + sdot
Δsdot = sdot + + = +
Δ= + Δ Δ
int int int
int int int
r r
r
e theorem)
j
xΔ
yΔ
zΔk
i
( ) P x y z center of cube=
MingsianMingsian R R BaiBai5
0 2 2
0 2 2
0
Similarly
ˆˆ ˆ ˆˆ ( ) ( )2
for some point ( ) on the - 2
Thus the net outflow per unit volume in -direction
ˆlim
x y z xx back x back
x face
V
xn v dS i v i v j v k dS v x y z y z
xx y z x back
x
n v dS
V
minus minus
minus
rarr
Δsdot = minus sdot + + = minus minus Δ Δ
Δminus
⎧ ⎫sdot⎪⎨ ⎬⎪⎩
int int
int
r
r0 0
0
( ) ( )2 2lim
x x
x y z
x xv x y z v x y z y z
Δ Δ Δ rarr
Δ Δ⎡ ⎤+ minus minus Δ Δ⎢ ⎥⎪ ⎣ ⎦=⎪⎭
x y zΔ Δ Δ
0 0
0
( ) ( )2 2 lim
Hence the net outflow per unit volume in the 3 directi
div
ons sums up to
(space operat ) or
x xx
x
yx zvv
x xv x y z v x y z vx x
vv vx y z
Δ rarr
Δ Δ+ minus minus part
=
partpart partnabla = + +
part part part
=Δ part
r r
MingsianMingsian R R BaiBai6
Gradient 梯度
g
R
r
ec
ad
all
ient
the divergence operator div
ˆˆ ˆIn Cartesian coordinates
Define the operator
a spatial differential operator
ˆˆ ˆgr
independent
ad
of coo
v v
i j kx
u u uu u i j kx y
y
z
z
part part part
=nablasdotpart part part
nabla + +part part part
rarr
nabla = + +part part part
r r
rdinates chosen ie is invariant wrt coordinates
directional derivative ( )ˆˆ ˆLet ( )
Physical interpretation
(position vector)
ˆ( ) (
u
u u x y z r xi yj zk
u u u u udu r dx dy dz ix y z x
nabla
= = + +
part part part part part= + + = +part part part part part
r
r
方向導數
ˆˆ ˆ
ˆ ˆˆ ˆ
( ) (
ˆ
)
) ( )
( )
dr r xi yj zk
uj k idx jdy kdzy z
du x u ddu r xu dr
= + +
=
part+ sdot + +part
primerarrsdot =nabla
r r
r r可視為 的推廣
MingsianMingsian R R BaiBai7
ˆ rate of change of in the direction Di
ˆ
rectional De
( )
ˆw
rivative
here
u sdu drr u u sdr dr
drsdr
=nabla sdot = nabla sdotr
r
r
s0
ˆ( ) ( )limh
du u r hs u rdr hrarr
+ minus=r r
drv
r dr+r r
rr
oNote
ˆ
ˆ )
ˆ
ˆ(
dudr
u s
s u
d sdr
s u
=nabla sdot
= sdotnabla
rArr
nabla
nabla
sdot=
sdotθ
s
s
( )0 yxx
y
u
unabla
drdu
duslopedr
=
( )yxfu =
MingsianMingsian R R BaiBai8
ˆEx Normal gradient outward normal
Note is often written as in the PDE literature
ˆˆ ˆˆPartial derivatives If then
ˆ
Physical interpretati
n
du udn n
du u u us i j kdr
du n ud
z
n
x y
partpart
part part part= rarrpart part part
sdotnabla
max
maximum rate of increase in a direction
ˆ ˆ ˆ ˆcos ( ) cos ( )
ˆ is independent of
ˆmax occurs when ( ) 0 and
points in the direction of ma
o
n
x a
du u s u s u s u u sdr
u sdu duu s udr dr
duudr
θ θ
θ ⎛ ⎞⎜ ⎟⎝ ⎠
=nabla sdot = nabla nabla = nabla nabla
nabla
there4 nabla = = nabla
rArrnabla
Q
( )max
(interpretation 1nd
gradient search steepest desce
)
nt
duudr
⎛ ⎞⎜ ⎟⎝ ⎠
nabla =
rArr minusnabla最佳化方法 的基礎
MingsianMingsian R R BaiBai9
(interpretatDirection of ion 2 normal to the level surface ( )
On the 0 ( )
ie
level surfac
o
e
n
)udu u dr
u cdu u dr
u dr u c
nabla= nabla sdot
== = nabla sdotnabla perp =
r
r
r
unabla
dr
unabla
Q
P
1cu=
2u c=
山
x
y
( )u x y
x
y
Hill-climbing search in optimization
MingsianMingsian R R BaiBai10
Ex Plane equationˆˆ ˆ( ) Normal vector
Ex 0 is the necessary condition for searching extrema (from multivariate Tylor expansion)
Ex Revisit of the exampl
u x y z ax by cz d u ai bj ck
u
= + + = rArr nabla = + +
nabla =v
多變數函數求極值
2 21 1 2 2
e in Linear Algebra using method of What is the maximum and minimum distance of the origin to the following curve
Lagrange multipli s
6 8
er
5 5x x x x+ + =
MingsianMingsian R R BaiBai11
1 2
2 21 2( )
2 21 1 2 2
Sol
This problem can be posed as a problemmin max ( )
constrained optimization
5 6 5 8 0 ( )which can
objectivecost function
constraint
equatioals
nx x
J x x
st g x x x x
= +
= + + minus =
2 2 2 21 2 1 1 2 2
o be converted into an unconstrained optimization problem by defining the ( ) (5 6 5 8)where is referred to as the thatLa
L
g ora
agr
nge ften h multi
angi
plie as
an
r ph
L x x x x x xλλ
= + + + + minus
ysical meaning in certain problemsThus the extreme values of will occur wh en or L J gJ nabla = nabla nabla0
MingsianMingsian R R BaiBai12
1 1 21
2 1 22
2 21 1 2 2
1 2 1 2
2 (10 6 ) 0 (1)
2 (6 10 ) 0 (2)
5 6 5 8 0 (constraint)
(1) (2) (2 16 )( ) 0 18 or (3)Substituting Eq (3) into Eqs (1) and (2) l
L x x xxL x x xxL x x x x
x x x x
λ
λ
λλ λ
part = + + =partpart = + + =partpart = + + minus =part+ + + = rArr = minus = minus
1 2 1 2eads to
and which corresponds to 14Thus minimummaximum distance 12x x x x J= = minus =
=
MingsianMingsian R R BaiBai13
22 2
1
1 1
2 2
The result above is identical to the previously obtained result in Linear Algebr
ellipse
45 rotatio
a
1 ( )4
1 11 1
n12
xx
x xx x
deg⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠
prime+ =prime
primeminus= rArr
prime2x prime
2x
1x prime
1x
2
MingsianMingsian R R BaiBai14
Curl 旋度
curl curl In Cartesian coordinates
spatial operator indep
ˆˆ ˆ
ˆˆ ˆ( )
ende
(
nt of coordi
) (
D
nates
ef
)
x y z
y x x xz z
i j k
vx y z
v v v
v v v vv vi j ky z x z x y
v v
part part partnablatimes =part part part
part part part partpart part= minus minus minus + minuspart
times
part part part part part
nabla
rarr
r
r r
0vnablatimes =rr 0vnablatimes ne
rr
MingsianMingsian R R BaiBai15
circulaPhysica
tion o spin ol interpretati
f fluidr or vortion
y
citΓ
( )0 0x dx y dy+ +
( )0 0x y ( )0 0x dx y+
Γ
1
2
3
4
( )0 0x y dy+
MingsianMingsian R R BaiBai16
yy
vv dy
ypart
+part
Γdyminus
dyyvv x
x partpart
+
dy
dx
dxminus
y yy
v vv dx dy
x ypart part
+ +part part
x xx
v vv dx dyx y
part part+ +part part
yy
vv dx
xpart
+part
xx
vv dxx
part+part
xv
yv
vv
dλ paddle weel
MingsianMingsian R R BaiBai17
contour
1 2 3 4
circulation
x x y y x x y y
x
v d
v d v d v d v d
v
λ
λ λ λ λrarr
Γ sdot
= + + +
asymp
intint int int int
vv
12
xv dxx
part+part ydx v
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
+ 12
y yv vdx dy
x ypart part
+ +part part
x
dy
v
⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦
+
12
xv dxx
part+part
( )xy
v dy dx vy
⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦
part+ minus +part
12
yvdy
ypart
+part
( )
( )
Similar process can be applied to the circulation in the and directionsThus the total circulation per unit area i curl s
y xz
dy
v v dxdy v dxdyx y
xv v
y
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎝ ⎠
minus
part part= minus = nablatimespart part
=nablatimes
v
v v
MingsianMingsian R R BaiBai18
Identities of differential operators ( p380 Greenberg )
( )
( ) ( )( )( )
( ) ( )( ) is required
Example of deriv
ˆ ˆˆ ˆ ˆ ˆ
ation
x y z
x y z
v v vx y z
uv u v u v
uv i j k uv i uv j uv kx y z
uv u v u v
u v v u u v
u v u vv v u u
+ +part part partpart part part
⎛ ⎞⎜ ⎟⎝ ⎠
nablasdot = nabla sdot + nablasdot
part part partnablasdot = + + sdot + + =part part part
nablatimes =nabla times + nablatimes
nablasdot times = sdotnablatimes minus sdotnablatimes
nablatimes times = nabla minus + minusnablanablasdot sdotsdot678 678
v
v v v
vL
v v v
v v v v v v
v v v v v v v v( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )2 cf
x y zu u ux y z
A B C A
u v
v w v w w v v w w v
v v v C B A B C
+ +part part partpart part part
sdotnabla
nabla sdot = sdotnabla + sdotnabla + times nablatimes + times
times
nablatimes
nablatimes nablatimes =nabla nabla times = sdot minussdot sdotminusnabla
678v
v v v
v v v v v vv v
v v v v v v
v v
v
v v
ˆ ˆˆ ˆ ˆ ˆ( ) ( ) ( )x y z x y zv i v j v k i j k u v v v ux y z x y zpart part part part part part+ + sdot + + = + +part part part part part part
v v
MingsianMingsian R R BaiBai19
2 2 22
2 2 2
0 (dc direct current) solenoidal0 (cg center of gravity) conservative potent
Laplacia
Zero identitie
n
s
ial
x y z
vu
⎧⎪⎨⎪⎩
part part partΔ =nabla nablasdotnabla = + +part part part
nablasdotnablatimes = rarr
nablatimesnabla = rarr
v
v
These differential operators will take more complicated and irregular form for the other coordinate systems
MingsianMingsian R R BaiBai20
2 2 3
2 2
ˆˆ ˆEx 2
ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ 2 0 2
2 0 0 2yx z
u x y v x yi z j k
u i j k u xyi x j k xyi x jx y z
vv vv xy xyx y z
⎛ ⎞⎜ ⎟⎝ ⎠
= = minus +
part part partnabla = + + = + + = +part part part
partpart partnablasdot = + + = + minus =part part part
v
v
MingsianMingsian R R BaiBai21
( ) ( ) ( )
( ) ( )
2 2 2 2
2 2 22
2 2 2
2
ˆˆ ˆ
ˆˆ ˆ
ˆ ˆˆ ˆ ˆ 0 3 0 0 0 3
2 0 0 2
ˆˆ ˆ
y yx xz z
x y z
i j kv vv vv vv i j k
x y z y z x z x yv v v
z i j x k z i x k
u u uu y yx y zv v v
i j kx y z
⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠ ⎝ ⎠
⎛
⎝
part partpart partpart partpart part partnablatimes = = minus minus minus + minuspart part part part part part part part part
= + minus minus + minus = minus
part part partnabla = + + = + + =part part part
nablatimes nablatimes =nabla nablasdot minusnabla
part part part= + +part part part
v
v v v
( ) ( )( ) ( ) ( ) ( )
( )
2 2 22 3
2 2 2ˆˆ ˆ2 2
ˆ ˆˆ ˆ ˆ ˆ 2 2 0 2 0 0 0 0 6 0 0 0
ˆ 2 6
xy x yi z j kx y z
yi xj k y i z j k
x z j
⎛ ⎞⎞⎜ ⎟⎜ ⎟
⎠ ⎝ ⎠⎡ ⎤⎣ ⎦
part part partminus + + minus +part part part
= + + minus + + + + minus + + minus
= +
MingsianMingsian R R BaiBai22
( )v w r w r r wnablatimes =nablatimes times = nablasdot minus nablasdotv v v v v v v ( )r w+ sdotnablav v ( ) constant w
w rminus sdotnablav
v v
Q
Ex Rigid body rotation
MingsianMingsian R R BaiBai23
( )( ) ( )
( ) ( )
ˆˆ ˆ
ˆ ˆˆ ˆ ˆ ˆ 1 1 1 3
ˆˆ ˆ
ˆˆ ˆ
Thus 3 2 is linked directly
x y z
x y z
r xi yj zk
r i j k xi yj zkx y z
w r w w w xi yj zkx y z
w i w j w k w
w r w r w r w w wv
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎝ ⎠
= + +
part part partnablasdot = + + + + = + + =part part part
part part partsdotnabla = + + + +part part part
= + + =
nablatimes times == nablasdot minus sdotnabla = minus =rArrnablatimes
v
v
v v
v
v v v v v v v v v
v spinn to ing
MingsianMingsian R R BaiBai24
Line surface and volume integrals
x
yC
B
A
isΔ
( )ii ηξ
Piecewise smooth curve
Line integral
MingsianMingsian R R BaiBai25
0 1lim ( ) ( ) or ( )
Intepretation of a line integral as an area ( )
i
Bn
i i isA
C
i C
F s F x y ds F x y d
x
s
F y ds
ξ ηΔ rarr =
Δ =sum int
int
int
x
y
z
C
( )z F x y=
MingsianMingsian R R BaiBai26
the area is always on the left of an obs
Positive dierver trave
rection of irsing on the
ntegrationcurve
RC
CCW CW
If is a regular closed curve piecewise smo
( ) ( )
(contou
oth simple ( )
r integral)
x y zC C
x y zC
F dR F F F dx dy dz
F dx F dy
C
F dz
sdot = + + + +
= + +
lowast
int int
int
i j k i j kv v
無交叉
work dW F dR= sdotv v
MingsianMingsian R R BaiBai27
0 1
Surface integral
lim ( ) ( )i
n
i i i iS i S
F S F x y z dSξ η ζΔ rarr =
Δ =sum intint( ) i i iξ η ζ
iSΔ
0 1
mass density
lim ( ) ( )
Ex
Volume integral
( )
i
n
i i i iV i V
V
F V F x y z dV
M x y z dV
ξ η ζ
ρ
Δ rarr =Δ =
=
sum intintint
intintint( ) i i iξ η ζ
iVΔ
MingsianMingsian R R BaiBai28
Ref (1)Hildebrand advanced Calculus for application Prentice-Hall 1976 p306-313(2) Greenberg p383-389 (3) Arfken ch2
1 2 3
1 2 3
1 2 3
1 2 3
(
Orthogonal coordinate systems base vectors are not constant in s
) ( )( )( )( )
ˆˆ ˆ
pacep x y z p u u ux x u u uy y u u uz z u u u
r xi yj zk
primerarr
= +
rarr
===
+v
1uv
2uv3uv
1u
2u
3uP coord curve
xy
z
O
X Curvilinear Coordinates
MingsianMingsian R R BaiBai29
1
11
1 1 1
1
1
1 1 1
11 1
1 1
2 2 2 3 3 3
Tangent vector to the at
where is the arc length
1
where unit vector
Similarly
curv
e
u psr rU
u s us
rs
U h uds rh udu sU h u U h u
partpart part= =part part part
part =part
there4 =part= =part
= =
v vv
vQ
v v
vv
v vv v
1u cood curve
1uv
1du 1dS
drv
P
MingsianMingsian R R BaiBai30
3 31 2 1 21 2 3
1 2 32
curv
In summary scale factor
If is the arc length along a in any directione
k k k
kk k
k k
k k k
U h u
ds rh Udu u
ds h dus
du dudu du du dudr r r r U U Uds u ds u ds u ds ds ds ds
dr drds ds
=
part= = =part
=
part part part= + + = + +part part part
= sdot
v v
v v
v v v v v v v
v v 3 3
1 1
23 3 32 2
21 1 1
2 2 2 2 2 2 2 2 2 21 1 2 2 3 3 1 2 3
1 (for orthogonal coord)
jii j
i j
ji ii j i
i j i
dududr U Uds ds ds
dudu duU U hds ds ds
ds h du h du h du ds ds ds
= =
= = =
⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
= sdot
rarr = sdot =
rarr = + + = + +
sum sum
sumsum sum
v v v
v v
MingsianMingsian R R BaiBai31
1 1 2 2 3 3
1 2 3
are mutually vectors having with arc length
k k k k kds h du U du
U du U du U duds ds ds
= =
rArr perp
v
v v v
1uv
2uv
3uv
1dS
2dS
3dS
dS
MingsianMingsian R R BaiBai32
( ) ( )
( ) ( )
1 1 2 2 3 3 1 2 3 1 2 3 1 2 3
1 2 3 1 2 3
1
1 2 2 3 3 2 2 2 3 3 3 2 3 2 3 1
Differential volume
Differential On const coord surface
area
Gradi t
en
d U du U du U du u u u h h h du du du
d h h h du du du
ud U du U du h u du h u du h h du du u
df
τ
τ
σ
= times sdot = times sdot
=
=
= times = times =
= nabla
v v v v v v
v vv v v v
1 2 31 2 3
1 2 3 1 1 2 2 3 31 2 3
1 1 1 2 2 2 3 3 3
f f ff dr df du du duu u u
r r rdr du du du U du U du U duu u u
h u du h u du h u du
part part partsdot rarr = + +part part part
part part part= + + = + +part part part
= + +
v
v v v v v vv
v v v
=1 (right hand rule)
MingsianMingsian R R BaiBai33
Rectangular Cartesian Coordinates
ij
k
MingsianMingsian R R BaiBai34
Circular Cylindrical Coordinatescossin
xyz z
ρ ϕρ ϕ
===
Ref Arfken
( )P zρ ϕ
MingsianMingsian R R BaiBai35
Polar Spherical Coordinatessin cossin sincos
x ry rz r
θ ϕθ ϕθ
===
Ref Arfken
0
0
0
sin cos sin sin coscos cos cos sin sin
sin cos
θ ϕ θ ϕ θθ ϕ θ ϕ θϕ ϕ
= + +
= + minus= minus +
r i j kθ i j kφ i j
2
2
sinsin
dS r d ddV r dr d d
θ θ ϕ
θ θ ϕ
=
=( )P r θ ϕ
MingsianMingsian R R BaiBai36
Integral TheoremsHeart of this chapter
Interchange of dimensionality Div Curl Grad in integrals High-dimensional integration
rarr
1
Thm1 Divergence theorem Let be a closed region bounded by piecewise smooth orientable surface and let (con
I Divergence theor
tinuous derivatives up to the first order
em (Gauss theorem)
)
VS v Cisin
r
ˆ If denotes the outward unit normal on
the ˆn V S
v dV n v dS
n S
nablasdot = sdotint intr r
MingsianMingsian R R BaiBai37
jS
jP
jV
S
0
ˆBy definition ( ) lim
ˆ( )
ˆ( )
j
j
j
j
Sj V
j
j j S
j j Sj j
n v dSv P
V
v P V n v dS
v P V n v dS
Δ rarr
⎧ ⎫sdot⎪ ⎪nabla sdot ⎨ ⎬Δ⎪ ⎪⎩ ⎭
nabla sdot Δ asymp sdot
nabla sdot Δ = sdot
int
intsum sumint
rr
r r
r r
Pf
MingsianMingsian R R BaiBai38
0
After interior surfaces cancel out only exterior surface (Why Opposite normal vectors)
ˆ( )
volumes exterior surfaces
ˆlim
QE
survive
j
j
j j Sj j
V SV
v p V n v dS
v dV n v dSΔ rarr
nabla sdot Δ = sdot
rArr nablasdot = sdot
sum sumint
int int
r r
r r
D
MingsianMingsian R R BaiBai39
( )
scalar variable constant vectorvector variable
Alternative formsLet be a an arbitrary and
a
ˆ
1 Le
(
ˆt
V
V S
S
V
v ap
v d
v dV vn d
V n v dS
v va
va dV v a v a
S
nablasdot = sdot
= rArr
nablasdot = nabla sdot + nabla
=
sdot
nablaint intint int
int
r
v
r r
r r
r r rQ ( )
( )
ˆ)
2 Let ˆV S
V
V S
dV n va dS
v a p
a p
p dV n p d
d p
S
V a
⎛ ⎞= sdot⎜ ⎟
⎝ ⎠
= times rArr
nablasdot times = sdot nabla
times
times
nabla = times
int int
int int
int
r
r r r
r r v
r
vQ
r
( ) ( ) ( )
( ) ( )
ˆ
ˆV S
V S
a p dV n a p dS
a p dV a n p dS
⎛ ⎞⎡ ⎤minus sdot nablatimes = sdot times⎣ ⎦⎜ ⎟⎜ ⎟⎜ ⎟rArr minus sdot nablatimes = minus sdot times⎜ ⎟⎝ ⎠
int int
int int
r rv v
r rv v
MingsianMingsian R R BaiBai40
( )
( )
From Leibnitzs rule
(1-D) ( )
( ) ( )
B t
A t
d f x t dxdt
f x t dx B ttpart prime= +part
int( )
( )( ( ) ) ( )
B t
A tf B t t A tprimeminusint ( ( ) )
(assume ( ) ( ) )
f A t t
A t B t const= =
ContinuityEx The Equation
rate of increase of mass
mass densityV
dM d dVdt dt
ρ
ρ
= int
flow velocityvv
MingsianMingsian R R BaiBai41
ˆ ˆ ( )
(
Conservation
) ( Divergence theorem)
of mass
( ) 0
V S S
V
V
dV n v dS n v dSt
v dV
v dVt
ρ ρ ρ
ρ
ρ ρ⎡ ⎤⎢ ⎥⎣ ⎦
part = minus sdot = minus sdotpart
= minus nablasdot
partrArr +nablasdot =part
int int int
int
int
v v
vQ
v
ˆOn the other hand rate influof mass through n ox i t
Extended to 3-D for time-independent fixed boundary
(3-D) ( ) ( )
Thus
S
V V
V
S V n v dS
Sd f x y z t dV f x y z t dVdt tdM dVdt t
ρ
ρ
= minus sdot
part=part
part=part
int
int int
intv
Fixed control volume (Eulerian coordinates)Otherwise Raynoldrsquos transport theoremMoving control volume Lagrangian coordinates
MingsianMingsian R R BaiBai42
But the control volime is only arbitrary
In particular for flow
( ) 0
con
(continuity equati
stant
on)
Divergence the
0 incompress
Note is t
ible
orem h
V
v
vt
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
e mother of the other integral theorems
MingsianMingsian R R BaiBai43
Ex Archimedes principle ( ) The buoyant force ( ) on a body with volume immersed in a fluid is equal to the weight of the fluid
ˆ displaced by the object ie In the followin
B
V
gVρ= minusf k
阿基米德原理
浮力
ˆg figure the ˆ with and being the fluid density gravitational
acceleration depth and outward normal unit vector to the surface
fhydraulic pressure g z
g z
S
ρ
ρ
= minusp n
n
1
the modified divergence theorem
ˆ
Hint
scalar fuctionV S
vdV v dS v Cnabla = isinint int n
MingsianMingsian R R BaiBai44
Levelsurfacefluid
V
S
airyx
z
ˆf g zρ= minusp n
ˆPf ( ) ( )
ˆ ˆ ˆ ˆ ˆ (0 0 ) v v
B fS S V
V V
g z g zdS dS dV
g dV g dV gV
ρ ρ
ρ ρ ρ
= = minus = minus nabla
= minus + + = minus = minus
int int int
int int
f p n
i j k k k
k
MingsianMingsian R R BaiBai45
Note This can also be thought of as simple balance of forces (resultant force that acts on and weight of filled with fluid)S V
2
~ a little history about George Green Recall the diveregence theorem
ˆ
Let
II th
e Green
are arbitrary fun
s identiti
ctions
es
V S
v dV n v dS
v u vu v C
nablasdot = sdot
= nablaisin
int intv v
v
ˆ ( ) ( )
ˆ ( )V S
V S
u v dV n u v dS
u v u v dV un v dS
nablasdot nabla = sdot nabla
nabla sdotnabla + nablasdotnabla = sdotnabla
int int
int int
MingsianMingsian R R BaiBai46
2 -( ) -(1)V S
vu v u v dV u dSnpart
nabla sdotnabla + nabla =partint int firs(Greens identt ity)
2
If
( ) --(2)
(1) (2) V S
u vuv u v u dV v dSn
harrpart
nabla sdotnabla + nabla =part
minus
int int
( )2 2
V S
dv duu v v u dV u v dSdn dn
⎛ ⎞nabla minus nabla = minus⎜ ⎟⎝ ⎠int int
second(Greens identity)
MingsianMingsian R R BaiBai47
( )
( )
( ) ( )
1st id ( )
Greens identities in the 1D case integration by parts
b b bb ba a
a a ab
ba
ab
ba
a
v udx v u v u dx u v uv dx uv
u v u v vu dx vu
uv vu dx uv vu
primeprime prime prime prime prime prime primeprime prime= minus rArr + =
prime prime primeprime primeharr + =
primeprime primeprime prime primerArr minus = minus
int int int
int
2
2nd id ( )
Good for any adjoint operators in addition to Good also for and integration by parts Important for theory amp reciprocity
From th
Gree
e 1s
2-
t
ns functi
identity
D 1-
tak 1
D
e
on
u
nablararr
=
int
2
V S
vv dV dSnpartnabla =part
rArr int int2Ex 0 st 0 on the boundary 0 ( 1st id u u unabla = = rArr equiv 以 証)
MingsianMingsian R R BaiBai48
1
Thm1 Stokes theorem Let be defined in Let be a piecewise-smooth orientable surface bounded by a piec
III Stokes theorem and
ewise-smooth simple clos
Greens
ed curve Then
theore
m
v C R S
C
isinv
( )
ˆwhere is a unit normal to according t right hand rulo the e
ˆS C
n S
n v dS v dRsdot nablatimes = sdotint intvv v
(not necessarily planar)S
Cn vnablatimes v vv
MingsianMingsian R R BaiBai49
Pf
( )( )
circulatioWe have shown in deriving thatthe in -direction
ˆor
ˆ
n
z
z
j j j j j
j j j j jj j
vz v dx dy
v dR n v S
v dR n v S
nablatimes
=nablatimes
sdot = sdot nablatimes Δ
sdot = sdot nablatimes Δ
intsum sumint
v
v
vv v
vv v
C
S
jS
jC
jP
ˆ jn
jSΔ
MingsianMingsian R R BaiBai50
( )
( )ext surface
After the interior boundaries cancel out only the exterior boundaries (Why)
ˆ
ˆ0 QED
Alternative forms scalar
survive
va
j
j j j j jC
jC S
v dR n v S
S v dR n v dS
φ
sdot = sdot nablatimes Δ
Δ rarr rArr sdot = sdot nablatimes
sum sumint
int int
vv v
vv v
( )
riable constant vector vector variable
1 Let
2 Let
ˆ
ˆ S C
S C
a p
v n dR dR
n p dS dR p
a
v a p
φ φφ= rArr timesnabla =
timesnabla times= rArr timestimes =
int int
int int
v v
v v
v
v v
vv vv v
MingsianMingsian R R BaiBai51
( ) ( )
( )
( )
( ) ( )
1
Thm2 Greens theorem ( planar version of Stokes theorem)ˆ ˆ Let
pf By Stokes theoremˆ
ˆ ˆ ˆ ˆ ˆ ˆ
S C
S C
S C
v P x y i Q x y j C
n v dS v dR
k Pi Qj dS Pi Qj dx i dy
Q P dS Pdx Qdyx y
⎛ ⎞part partminus = +⎜ ⎟
= + isin
sdot nablatimes = sdot
⎡ ⎤sdot nablatimes + =
part part⎝ ⎠
+ sdot +⎣ ⎦
int int
int
int
int
int
v
vv v
( )
( )
( )
ˆ
ˆˆ ˆ
ˆˆ ˆ ˆ ˆ( ) ( )
0
QEDS C
j
i j kQ PPi Qj i j k
x y z x yP Q
Q P dS Pdx Qdyx y
⎛ ⎞part part part part partnablatimes + = = + + minus⎜ ⎟part part part part part⎝ ⎠
⎛ ⎞part partrArr minus = +⎜ ⎟part part⎝ ⎠int int
L L
S
C
i
j
x
y
MingsianMingsian R R BaiBai52
Potential TheorypotentiGeorge Green coined the term (l
a )
rarr 位能 勢能
例子 重力場 電磁場 流場 聲場 彈性力學
1
2
Thm1 If is in a simply connected region then the following statements are equivalent(i) There exists a scala
Conservative Field
potential functior such that
(ii
n
) 0
v
C
C
v
v R
φφ=nabla
nablatimes
isin
isin
=r
r
r
r in (Irrotational field curl-free)
cf 0 st solenoidal field divergence-free
0
(iii) in (Conservative field)C
Rv w v w
v R Rd
nablasdot = exist nabla
sdot =
= times
int
r r r
r
r
r
MingsianMingsian R R BaiBai53
(iv) is connecting and
independent o
Ex Think
f the pa
of dynamics is the force
thB
A
v dR A B
v
sdotintrr
r
1 2
1
0
Pf
(i) (ii) ( ) 0 (zero identity cg rule) irrotational
ˆ(ii) (iii) ( ) 0 (Stokes theorem)
(iii) (iv) ( ) 0
C S
C C C
C
v
v dR n v dS
v dR v dR
v dR
φ
=
rarr nablatimes = nablatimes nabla =
rarr sdot = sdot nablatimes =
rarr sdot = minus sdot =
rArr sdot =
int int
int int int
int
r
rr
rr r
r rr r
rr r
2
independent of pathC
v dRsdotintr
A
B
1C
2C
MingsianMingsian R R BaiBai54
1 2
(iv) (i) constant ( ) ( )
--(1)
But (2)
(1) (2) QED
Note on application
Check
B
C C A
v dR v dR B A d
v dR d
d dx dy dz dRx y z
v dR dR v R
φ φ φ
φφ φ φφ φ
φ φ
rarr sdot = sdot = = minus =
rArr sdot =part part part
= + + = nabla sdot minus minus minus minus minuspart part part
sdot = nabla sdot rArr = nabla forall
int int intr rr r
rr
r
r r rr r
use simpler path (iv)if 0
find (i)v
φ⎧
nablatimes = ⎨⎩
rr
A
B
x
y
Cz
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai2
Vector Calculus (field theory)
Scalar field ( )Vector field ( ) in fluid mechanics (historical origin)
Key elements in this chapter + field integrals + curvilinedifferential opera ar coordinates
+ tors
integral theorem
p p tv v t==
xxr r
s potential t+ heory
MingsianMingsian R R BaiBai3
Divergence 散度
v velocityv
ˆ n outward normal
control volume piecewise smooth orientable closely connected surface
Net outflow (flux) per unit voluInterpr me per etat unio i in t t me
BS
rarr
0
ˆdiv ( ) ( ) lim the volume of
a spatial differential operator independent of coordinate system unique value
SV
n v dSv p v p V B
Vrarr
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
sdot= nabla
lowastrarr
intr
r r
MingsianMingsian R R BaiBai4
For the Cartesian coordinates
0 1 1
0
0 1 1
for some point ( ) on the - (by 2
mean valu
ˆ ˆ
ˆˆ ˆ ˆˆ ( ) ( )2
( )2
x face x front x back
x y z xx front x front
x
xx y z x front
n v dS n v dS
xn v dS i v i v j v k dS v x y z dydz
xv x y z y z
minus minus minus
minus minus
Δ+
sdot = + sdot
Δsdot = sdot + + = +
Δ= + Δ Δ
int int int
int int int
r r
r
e theorem)
j
xΔ
yΔ
zΔk
i
( ) P x y z center of cube=
MingsianMingsian R R BaiBai5
0 2 2
0 2 2
0
Similarly
ˆˆ ˆ ˆˆ ( ) ( )2
for some point ( ) on the - 2
Thus the net outflow per unit volume in -direction
ˆlim
x y z xx back x back
x face
V
xn v dS i v i v j v k dS v x y z y z
xx y z x back
x
n v dS
V
minus minus
minus
rarr
Δsdot = minus sdot + + = minus minus Δ Δ
Δminus
⎧ ⎫sdot⎪⎨ ⎬⎪⎩
int int
int
r
r0 0
0
( ) ( )2 2lim
x x
x y z
x xv x y z v x y z y z
Δ Δ Δ rarr
Δ Δ⎡ ⎤+ minus minus Δ Δ⎢ ⎥⎪ ⎣ ⎦=⎪⎭
x y zΔ Δ Δ
0 0
0
( ) ( )2 2 lim
Hence the net outflow per unit volume in the 3 directi
div
ons sums up to
(space operat ) or
x xx
x
yx zvv
x xv x y z v x y z vx x
vv vx y z
Δ rarr
Δ Δ+ minus minus part
=
partpart partnabla = + +
part part part
=Δ part
r r
MingsianMingsian R R BaiBai6
Gradient 梯度
g
R
r
ec
ad
all
ient
the divergence operator div
ˆˆ ˆIn Cartesian coordinates
Define the operator
a spatial differential operator
ˆˆ ˆgr
independent
ad
of coo
v v
i j kx
u u uu u i j kx y
y
z
z
part part part
=nablasdotpart part part
nabla + +part part part
rarr
nabla = + +part part part
r r
rdinates chosen ie is invariant wrt coordinates
directional derivative ( )ˆˆ ˆLet ( )
Physical interpretation
(position vector)
ˆ( ) (
u
u u x y z r xi yj zk
u u u u udu r dx dy dz ix y z x
nabla
= = + +
part part part part part= + + = +part part part part part
r
r
方向導數
ˆˆ ˆ
ˆ ˆˆ ˆ
( ) (
ˆ
)
) ( )
( )
dr r xi yj zk
uj k idx jdy kdzy z
du x u ddu r xu dr
= + +
=
part+ sdot + +part
primerarrsdot =nabla
r r
r r可視為 的推廣
MingsianMingsian R R BaiBai7
ˆ rate of change of in the direction Di
ˆ
rectional De
( )
ˆw
rivative
here
u sdu drr u u sdr dr
drsdr
=nabla sdot = nabla sdotr
r
r
s0
ˆ( ) ( )limh
du u r hs u rdr hrarr
+ minus=r r
drv
r dr+r r
rr
oNote
ˆ
ˆ )
ˆ
ˆ(
dudr
u s
s u
d sdr
s u
=nabla sdot
= sdotnabla
rArr
nabla
nabla
sdot=
sdotθ
s
s
( )0 yxx
y
u
unabla
drdu
duslopedr
=
( )yxfu =
MingsianMingsian R R BaiBai8
ˆEx Normal gradient outward normal
Note is often written as in the PDE literature
ˆˆ ˆˆPartial derivatives If then
ˆ
Physical interpretati
n
du udn n
du u u us i j kdr
du n ud
z
n
x y
partpart
part part part= rarrpart part part
sdotnabla
max
maximum rate of increase in a direction
ˆ ˆ ˆ ˆcos ( ) cos ( )
ˆ is independent of
ˆmax occurs when ( ) 0 and
points in the direction of ma
o
n
x a
du u s u s u s u u sdr
u sdu duu s udr dr
duudr
θ θ
θ ⎛ ⎞⎜ ⎟⎝ ⎠
=nabla sdot = nabla nabla = nabla nabla
nabla
there4 nabla = = nabla
rArrnabla
Q
( )max
(interpretation 1nd
gradient search steepest desce
)
nt
duudr
⎛ ⎞⎜ ⎟⎝ ⎠
nabla =
rArr minusnabla最佳化方法 的基礎
MingsianMingsian R R BaiBai9
(interpretatDirection of ion 2 normal to the level surface ( )
On the 0 ( )
ie
level surfac
o
e
n
)udu u dr
u cdu u dr
u dr u c
nabla= nabla sdot
== = nabla sdotnabla perp =
r
r
r
unabla
dr
unabla
Q
P
1cu=
2u c=
山
x
y
( )u x y
x
y
Hill-climbing search in optimization
MingsianMingsian R R BaiBai10
Ex Plane equationˆˆ ˆ( ) Normal vector
Ex 0 is the necessary condition for searching extrema (from multivariate Tylor expansion)
Ex Revisit of the exampl
u x y z ax by cz d u ai bj ck
u
= + + = rArr nabla = + +
nabla =v
多變數函數求極值
2 21 1 2 2
e in Linear Algebra using method of What is the maximum and minimum distance of the origin to the following curve
Lagrange multipli s
6 8
er
5 5x x x x+ + =
MingsianMingsian R R BaiBai11
1 2
2 21 2( )
2 21 1 2 2
Sol
This problem can be posed as a problemmin max ( )
constrained optimization
5 6 5 8 0 ( )which can
objectivecost function
constraint
equatioals
nx x
J x x
st g x x x x
= +
= + + minus =
2 2 2 21 2 1 1 2 2
o be converted into an unconstrained optimization problem by defining the ( ) (5 6 5 8)where is referred to as the thatLa
L
g ora
agr
nge ften h multi
angi
plie as
an
r ph
L x x x x x xλλ
= + + + + minus
ysical meaning in certain problemsThus the extreme values of will occur wh en or L J gJ nabla = nabla nabla0
MingsianMingsian R R BaiBai12
1 1 21
2 1 22
2 21 1 2 2
1 2 1 2
2 (10 6 ) 0 (1)
2 (6 10 ) 0 (2)
5 6 5 8 0 (constraint)
(1) (2) (2 16 )( ) 0 18 or (3)Substituting Eq (3) into Eqs (1) and (2) l
L x x xxL x x xxL x x x x
x x x x
λ
λ
λλ λ
part = + + =partpart = + + =partpart = + + minus =part+ + + = rArr = minus = minus
1 2 1 2eads to
and which corresponds to 14Thus minimummaximum distance 12x x x x J= = minus =
=
MingsianMingsian R R BaiBai13
22 2
1
1 1
2 2
The result above is identical to the previously obtained result in Linear Algebr
ellipse
45 rotatio
a
1 ( )4
1 11 1
n12
xx
x xx x
deg⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠
prime+ =prime
primeminus= rArr
prime2x prime
2x
1x prime
1x
2
MingsianMingsian R R BaiBai14
Curl 旋度
curl curl In Cartesian coordinates
spatial operator indep
ˆˆ ˆ
ˆˆ ˆ( )
ende
(
nt of coordi
) (
D
nates
ef
)
x y z
y x x xz z
i j k
vx y z
v v v
v v v vv vi j ky z x z x y
v v
part part partnablatimes =part part part
part part part partpart part= minus minus minus + minuspart
times
part part part part part
nabla
rarr
r
r r
0vnablatimes =rr 0vnablatimes ne
rr
MingsianMingsian R R BaiBai15
circulaPhysica
tion o spin ol interpretati
f fluidr or vortion
y
citΓ
( )0 0x dx y dy+ +
( )0 0x y ( )0 0x dx y+
Γ
1
2
3
4
( )0 0x y dy+
MingsianMingsian R R BaiBai16
yy
vv dy
ypart
+part
Γdyminus
dyyvv x
x partpart
+
dy
dx
dxminus
y yy
v vv dx dy
x ypart part
+ +part part
x xx
v vv dx dyx y
part part+ +part part
yy
vv dx
xpart
+part
xx
vv dxx
part+part
xv
yv
vv
dλ paddle weel
MingsianMingsian R R BaiBai17
contour
1 2 3 4
circulation
x x y y x x y y
x
v d
v d v d v d v d
v
λ
λ λ λ λrarr
Γ sdot
= + + +
asymp
intint int int int
vv
12
xv dxx
part+part ydx v
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
+ 12
y yv vdx dy
x ypart part
+ +part part
x
dy
v
⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦
+
12
xv dxx
part+part
( )xy
v dy dx vy
⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦
part+ minus +part
12
yvdy
ypart
+part
( )
( )
Similar process can be applied to the circulation in the and directionsThus the total circulation per unit area i curl s
y xz
dy
v v dxdy v dxdyx y
xv v
y
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎝ ⎠
minus
part part= minus = nablatimespart part
=nablatimes
v
v v
MingsianMingsian R R BaiBai18
Identities of differential operators ( p380 Greenberg )
( )
( ) ( )( )( )
( ) ( )( ) is required
Example of deriv
ˆ ˆˆ ˆ ˆ ˆ
ation
x y z
x y z
v v vx y z
uv u v u v
uv i j k uv i uv j uv kx y z
uv u v u v
u v v u u v
u v u vv v u u
+ +part part partpart part part
⎛ ⎞⎜ ⎟⎝ ⎠
nablasdot = nabla sdot + nablasdot
part part partnablasdot = + + sdot + + =part part part
nablatimes =nabla times + nablatimes
nablasdot times = sdotnablatimes minus sdotnablatimes
nablatimes times = nabla minus + minusnablanablasdot sdotsdot678 678
v
v v v
vL
v v v
v v v v v v
v v v v v v v v( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )2 cf
x y zu u ux y z
A B C A
u v
v w v w w v v w w v
v v v C B A B C
+ +part part partpart part part
sdotnabla
nabla sdot = sdotnabla + sdotnabla + times nablatimes + times
times
nablatimes
nablatimes nablatimes =nabla nabla times = sdot minussdot sdotminusnabla
678v
v v v
v v v v v vv v
v v v v v v
v v
v
v v
ˆ ˆˆ ˆ ˆ ˆ( ) ( ) ( )x y z x y zv i v j v k i j k u v v v ux y z x y zpart part part part part part+ + sdot + + = + +part part part part part part
v v
MingsianMingsian R R BaiBai19
2 2 22
2 2 2
0 (dc direct current) solenoidal0 (cg center of gravity) conservative potent
Laplacia
Zero identitie
n
s
ial
x y z
vu
⎧⎪⎨⎪⎩
part part partΔ =nabla nablasdotnabla = + +part part part
nablasdotnablatimes = rarr
nablatimesnabla = rarr
v
v
These differential operators will take more complicated and irregular form for the other coordinate systems
MingsianMingsian R R BaiBai20
2 2 3
2 2
ˆˆ ˆEx 2
ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ 2 0 2
2 0 0 2yx z
u x y v x yi z j k
u i j k u xyi x j k xyi x jx y z
vv vv xy xyx y z
⎛ ⎞⎜ ⎟⎝ ⎠
= = minus +
part part partnabla = + + = + + = +part part part
partpart partnablasdot = + + = + minus =part part part
v
v
MingsianMingsian R R BaiBai21
( ) ( ) ( )
( ) ( )
2 2 2 2
2 2 22
2 2 2
2
ˆˆ ˆ
ˆˆ ˆ
ˆ ˆˆ ˆ ˆ 0 3 0 0 0 3
2 0 0 2
ˆˆ ˆ
y yx xz z
x y z
i j kv vv vv vv i j k
x y z y z x z x yv v v
z i j x k z i x k
u u uu y yx y zv v v
i j kx y z
⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠ ⎝ ⎠
⎛
⎝
part partpart partpart partpart part partnablatimes = = minus minus minus + minuspart part part part part part part part part
= + minus minus + minus = minus
part part partnabla = + + = + + =part part part
nablatimes nablatimes =nabla nablasdot minusnabla
part part part= + +part part part
v
v v v
( ) ( )( ) ( ) ( ) ( )
( )
2 2 22 3
2 2 2ˆˆ ˆ2 2
ˆ ˆˆ ˆ ˆ ˆ 2 2 0 2 0 0 0 0 6 0 0 0
ˆ 2 6
xy x yi z j kx y z
yi xj k y i z j k
x z j
⎛ ⎞⎞⎜ ⎟⎜ ⎟
⎠ ⎝ ⎠⎡ ⎤⎣ ⎦
part part partminus + + minus +part part part
= + + minus + + + + minus + + minus
= +
MingsianMingsian R R BaiBai22
( )v w r w r r wnablatimes =nablatimes times = nablasdot minus nablasdotv v v v v v v ( )r w+ sdotnablav v ( ) constant w
w rminus sdotnablav
v v
Q
Ex Rigid body rotation
MingsianMingsian R R BaiBai23
( )( ) ( )
( ) ( )
ˆˆ ˆ
ˆ ˆˆ ˆ ˆ ˆ 1 1 1 3
ˆˆ ˆ
ˆˆ ˆ
Thus 3 2 is linked directly
x y z
x y z
r xi yj zk
r i j k xi yj zkx y z
w r w w w xi yj zkx y z
w i w j w k w
w r w r w r w w wv
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎝ ⎠
= + +
part part partnablasdot = + + + + = + + =part part part
part part partsdotnabla = + + + +part part part
= + + =
nablatimes times == nablasdot minus sdotnabla = minus =rArrnablatimes
v
v
v v
v
v v v v v v v v v
v spinn to ing
MingsianMingsian R R BaiBai24
Line surface and volume integrals
x
yC
B
A
isΔ
( )ii ηξ
Piecewise smooth curve
Line integral
MingsianMingsian R R BaiBai25
0 1lim ( ) ( ) or ( )
Intepretation of a line integral as an area ( )
i
Bn
i i isA
C
i C
F s F x y ds F x y d
x
s
F y ds
ξ ηΔ rarr =
Δ =sum int
int
int
x
y
z
C
( )z F x y=
MingsianMingsian R R BaiBai26
the area is always on the left of an obs
Positive dierver trave
rection of irsing on the
ntegrationcurve
RC
CCW CW
If is a regular closed curve piecewise smo
( ) ( )
(contou
oth simple ( )
r integral)
x y zC C
x y zC
F dR F F F dx dy dz
F dx F dy
C
F dz
sdot = + + + +
= + +
lowast
int int
int
i j k i j kv v
無交叉
work dW F dR= sdotv v
MingsianMingsian R R BaiBai27
0 1
Surface integral
lim ( ) ( )i
n
i i i iS i S
F S F x y z dSξ η ζΔ rarr =
Δ =sum intint( ) i i iξ η ζ
iSΔ
0 1
mass density
lim ( ) ( )
Ex
Volume integral
( )
i
n
i i i iV i V
V
F V F x y z dV
M x y z dV
ξ η ζ
ρ
Δ rarr =Δ =
=
sum intintint
intintint( ) i i iξ η ζ
iVΔ
MingsianMingsian R R BaiBai28
Ref (1)Hildebrand advanced Calculus for application Prentice-Hall 1976 p306-313(2) Greenberg p383-389 (3) Arfken ch2
1 2 3
1 2 3
1 2 3
1 2 3
(
Orthogonal coordinate systems base vectors are not constant in s
) ( )( )( )( )
ˆˆ ˆ
pacep x y z p u u ux x u u uy y u u uz z u u u
r xi yj zk
primerarr
= +
rarr
===
+v
1uv
2uv3uv
1u
2u
3uP coord curve
xy
z
O
X Curvilinear Coordinates
MingsianMingsian R R BaiBai29
1
11
1 1 1
1
1
1 1 1
11 1
1 1
2 2 2 3 3 3
Tangent vector to the at
where is the arc length
1
where unit vector
Similarly
curv
e
u psr rU
u s us
rs
U h uds rh udu sU h u U h u
partpart part= =part part part
part =part
there4 =part= =part
= =
v vv
vQ
v v
vv
v vv v
1u cood curve
1uv
1du 1dS
drv
P
MingsianMingsian R R BaiBai30
3 31 2 1 21 2 3
1 2 32
curv
In summary scale factor
If is the arc length along a in any directione
k k k
kk k
k k
k k k
U h u
ds rh Udu u
ds h dus
du dudu du du dudr r r r U U Uds u ds u ds u ds ds ds ds
dr drds ds
=
part= = =part
=
part part part= + + = + +part part part
= sdot
v v
v v
v v v v v v v
v v 3 3
1 1
23 3 32 2
21 1 1
2 2 2 2 2 2 2 2 2 21 1 2 2 3 3 1 2 3
1 (for orthogonal coord)
jii j
i j
ji ii j i
i j i
dududr U Uds ds ds
dudu duU U hds ds ds
ds h du h du h du ds ds ds
= =
= = =
⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
= sdot
rarr = sdot =
rarr = + + = + +
sum sum
sumsum sum
v v v
v v
MingsianMingsian R R BaiBai31
1 1 2 2 3 3
1 2 3
are mutually vectors having with arc length
k k k k kds h du U du
U du U du U duds ds ds
= =
rArr perp
v
v v v
1uv
2uv
3uv
1dS
2dS
3dS
dS
MingsianMingsian R R BaiBai32
( ) ( )
( ) ( )
1 1 2 2 3 3 1 2 3 1 2 3 1 2 3
1 2 3 1 2 3
1
1 2 2 3 3 2 2 2 3 3 3 2 3 2 3 1
Differential volume
Differential On const coord surface
area
Gradi t
en
d U du U du U du u u u h h h du du du
d h h h du du du
ud U du U du h u du h u du h h du du u
df
τ
τ
σ
= times sdot = times sdot
=
=
= times = times =
= nabla
v v v v v v
v vv v v v
1 2 31 2 3
1 2 3 1 1 2 2 3 31 2 3
1 1 1 2 2 2 3 3 3
f f ff dr df du du duu u u
r r rdr du du du U du U du U duu u u
h u du h u du h u du
part part partsdot rarr = + +part part part
part part part= + + = + +part part part
= + +
v
v v v v v vv
v v v
=1 (right hand rule)
MingsianMingsian R R BaiBai33
Rectangular Cartesian Coordinates
ij
k
MingsianMingsian R R BaiBai34
Circular Cylindrical Coordinatescossin
xyz z
ρ ϕρ ϕ
===
Ref Arfken
( )P zρ ϕ
MingsianMingsian R R BaiBai35
Polar Spherical Coordinatessin cossin sincos
x ry rz r
θ ϕθ ϕθ
===
Ref Arfken
0
0
0
sin cos sin sin coscos cos cos sin sin
sin cos
θ ϕ θ ϕ θθ ϕ θ ϕ θϕ ϕ
= + +
= + minus= minus +
r i j kθ i j kφ i j
2
2
sinsin
dS r d ddV r dr d d
θ θ ϕ
θ θ ϕ
=
=( )P r θ ϕ
MingsianMingsian R R BaiBai36
Integral TheoremsHeart of this chapter
Interchange of dimensionality Div Curl Grad in integrals High-dimensional integration
rarr
1
Thm1 Divergence theorem Let be a closed region bounded by piecewise smooth orientable surface and let (con
I Divergence theor
tinuous derivatives up to the first order
em (Gauss theorem)
)
VS v Cisin
r
ˆ If denotes the outward unit normal on
the ˆn V S
v dV n v dS
n S
nablasdot = sdotint intr r
MingsianMingsian R R BaiBai37
jS
jP
jV
S
0
ˆBy definition ( ) lim
ˆ( )
ˆ( )
j
j
j
j
Sj V
j
j j S
j j Sj j
n v dSv P
V
v P V n v dS
v P V n v dS
Δ rarr
⎧ ⎫sdot⎪ ⎪nabla sdot ⎨ ⎬Δ⎪ ⎪⎩ ⎭
nabla sdot Δ asymp sdot
nabla sdot Δ = sdot
int
intsum sumint
rr
r r
r r
Pf
MingsianMingsian R R BaiBai38
0
After interior surfaces cancel out only exterior surface (Why Opposite normal vectors)
ˆ( )
volumes exterior surfaces
ˆlim
QE
survive
j
j
j j Sj j
V SV
v p V n v dS
v dV n v dSΔ rarr
nabla sdot Δ = sdot
rArr nablasdot = sdot
sum sumint
int int
r r
r r
D
MingsianMingsian R R BaiBai39
( )
scalar variable constant vectorvector variable
Alternative formsLet be a an arbitrary and
a
ˆ
1 Le
(
ˆt
V
V S
S
V
v ap
v d
v dV vn d
V n v dS
v va
va dV v a v a
S
nablasdot = sdot
= rArr
nablasdot = nabla sdot + nabla
=
sdot
nablaint intint int
int
r
v
r r
r r
r r rQ ( )
( )
ˆ)
2 Let ˆV S
V
V S
dV n va dS
v a p
a p
p dV n p d
d p
S
V a
⎛ ⎞= sdot⎜ ⎟
⎝ ⎠
= times rArr
nablasdot times = sdot nabla
times
times
nabla = times
int int
int int
int
r
r r r
r r v
r
vQ
r
( ) ( ) ( )
( ) ( )
ˆ
ˆV S
V S
a p dV n a p dS
a p dV a n p dS
⎛ ⎞⎡ ⎤minus sdot nablatimes = sdot times⎣ ⎦⎜ ⎟⎜ ⎟⎜ ⎟rArr minus sdot nablatimes = minus sdot times⎜ ⎟⎝ ⎠
int int
int int
r rv v
r rv v
MingsianMingsian R R BaiBai40
( )
( )
From Leibnitzs rule
(1-D) ( )
( ) ( )
B t
A t
d f x t dxdt
f x t dx B ttpart prime= +part
int( )
( )( ( ) ) ( )
B t
A tf B t t A tprimeminusint ( ( ) )
(assume ( ) ( ) )
f A t t
A t B t const= =
ContinuityEx The Equation
rate of increase of mass
mass densityV
dM d dVdt dt
ρ
ρ
= int
flow velocityvv
MingsianMingsian R R BaiBai41
ˆ ˆ ( )
(
Conservation
) ( Divergence theorem)
of mass
( ) 0
V S S
V
V
dV n v dS n v dSt
v dV
v dVt
ρ ρ ρ
ρ
ρ ρ⎡ ⎤⎢ ⎥⎣ ⎦
part = minus sdot = minus sdotpart
= minus nablasdot
partrArr +nablasdot =part
int int int
int
int
v v
vQ
v
ˆOn the other hand rate influof mass through n ox i t
Extended to 3-D for time-independent fixed boundary
(3-D) ( ) ( )
Thus
S
V V
V
S V n v dS
Sd f x y z t dV f x y z t dVdt tdM dVdt t
ρ
ρ
= minus sdot
part=part
part=part
int
int int
intv
Fixed control volume (Eulerian coordinates)Otherwise Raynoldrsquos transport theoremMoving control volume Lagrangian coordinates
MingsianMingsian R R BaiBai42
But the control volime is only arbitrary
In particular for flow
( ) 0
con
(continuity equati
stant
on)
Divergence the
0 incompress
Note is t
ible
orem h
V
v
vt
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
e mother of the other integral theorems
MingsianMingsian R R BaiBai43
Ex Archimedes principle ( ) The buoyant force ( ) on a body with volume immersed in a fluid is equal to the weight of the fluid
ˆ displaced by the object ie In the followin
B
V
gVρ= minusf k
阿基米德原理
浮力
ˆg figure the ˆ with and being the fluid density gravitational
acceleration depth and outward normal unit vector to the surface
fhydraulic pressure g z
g z
S
ρ
ρ
= minusp n
n
1
the modified divergence theorem
ˆ
Hint
scalar fuctionV S
vdV v dS v Cnabla = isinint int n
MingsianMingsian R R BaiBai44
Levelsurfacefluid
V
S
airyx
z
ˆf g zρ= minusp n
ˆPf ( ) ( )
ˆ ˆ ˆ ˆ ˆ (0 0 ) v v
B fS S V
V V
g z g zdS dS dV
g dV g dV gV
ρ ρ
ρ ρ ρ
= = minus = minus nabla
= minus + + = minus = minus
int int int
int int
f p n
i j k k k
k
MingsianMingsian R R BaiBai45
Note This can also be thought of as simple balance of forces (resultant force that acts on and weight of filled with fluid)S V
2
~ a little history about George Green Recall the diveregence theorem
ˆ
Let
II th
e Green
are arbitrary fun
s identiti
ctions
es
V S
v dV n v dS
v u vu v C
nablasdot = sdot
= nablaisin
int intv v
v
ˆ ( ) ( )
ˆ ( )V S
V S
u v dV n u v dS
u v u v dV un v dS
nablasdot nabla = sdot nabla
nabla sdotnabla + nablasdotnabla = sdotnabla
int int
int int
MingsianMingsian R R BaiBai46
2 -( ) -(1)V S
vu v u v dV u dSnpart
nabla sdotnabla + nabla =partint int firs(Greens identt ity)
2
If
( ) --(2)
(1) (2) V S
u vuv u v u dV v dSn
harrpart
nabla sdotnabla + nabla =part
minus
int int
( )2 2
V S
dv duu v v u dV u v dSdn dn
⎛ ⎞nabla minus nabla = minus⎜ ⎟⎝ ⎠int int
second(Greens identity)
MingsianMingsian R R BaiBai47
( )
( )
( ) ( )
1st id ( )
Greens identities in the 1D case integration by parts
b b bb ba a
a a ab
ba
ab
ba
a
v udx v u v u dx u v uv dx uv
u v u v vu dx vu
uv vu dx uv vu
primeprime prime prime prime prime prime primeprime prime= minus rArr + =
prime prime primeprime primeharr + =
primeprime primeprime prime primerArr minus = minus
int int int
int
2
2nd id ( )
Good for any adjoint operators in addition to Good also for and integration by parts Important for theory amp reciprocity
From th
Gree
e 1s
2-
t
ns functi
identity
D 1-
tak 1
D
e
on
u
nablararr
=
int
2
V S
vv dV dSnpartnabla =part
rArr int int2Ex 0 st 0 on the boundary 0 ( 1st id u u unabla = = rArr equiv 以 証)
MingsianMingsian R R BaiBai48
1
Thm1 Stokes theorem Let be defined in Let be a piecewise-smooth orientable surface bounded by a piec
III Stokes theorem and
ewise-smooth simple clos
Greens
ed curve Then
theore
m
v C R S
C
isinv
( )
ˆwhere is a unit normal to according t right hand rulo the e
ˆS C
n S
n v dS v dRsdot nablatimes = sdotint intvv v
(not necessarily planar)S
Cn vnablatimes v vv
MingsianMingsian R R BaiBai49
Pf
( )( )
circulatioWe have shown in deriving thatthe in -direction
ˆor
ˆ
n
z
z
j j j j j
j j j j jj j
vz v dx dy
v dR n v S
v dR n v S
nablatimes
=nablatimes
sdot = sdot nablatimes Δ
sdot = sdot nablatimes Δ
intsum sumint
v
v
vv v
vv v
C
S
jS
jC
jP
ˆ jn
jSΔ
MingsianMingsian R R BaiBai50
( )
( )ext surface
After the interior boundaries cancel out only the exterior boundaries (Why)
ˆ
ˆ0 QED
Alternative forms scalar
survive
va
j
j j j j jC
jC S
v dR n v S
S v dR n v dS
φ
sdot = sdot nablatimes Δ
Δ rarr rArr sdot = sdot nablatimes
sum sumint
int int
vv v
vv v
( )
riable constant vector vector variable
1 Let
2 Let
ˆ
ˆ S C
S C
a p
v n dR dR
n p dS dR p
a
v a p
φ φφ= rArr timesnabla =
timesnabla times= rArr timestimes =
int int
int int
v v
v v
v
v v
vv vv v
MingsianMingsian R R BaiBai51
( ) ( )
( )
( )
( ) ( )
1
Thm2 Greens theorem ( planar version of Stokes theorem)ˆ ˆ Let
pf By Stokes theoremˆ
ˆ ˆ ˆ ˆ ˆ ˆ
S C
S C
S C
v P x y i Q x y j C
n v dS v dR
k Pi Qj dS Pi Qj dx i dy
Q P dS Pdx Qdyx y
⎛ ⎞part partminus = +⎜ ⎟
= + isin
sdot nablatimes = sdot
⎡ ⎤sdot nablatimes + =
part part⎝ ⎠
+ sdot +⎣ ⎦
int int
int
int
int
int
v
vv v
( )
( )
( )
ˆ
ˆˆ ˆ
ˆˆ ˆ ˆ ˆ( ) ( )
0
QEDS C
j
i j kQ PPi Qj i j k
x y z x yP Q
Q P dS Pdx Qdyx y
⎛ ⎞part part part part partnablatimes + = = + + minus⎜ ⎟part part part part part⎝ ⎠
⎛ ⎞part partrArr minus = +⎜ ⎟part part⎝ ⎠int int
L L
S
C
i
j
x
y
MingsianMingsian R R BaiBai52
Potential TheorypotentiGeorge Green coined the term (l
a )
rarr 位能 勢能
例子 重力場 電磁場 流場 聲場 彈性力學
1
2
Thm1 If is in a simply connected region then the following statements are equivalent(i) There exists a scala
Conservative Field
potential functior such that
(ii
n
) 0
v
C
C
v
v R
φφ=nabla
nablatimes
isin
isin
=r
r
r
r in (Irrotational field curl-free)
cf 0 st solenoidal field divergence-free
0
(iii) in (Conservative field)C
Rv w v w
v R Rd
nablasdot = exist nabla
sdot =
= times
int
r r r
r
r
r
MingsianMingsian R R BaiBai53
(iv) is connecting and
independent o
Ex Think
f the pa
of dynamics is the force
thB
A
v dR A B
v
sdotintrr
r
1 2
1
0
Pf
(i) (ii) ( ) 0 (zero identity cg rule) irrotational
ˆ(ii) (iii) ( ) 0 (Stokes theorem)
(iii) (iv) ( ) 0
C S
C C C
C
v
v dR n v dS
v dR v dR
v dR
φ
=
rarr nablatimes = nablatimes nabla =
rarr sdot = sdot nablatimes =
rarr sdot = minus sdot =
rArr sdot =
int int
int int int
int
r
rr
rr r
r rr r
rr r
2
independent of pathC
v dRsdotintr
A
B
1C
2C
MingsianMingsian R R BaiBai54
1 2
(iv) (i) constant ( ) ( )
--(1)
But (2)
(1) (2) QED
Note on application
Check
B
C C A
v dR v dR B A d
v dR d
d dx dy dz dRx y z
v dR dR v R
φ φ φ
φφ φ φφ φ
φ φ
rarr sdot = sdot = = minus =
rArr sdot =part part part
= + + = nabla sdot minus minus minus minus minuspart part part
sdot = nabla sdot rArr = nabla forall
int int intr rr r
rr
r
r r rr r
use simpler path (iv)if 0
find (i)v
φ⎧
nablatimes = ⎨⎩
rr
A
B
x
y
Cz
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai3
Divergence 散度
v velocityv
ˆ n outward normal
control volume piecewise smooth orientable closely connected surface
Net outflow (flux) per unit voluInterpr me per etat unio i in t t me
BS
rarr
0
ˆdiv ( ) ( ) lim the volume of
a spatial differential operator independent of coordinate system unique value
SV
n v dSv p v p V B
Vrarr
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
sdot= nabla
lowastrarr
intr
r r
MingsianMingsian R R BaiBai4
For the Cartesian coordinates
0 1 1
0
0 1 1
for some point ( ) on the - (by 2
mean valu
ˆ ˆ
ˆˆ ˆ ˆˆ ( ) ( )2
( )2
x face x front x back
x y z xx front x front
x
xx y z x front
n v dS n v dS
xn v dS i v i v j v k dS v x y z dydz
xv x y z y z
minus minus minus
minus minus
Δ+
sdot = + sdot
Δsdot = sdot + + = +
Δ= + Δ Δ
int int int
int int int
r r
r
e theorem)
j
xΔ
yΔ
zΔk
i
( ) P x y z center of cube=
MingsianMingsian R R BaiBai5
0 2 2
0 2 2
0
Similarly
ˆˆ ˆ ˆˆ ( ) ( )2
for some point ( ) on the - 2
Thus the net outflow per unit volume in -direction
ˆlim
x y z xx back x back
x face
V
xn v dS i v i v j v k dS v x y z y z
xx y z x back
x
n v dS
V
minus minus
minus
rarr
Δsdot = minus sdot + + = minus minus Δ Δ
Δminus
⎧ ⎫sdot⎪⎨ ⎬⎪⎩
int int
int
r
r0 0
0
( ) ( )2 2lim
x x
x y z
x xv x y z v x y z y z
Δ Δ Δ rarr
Δ Δ⎡ ⎤+ minus minus Δ Δ⎢ ⎥⎪ ⎣ ⎦=⎪⎭
x y zΔ Δ Δ
0 0
0
( ) ( )2 2 lim
Hence the net outflow per unit volume in the 3 directi
div
ons sums up to
(space operat ) or
x xx
x
yx zvv
x xv x y z v x y z vx x
vv vx y z
Δ rarr
Δ Δ+ minus minus part
=
partpart partnabla = + +
part part part
=Δ part
r r
MingsianMingsian R R BaiBai6
Gradient 梯度
g
R
r
ec
ad
all
ient
the divergence operator div
ˆˆ ˆIn Cartesian coordinates
Define the operator
a spatial differential operator
ˆˆ ˆgr
independent
ad
of coo
v v
i j kx
u u uu u i j kx y
y
z
z
part part part
=nablasdotpart part part
nabla + +part part part
rarr
nabla = + +part part part
r r
rdinates chosen ie is invariant wrt coordinates
directional derivative ( )ˆˆ ˆLet ( )
Physical interpretation
(position vector)
ˆ( ) (
u
u u x y z r xi yj zk
u u u u udu r dx dy dz ix y z x
nabla
= = + +
part part part part part= + + = +part part part part part
r
r
方向導數
ˆˆ ˆ
ˆ ˆˆ ˆ
( ) (
ˆ
)
) ( )
( )
dr r xi yj zk
uj k idx jdy kdzy z
du x u ddu r xu dr
= + +
=
part+ sdot + +part
primerarrsdot =nabla
r r
r r可視為 的推廣
MingsianMingsian R R BaiBai7
ˆ rate of change of in the direction Di
ˆ
rectional De
( )
ˆw
rivative
here
u sdu drr u u sdr dr
drsdr
=nabla sdot = nabla sdotr
r
r
s0
ˆ( ) ( )limh
du u r hs u rdr hrarr
+ minus=r r
drv
r dr+r r
rr
oNote
ˆ
ˆ )
ˆ
ˆ(
dudr
u s
s u
d sdr
s u
=nabla sdot
= sdotnabla
rArr
nabla
nabla
sdot=
sdotθ
s
s
( )0 yxx
y
u
unabla
drdu
duslopedr
=
( )yxfu =
MingsianMingsian R R BaiBai8
ˆEx Normal gradient outward normal
Note is often written as in the PDE literature
ˆˆ ˆˆPartial derivatives If then
ˆ
Physical interpretati
n
du udn n
du u u us i j kdr
du n ud
z
n
x y
partpart
part part part= rarrpart part part
sdotnabla
max
maximum rate of increase in a direction
ˆ ˆ ˆ ˆcos ( ) cos ( )
ˆ is independent of
ˆmax occurs when ( ) 0 and
points in the direction of ma
o
n
x a
du u s u s u s u u sdr
u sdu duu s udr dr
duudr
θ θ
θ ⎛ ⎞⎜ ⎟⎝ ⎠
=nabla sdot = nabla nabla = nabla nabla
nabla
there4 nabla = = nabla
rArrnabla
Q
( )max
(interpretation 1nd
gradient search steepest desce
)
nt
duudr
⎛ ⎞⎜ ⎟⎝ ⎠
nabla =
rArr minusnabla最佳化方法 的基礎
MingsianMingsian R R BaiBai9
(interpretatDirection of ion 2 normal to the level surface ( )
On the 0 ( )
ie
level surfac
o
e
n
)udu u dr
u cdu u dr
u dr u c
nabla= nabla sdot
== = nabla sdotnabla perp =
r
r
r
unabla
dr
unabla
Q
P
1cu=
2u c=
山
x
y
( )u x y
x
y
Hill-climbing search in optimization
MingsianMingsian R R BaiBai10
Ex Plane equationˆˆ ˆ( ) Normal vector
Ex 0 is the necessary condition for searching extrema (from multivariate Tylor expansion)
Ex Revisit of the exampl
u x y z ax by cz d u ai bj ck
u
= + + = rArr nabla = + +
nabla =v
多變數函數求極值
2 21 1 2 2
e in Linear Algebra using method of What is the maximum and minimum distance of the origin to the following curve
Lagrange multipli s
6 8
er
5 5x x x x+ + =
MingsianMingsian R R BaiBai11
1 2
2 21 2( )
2 21 1 2 2
Sol
This problem can be posed as a problemmin max ( )
constrained optimization
5 6 5 8 0 ( )which can
objectivecost function
constraint
equatioals
nx x
J x x
st g x x x x
= +
= + + minus =
2 2 2 21 2 1 1 2 2
o be converted into an unconstrained optimization problem by defining the ( ) (5 6 5 8)where is referred to as the thatLa
L
g ora
agr
nge ften h multi
angi
plie as
an
r ph
L x x x x x xλλ
= + + + + minus
ysical meaning in certain problemsThus the extreme values of will occur wh en or L J gJ nabla = nabla nabla0
MingsianMingsian R R BaiBai12
1 1 21
2 1 22
2 21 1 2 2
1 2 1 2
2 (10 6 ) 0 (1)
2 (6 10 ) 0 (2)
5 6 5 8 0 (constraint)
(1) (2) (2 16 )( ) 0 18 or (3)Substituting Eq (3) into Eqs (1) and (2) l
L x x xxL x x xxL x x x x
x x x x
λ
λ
λλ λ
part = + + =partpart = + + =partpart = + + minus =part+ + + = rArr = minus = minus
1 2 1 2eads to
and which corresponds to 14Thus minimummaximum distance 12x x x x J= = minus =
=
MingsianMingsian R R BaiBai13
22 2
1
1 1
2 2
The result above is identical to the previously obtained result in Linear Algebr
ellipse
45 rotatio
a
1 ( )4
1 11 1
n12
xx
x xx x
deg⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠
prime+ =prime
primeminus= rArr
prime2x prime
2x
1x prime
1x
2
MingsianMingsian R R BaiBai14
Curl 旋度
curl curl In Cartesian coordinates
spatial operator indep
ˆˆ ˆ
ˆˆ ˆ( )
ende
(
nt of coordi
) (
D
nates
ef
)
x y z
y x x xz z
i j k
vx y z
v v v
v v v vv vi j ky z x z x y
v v
part part partnablatimes =part part part
part part part partpart part= minus minus minus + minuspart
times
part part part part part
nabla
rarr
r
r r
0vnablatimes =rr 0vnablatimes ne
rr
MingsianMingsian R R BaiBai15
circulaPhysica
tion o spin ol interpretati
f fluidr or vortion
y
citΓ
( )0 0x dx y dy+ +
( )0 0x y ( )0 0x dx y+
Γ
1
2
3
4
( )0 0x y dy+
MingsianMingsian R R BaiBai16
yy
vv dy
ypart
+part
Γdyminus
dyyvv x
x partpart
+
dy
dx
dxminus
y yy
v vv dx dy
x ypart part
+ +part part
x xx
v vv dx dyx y
part part+ +part part
yy
vv dx
xpart
+part
xx
vv dxx
part+part
xv
yv
vv
dλ paddle weel
MingsianMingsian R R BaiBai17
contour
1 2 3 4
circulation
x x y y x x y y
x
v d
v d v d v d v d
v
λ
λ λ λ λrarr
Γ sdot
= + + +
asymp
intint int int int
vv
12
xv dxx
part+part ydx v
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
+ 12
y yv vdx dy
x ypart part
+ +part part
x
dy
v
⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦
+
12
xv dxx
part+part
( )xy
v dy dx vy
⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦
part+ minus +part
12
yvdy
ypart
+part
( )
( )
Similar process can be applied to the circulation in the and directionsThus the total circulation per unit area i curl s
y xz
dy
v v dxdy v dxdyx y
xv v
y
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎝ ⎠
minus
part part= minus = nablatimespart part
=nablatimes
v
v v
MingsianMingsian R R BaiBai18
Identities of differential operators ( p380 Greenberg )
( )
( ) ( )( )( )
( ) ( )( ) is required
Example of deriv
ˆ ˆˆ ˆ ˆ ˆ
ation
x y z
x y z
v v vx y z
uv u v u v
uv i j k uv i uv j uv kx y z
uv u v u v
u v v u u v
u v u vv v u u
+ +part part partpart part part
⎛ ⎞⎜ ⎟⎝ ⎠
nablasdot = nabla sdot + nablasdot
part part partnablasdot = + + sdot + + =part part part
nablatimes =nabla times + nablatimes
nablasdot times = sdotnablatimes minus sdotnablatimes
nablatimes times = nabla minus + minusnablanablasdot sdotsdot678 678
v
v v v
vL
v v v
v v v v v v
v v v v v v v v( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )2 cf
x y zu u ux y z
A B C A
u v
v w v w w v v w w v
v v v C B A B C
+ +part part partpart part part
sdotnabla
nabla sdot = sdotnabla + sdotnabla + times nablatimes + times
times
nablatimes
nablatimes nablatimes =nabla nabla times = sdot minussdot sdotminusnabla
678v
v v v
v v v v v vv v
v v v v v v
v v
v
v v
ˆ ˆˆ ˆ ˆ ˆ( ) ( ) ( )x y z x y zv i v j v k i j k u v v v ux y z x y zpart part part part part part+ + sdot + + = + +part part part part part part
v v
MingsianMingsian R R BaiBai19
2 2 22
2 2 2
0 (dc direct current) solenoidal0 (cg center of gravity) conservative potent
Laplacia
Zero identitie
n
s
ial
x y z
vu
⎧⎪⎨⎪⎩
part part partΔ =nabla nablasdotnabla = + +part part part
nablasdotnablatimes = rarr
nablatimesnabla = rarr
v
v
These differential operators will take more complicated and irregular form for the other coordinate systems
MingsianMingsian R R BaiBai20
2 2 3
2 2
ˆˆ ˆEx 2
ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ 2 0 2
2 0 0 2yx z
u x y v x yi z j k
u i j k u xyi x j k xyi x jx y z
vv vv xy xyx y z
⎛ ⎞⎜ ⎟⎝ ⎠
= = minus +
part part partnabla = + + = + + = +part part part
partpart partnablasdot = + + = + minus =part part part
v
v
MingsianMingsian R R BaiBai21
( ) ( ) ( )
( ) ( )
2 2 2 2
2 2 22
2 2 2
2
ˆˆ ˆ
ˆˆ ˆ
ˆ ˆˆ ˆ ˆ 0 3 0 0 0 3
2 0 0 2
ˆˆ ˆ
y yx xz z
x y z
i j kv vv vv vv i j k
x y z y z x z x yv v v
z i j x k z i x k
u u uu y yx y zv v v
i j kx y z
⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠ ⎝ ⎠
⎛
⎝
part partpart partpart partpart part partnablatimes = = minus minus minus + minuspart part part part part part part part part
= + minus minus + minus = minus
part part partnabla = + + = + + =part part part
nablatimes nablatimes =nabla nablasdot minusnabla
part part part= + +part part part
v
v v v
( ) ( )( ) ( ) ( ) ( )
( )
2 2 22 3
2 2 2ˆˆ ˆ2 2
ˆ ˆˆ ˆ ˆ ˆ 2 2 0 2 0 0 0 0 6 0 0 0
ˆ 2 6
xy x yi z j kx y z
yi xj k y i z j k
x z j
⎛ ⎞⎞⎜ ⎟⎜ ⎟
⎠ ⎝ ⎠⎡ ⎤⎣ ⎦
part part partminus + + minus +part part part
= + + minus + + + + minus + + minus
= +
MingsianMingsian R R BaiBai22
( )v w r w r r wnablatimes =nablatimes times = nablasdot minus nablasdotv v v v v v v ( )r w+ sdotnablav v ( ) constant w
w rminus sdotnablav
v v
Q
Ex Rigid body rotation
MingsianMingsian R R BaiBai23
( )( ) ( )
( ) ( )
ˆˆ ˆ
ˆ ˆˆ ˆ ˆ ˆ 1 1 1 3
ˆˆ ˆ
ˆˆ ˆ
Thus 3 2 is linked directly
x y z
x y z
r xi yj zk
r i j k xi yj zkx y z
w r w w w xi yj zkx y z
w i w j w k w
w r w r w r w w wv
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎝ ⎠
= + +
part part partnablasdot = + + + + = + + =part part part
part part partsdotnabla = + + + +part part part
= + + =
nablatimes times == nablasdot minus sdotnabla = minus =rArrnablatimes
v
v
v v
v
v v v v v v v v v
v spinn to ing
MingsianMingsian R R BaiBai24
Line surface and volume integrals
x
yC
B
A
isΔ
( )ii ηξ
Piecewise smooth curve
Line integral
MingsianMingsian R R BaiBai25
0 1lim ( ) ( ) or ( )
Intepretation of a line integral as an area ( )
i
Bn
i i isA
C
i C
F s F x y ds F x y d
x
s
F y ds
ξ ηΔ rarr =
Δ =sum int
int
int
x
y
z
C
( )z F x y=
MingsianMingsian R R BaiBai26
the area is always on the left of an obs
Positive dierver trave
rection of irsing on the
ntegrationcurve
RC
CCW CW
If is a regular closed curve piecewise smo
( ) ( )
(contou
oth simple ( )
r integral)
x y zC C
x y zC
F dR F F F dx dy dz
F dx F dy
C
F dz
sdot = + + + +
= + +
lowast
int int
int
i j k i j kv v
無交叉
work dW F dR= sdotv v
MingsianMingsian R R BaiBai27
0 1
Surface integral
lim ( ) ( )i
n
i i i iS i S
F S F x y z dSξ η ζΔ rarr =
Δ =sum intint( ) i i iξ η ζ
iSΔ
0 1
mass density
lim ( ) ( )
Ex
Volume integral
( )
i
n
i i i iV i V
V
F V F x y z dV
M x y z dV
ξ η ζ
ρ
Δ rarr =Δ =
=
sum intintint
intintint( ) i i iξ η ζ
iVΔ
MingsianMingsian R R BaiBai28
Ref (1)Hildebrand advanced Calculus for application Prentice-Hall 1976 p306-313(2) Greenberg p383-389 (3) Arfken ch2
1 2 3
1 2 3
1 2 3
1 2 3
(
Orthogonal coordinate systems base vectors are not constant in s
) ( )( )( )( )
ˆˆ ˆ
pacep x y z p u u ux x u u uy y u u uz z u u u
r xi yj zk
primerarr
= +
rarr
===
+v
1uv
2uv3uv
1u
2u
3uP coord curve
xy
z
O
X Curvilinear Coordinates
MingsianMingsian R R BaiBai29
1
11
1 1 1
1
1
1 1 1
11 1
1 1
2 2 2 3 3 3
Tangent vector to the at
where is the arc length
1
where unit vector
Similarly
curv
e
u psr rU
u s us
rs
U h uds rh udu sU h u U h u
partpart part= =part part part
part =part
there4 =part= =part
= =
v vv
vQ
v v
vv
v vv v
1u cood curve
1uv
1du 1dS
drv
P
MingsianMingsian R R BaiBai30
3 31 2 1 21 2 3
1 2 32
curv
In summary scale factor
If is the arc length along a in any directione
k k k
kk k
k k
k k k
U h u
ds rh Udu u
ds h dus
du dudu du du dudr r r r U U Uds u ds u ds u ds ds ds ds
dr drds ds
=
part= = =part
=
part part part= + + = + +part part part
= sdot
v v
v v
v v v v v v v
v v 3 3
1 1
23 3 32 2
21 1 1
2 2 2 2 2 2 2 2 2 21 1 2 2 3 3 1 2 3
1 (for orthogonal coord)
jii j
i j
ji ii j i
i j i
dududr U Uds ds ds
dudu duU U hds ds ds
ds h du h du h du ds ds ds
= =
= = =
⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
= sdot
rarr = sdot =
rarr = + + = + +
sum sum
sumsum sum
v v v
v v
MingsianMingsian R R BaiBai31
1 1 2 2 3 3
1 2 3
are mutually vectors having with arc length
k k k k kds h du U du
U du U du U duds ds ds
= =
rArr perp
v
v v v
1uv
2uv
3uv
1dS
2dS
3dS
dS
MingsianMingsian R R BaiBai32
( ) ( )
( ) ( )
1 1 2 2 3 3 1 2 3 1 2 3 1 2 3
1 2 3 1 2 3
1
1 2 2 3 3 2 2 2 3 3 3 2 3 2 3 1
Differential volume
Differential On const coord surface
area
Gradi t
en
d U du U du U du u u u h h h du du du
d h h h du du du
ud U du U du h u du h u du h h du du u
df
τ
τ
σ
= times sdot = times sdot
=
=
= times = times =
= nabla
v v v v v v
v vv v v v
1 2 31 2 3
1 2 3 1 1 2 2 3 31 2 3
1 1 1 2 2 2 3 3 3
f f ff dr df du du duu u u
r r rdr du du du U du U du U duu u u
h u du h u du h u du
part part partsdot rarr = + +part part part
part part part= + + = + +part part part
= + +
v
v v v v v vv
v v v
=1 (right hand rule)
MingsianMingsian R R BaiBai33
Rectangular Cartesian Coordinates
ij
k
MingsianMingsian R R BaiBai34
Circular Cylindrical Coordinatescossin
xyz z
ρ ϕρ ϕ
===
Ref Arfken
( )P zρ ϕ
MingsianMingsian R R BaiBai35
Polar Spherical Coordinatessin cossin sincos
x ry rz r
θ ϕθ ϕθ
===
Ref Arfken
0
0
0
sin cos sin sin coscos cos cos sin sin
sin cos
θ ϕ θ ϕ θθ ϕ θ ϕ θϕ ϕ
= + +
= + minus= minus +
r i j kθ i j kφ i j
2
2
sinsin
dS r d ddV r dr d d
θ θ ϕ
θ θ ϕ
=
=( )P r θ ϕ
MingsianMingsian R R BaiBai36
Integral TheoremsHeart of this chapter
Interchange of dimensionality Div Curl Grad in integrals High-dimensional integration
rarr
1
Thm1 Divergence theorem Let be a closed region bounded by piecewise smooth orientable surface and let (con
I Divergence theor
tinuous derivatives up to the first order
em (Gauss theorem)
)
VS v Cisin
r
ˆ If denotes the outward unit normal on
the ˆn V S
v dV n v dS
n S
nablasdot = sdotint intr r
MingsianMingsian R R BaiBai37
jS
jP
jV
S
0
ˆBy definition ( ) lim
ˆ( )
ˆ( )
j
j
j
j
Sj V
j
j j S
j j Sj j
n v dSv P
V
v P V n v dS
v P V n v dS
Δ rarr
⎧ ⎫sdot⎪ ⎪nabla sdot ⎨ ⎬Δ⎪ ⎪⎩ ⎭
nabla sdot Δ asymp sdot
nabla sdot Δ = sdot
int
intsum sumint
rr
r r
r r
Pf
MingsianMingsian R R BaiBai38
0
After interior surfaces cancel out only exterior surface (Why Opposite normal vectors)
ˆ( )
volumes exterior surfaces
ˆlim
QE
survive
j
j
j j Sj j
V SV
v p V n v dS
v dV n v dSΔ rarr
nabla sdot Δ = sdot
rArr nablasdot = sdot
sum sumint
int int
r r
r r
D
MingsianMingsian R R BaiBai39
( )
scalar variable constant vectorvector variable
Alternative formsLet be a an arbitrary and
a
ˆ
1 Le
(
ˆt
V
V S
S
V
v ap
v d
v dV vn d
V n v dS
v va
va dV v a v a
S
nablasdot = sdot
= rArr
nablasdot = nabla sdot + nabla
=
sdot
nablaint intint int
int
r
v
r r
r r
r r rQ ( )
( )
ˆ)
2 Let ˆV S
V
V S
dV n va dS
v a p
a p
p dV n p d
d p
S
V a
⎛ ⎞= sdot⎜ ⎟
⎝ ⎠
= times rArr
nablasdot times = sdot nabla
times
times
nabla = times
int int
int int
int
r
r r r
r r v
r
vQ
r
( ) ( ) ( )
( ) ( )
ˆ
ˆV S
V S
a p dV n a p dS
a p dV a n p dS
⎛ ⎞⎡ ⎤minus sdot nablatimes = sdot times⎣ ⎦⎜ ⎟⎜ ⎟⎜ ⎟rArr minus sdot nablatimes = minus sdot times⎜ ⎟⎝ ⎠
int int
int int
r rv v
r rv v
MingsianMingsian R R BaiBai40
( )
( )
From Leibnitzs rule
(1-D) ( )
( ) ( )
B t
A t
d f x t dxdt
f x t dx B ttpart prime= +part
int( )
( )( ( ) ) ( )
B t
A tf B t t A tprimeminusint ( ( ) )
(assume ( ) ( ) )
f A t t
A t B t const= =
ContinuityEx The Equation
rate of increase of mass
mass densityV
dM d dVdt dt
ρ
ρ
= int
flow velocityvv
MingsianMingsian R R BaiBai41
ˆ ˆ ( )
(
Conservation
) ( Divergence theorem)
of mass
( ) 0
V S S
V
V
dV n v dS n v dSt
v dV
v dVt
ρ ρ ρ
ρ
ρ ρ⎡ ⎤⎢ ⎥⎣ ⎦
part = minus sdot = minus sdotpart
= minus nablasdot
partrArr +nablasdot =part
int int int
int
int
v v
vQ
v
ˆOn the other hand rate influof mass through n ox i t
Extended to 3-D for time-independent fixed boundary
(3-D) ( ) ( )
Thus
S
V V
V
S V n v dS
Sd f x y z t dV f x y z t dVdt tdM dVdt t
ρ
ρ
= minus sdot
part=part
part=part
int
int int
intv
Fixed control volume (Eulerian coordinates)Otherwise Raynoldrsquos transport theoremMoving control volume Lagrangian coordinates
MingsianMingsian R R BaiBai42
But the control volime is only arbitrary
In particular for flow
( ) 0
con
(continuity equati
stant
on)
Divergence the
0 incompress
Note is t
ible
orem h
V
v
vt
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
e mother of the other integral theorems
MingsianMingsian R R BaiBai43
Ex Archimedes principle ( ) The buoyant force ( ) on a body with volume immersed in a fluid is equal to the weight of the fluid
ˆ displaced by the object ie In the followin
B
V
gVρ= minusf k
阿基米德原理
浮力
ˆg figure the ˆ with and being the fluid density gravitational
acceleration depth and outward normal unit vector to the surface
fhydraulic pressure g z
g z
S
ρ
ρ
= minusp n
n
1
the modified divergence theorem
ˆ
Hint
scalar fuctionV S
vdV v dS v Cnabla = isinint int n
MingsianMingsian R R BaiBai44
Levelsurfacefluid
V
S
airyx
z
ˆf g zρ= minusp n
ˆPf ( ) ( )
ˆ ˆ ˆ ˆ ˆ (0 0 ) v v
B fS S V
V V
g z g zdS dS dV
g dV g dV gV
ρ ρ
ρ ρ ρ
= = minus = minus nabla
= minus + + = minus = minus
int int int
int int
f p n
i j k k k
k
MingsianMingsian R R BaiBai45
Note This can also be thought of as simple balance of forces (resultant force that acts on and weight of filled with fluid)S V
2
~ a little history about George Green Recall the diveregence theorem
ˆ
Let
II th
e Green
are arbitrary fun
s identiti
ctions
es
V S
v dV n v dS
v u vu v C
nablasdot = sdot
= nablaisin
int intv v
v
ˆ ( ) ( )
ˆ ( )V S
V S
u v dV n u v dS
u v u v dV un v dS
nablasdot nabla = sdot nabla
nabla sdotnabla + nablasdotnabla = sdotnabla
int int
int int
MingsianMingsian R R BaiBai46
2 -( ) -(1)V S
vu v u v dV u dSnpart
nabla sdotnabla + nabla =partint int firs(Greens identt ity)
2
If
( ) --(2)
(1) (2) V S
u vuv u v u dV v dSn
harrpart
nabla sdotnabla + nabla =part
minus
int int
( )2 2
V S
dv duu v v u dV u v dSdn dn
⎛ ⎞nabla minus nabla = minus⎜ ⎟⎝ ⎠int int
second(Greens identity)
MingsianMingsian R R BaiBai47
( )
( )
( ) ( )
1st id ( )
Greens identities in the 1D case integration by parts
b b bb ba a
a a ab
ba
ab
ba
a
v udx v u v u dx u v uv dx uv
u v u v vu dx vu
uv vu dx uv vu
primeprime prime prime prime prime prime primeprime prime= minus rArr + =
prime prime primeprime primeharr + =
primeprime primeprime prime primerArr minus = minus
int int int
int
2
2nd id ( )
Good for any adjoint operators in addition to Good also for and integration by parts Important for theory amp reciprocity
From th
Gree
e 1s
2-
t
ns functi
identity
D 1-
tak 1
D
e
on
u
nablararr
=
int
2
V S
vv dV dSnpartnabla =part
rArr int int2Ex 0 st 0 on the boundary 0 ( 1st id u u unabla = = rArr equiv 以 証)
MingsianMingsian R R BaiBai48
1
Thm1 Stokes theorem Let be defined in Let be a piecewise-smooth orientable surface bounded by a piec
III Stokes theorem and
ewise-smooth simple clos
Greens
ed curve Then
theore
m
v C R S
C
isinv
( )
ˆwhere is a unit normal to according t right hand rulo the e
ˆS C
n S
n v dS v dRsdot nablatimes = sdotint intvv v
(not necessarily planar)S
Cn vnablatimes v vv
MingsianMingsian R R BaiBai49
Pf
( )( )
circulatioWe have shown in deriving thatthe in -direction
ˆor
ˆ
n
z
z
j j j j j
j j j j jj j
vz v dx dy
v dR n v S
v dR n v S
nablatimes
=nablatimes
sdot = sdot nablatimes Δ
sdot = sdot nablatimes Δ
intsum sumint
v
v
vv v
vv v
C
S
jS
jC
jP
ˆ jn
jSΔ
MingsianMingsian R R BaiBai50
( )
( )ext surface
After the interior boundaries cancel out only the exterior boundaries (Why)
ˆ
ˆ0 QED
Alternative forms scalar
survive
va
j
j j j j jC
jC S
v dR n v S
S v dR n v dS
φ
sdot = sdot nablatimes Δ
Δ rarr rArr sdot = sdot nablatimes
sum sumint
int int
vv v
vv v
( )
riable constant vector vector variable
1 Let
2 Let
ˆ
ˆ S C
S C
a p
v n dR dR
n p dS dR p
a
v a p
φ φφ= rArr timesnabla =
timesnabla times= rArr timestimes =
int int
int int
v v
v v
v
v v
vv vv v
MingsianMingsian R R BaiBai51
( ) ( )
( )
( )
( ) ( )
1
Thm2 Greens theorem ( planar version of Stokes theorem)ˆ ˆ Let
pf By Stokes theoremˆ
ˆ ˆ ˆ ˆ ˆ ˆ
S C
S C
S C
v P x y i Q x y j C
n v dS v dR
k Pi Qj dS Pi Qj dx i dy
Q P dS Pdx Qdyx y
⎛ ⎞part partminus = +⎜ ⎟
= + isin
sdot nablatimes = sdot
⎡ ⎤sdot nablatimes + =
part part⎝ ⎠
+ sdot +⎣ ⎦
int int
int
int
int
int
v
vv v
( )
( )
( )
ˆ
ˆˆ ˆ
ˆˆ ˆ ˆ ˆ( ) ( )
0
QEDS C
j
i j kQ PPi Qj i j k
x y z x yP Q
Q P dS Pdx Qdyx y
⎛ ⎞part part part part partnablatimes + = = + + minus⎜ ⎟part part part part part⎝ ⎠
⎛ ⎞part partrArr minus = +⎜ ⎟part part⎝ ⎠int int
L L
S
C
i
j
x
y
MingsianMingsian R R BaiBai52
Potential TheorypotentiGeorge Green coined the term (l
a )
rarr 位能 勢能
例子 重力場 電磁場 流場 聲場 彈性力學
1
2
Thm1 If is in a simply connected region then the following statements are equivalent(i) There exists a scala
Conservative Field
potential functior such that
(ii
n
) 0
v
C
C
v
v R
φφ=nabla
nablatimes
isin
isin
=r
r
r
r in (Irrotational field curl-free)
cf 0 st solenoidal field divergence-free
0
(iii) in (Conservative field)C
Rv w v w
v R Rd
nablasdot = exist nabla
sdot =
= times
int
r r r
r
r
r
MingsianMingsian R R BaiBai53
(iv) is connecting and
independent o
Ex Think
f the pa
of dynamics is the force
thB
A
v dR A B
v
sdotintrr
r
1 2
1
0
Pf
(i) (ii) ( ) 0 (zero identity cg rule) irrotational
ˆ(ii) (iii) ( ) 0 (Stokes theorem)
(iii) (iv) ( ) 0
C S
C C C
C
v
v dR n v dS
v dR v dR
v dR
φ
=
rarr nablatimes = nablatimes nabla =
rarr sdot = sdot nablatimes =
rarr sdot = minus sdot =
rArr sdot =
int int
int int int
int
r
rr
rr r
r rr r
rr r
2
independent of pathC
v dRsdotintr
A
B
1C
2C
MingsianMingsian R R BaiBai54
1 2
(iv) (i) constant ( ) ( )
--(1)
But (2)
(1) (2) QED
Note on application
Check
B
C C A
v dR v dR B A d
v dR d
d dx dy dz dRx y z
v dR dR v R
φ φ φ
φφ φ φφ φ
φ φ
rarr sdot = sdot = = minus =
rArr sdot =part part part
= + + = nabla sdot minus minus minus minus minuspart part part
sdot = nabla sdot rArr = nabla forall
int int intr rr r
rr
r
r r rr r
use simpler path (iv)if 0
find (i)v
φ⎧
nablatimes = ⎨⎩
rr
A
B
x
y
Cz
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai4
For the Cartesian coordinates
0 1 1
0
0 1 1
for some point ( ) on the - (by 2
mean valu
ˆ ˆ
ˆˆ ˆ ˆˆ ( ) ( )2
( )2
x face x front x back
x y z xx front x front
x
xx y z x front
n v dS n v dS
xn v dS i v i v j v k dS v x y z dydz
xv x y z y z
minus minus minus
minus minus
Δ+
sdot = + sdot
Δsdot = sdot + + = +
Δ= + Δ Δ
int int int
int int int
r r
r
e theorem)
j
xΔ
yΔ
zΔk
i
( ) P x y z center of cube=
MingsianMingsian R R BaiBai5
0 2 2
0 2 2
0
Similarly
ˆˆ ˆ ˆˆ ( ) ( )2
for some point ( ) on the - 2
Thus the net outflow per unit volume in -direction
ˆlim
x y z xx back x back
x face
V
xn v dS i v i v j v k dS v x y z y z
xx y z x back
x
n v dS
V
minus minus
minus
rarr
Δsdot = minus sdot + + = minus minus Δ Δ
Δminus
⎧ ⎫sdot⎪⎨ ⎬⎪⎩
int int
int
r
r0 0
0
( ) ( )2 2lim
x x
x y z
x xv x y z v x y z y z
Δ Δ Δ rarr
Δ Δ⎡ ⎤+ minus minus Δ Δ⎢ ⎥⎪ ⎣ ⎦=⎪⎭
x y zΔ Δ Δ
0 0
0
( ) ( )2 2 lim
Hence the net outflow per unit volume in the 3 directi
div
ons sums up to
(space operat ) or
x xx
x
yx zvv
x xv x y z v x y z vx x
vv vx y z
Δ rarr
Δ Δ+ minus minus part
=
partpart partnabla = + +
part part part
=Δ part
r r
MingsianMingsian R R BaiBai6
Gradient 梯度
g
R
r
ec
ad
all
ient
the divergence operator div
ˆˆ ˆIn Cartesian coordinates
Define the operator
a spatial differential operator
ˆˆ ˆgr
independent
ad
of coo
v v
i j kx
u u uu u i j kx y
y
z
z
part part part
=nablasdotpart part part
nabla + +part part part
rarr
nabla = + +part part part
r r
rdinates chosen ie is invariant wrt coordinates
directional derivative ( )ˆˆ ˆLet ( )
Physical interpretation
(position vector)
ˆ( ) (
u
u u x y z r xi yj zk
u u u u udu r dx dy dz ix y z x
nabla
= = + +
part part part part part= + + = +part part part part part
r
r
方向導數
ˆˆ ˆ
ˆ ˆˆ ˆ
( ) (
ˆ
)
) ( )
( )
dr r xi yj zk
uj k idx jdy kdzy z
du x u ddu r xu dr
= + +
=
part+ sdot + +part
primerarrsdot =nabla
r r
r r可視為 的推廣
MingsianMingsian R R BaiBai7
ˆ rate of change of in the direction Di
ˆ
rectional De
( )
ˆw
rivative
here
u sdu drr u u sdr dr
drsdr
=nabla sdot = nabla sdotr
r
r
s0
ˆ( ) ( )limh
du u r hs u rdr hrarr
+ minus=r r
drv
r dr+r r
rr
oNote
ˆ
ˆ )
ˆ
ˆ(
dudr
u s
s u
d sdr
s u
=nabla sdot
= sdotnabla
rArr
nabla
nabla
sdot=
sdotθ
s
s
( )0 yxx
y
u
unabla
drdu
duslopedr
=
( )yxfu =
MingsianMingsian R R BaiBai8
ˆEx Normal gradient outward normal
Note is often written as in the PDE literature
ˆˆ ˆˆPartial derivatives If then
ˆ
Physical interpretati
n
du udn n
du u u us i j kdr
du n ud
z
n
x y
partpart
part part part= rarrpart part part
sdotnabla
max
maximum rate of increase in a direction
ˆ ˆ ˆ ˆcos ( ) cos ( )
ˆ is independent of
ˆmax occurs when ( ) 0 and
points in the direction of ma
o
n
x a
du u s u s u s u u sdr
u sdu duu s udr dr
duudr
θ θ
θ ⎛ ⎞⎜ ⎟⎝ ⎠
=nabla sdot = nabla nabla = nabla nabla
nabla
there4 nabla = = nabla
rArrnabla
Q
( )max
(interpretation 1nd
gradient search steepest desce
)
nt
duudr
⎛ ⎞⎜ ⎟⎝ ⎠
nabla =
rArr minusnabla最佳化方法 的基礎
MingsianMingsian R R BaiBai9
(interpretatDirection of ion 2 normal to the level surface ( )
On the 0 ( )
ie
level surfac
o
e
n
)udu u dr
u cdu u dr
u dr u c
nabla= nabla sdot
== = nabla sdotnabla perp =
r
r
r
unabla
dr
unabla
Q
P
1cu=
2u c=
山
x
y
( )u x y
x
y
Hill-climbing search in optimization
MingsianMingsian R R BaiBai10
Ex Plane equationˆˆ ˆ( ) Normal vector
Ex 0 is the necessary condition for searching extrema (from multivariate Tylor expansion)
Ex Revisit of the exampl
u x y z ax by cz d u ai bj ck
u
= + + = rArr nabla = + +
nabla =v
多變數函數求極值
2 21 1 2 2
e in Linear Algebra using method of What is the maximum and minimum distance of the origin to the following curve
Lagrange multipli s
6 8
er
5 5x x x x+ + =
MingsianMingsian R R BaiBai11
1 2
2 21 2( )
2 21 1 2 2
Sol
This problem can be posed as a problemmin max ( )
constrained optimization
5 6 5 8 0 ( )which can
objectivecost function
constraint
equatioals
nx x
J x x
st g x x x x
= +
= + + minus =
2 2 2 21 2 1 1 2 2
o be converted into an unconstrained optimization problem by defining the ( ) (5 6 5 8)where is referred to as the thatLa
L
g ora
agr
nge ften h multi
angi
plie as
an
r ph
L x x x x x xλλ
= + + + + minus
ysical meaning in certain problemsThus the extreme values of will occur wh en or L J gJ nabla = nabla nabla0
MingsianMingsian R R BaiBai12
1 1 21
2 1 22
2 21 1 2 2
1 2 1 2
2 (10 6 ) 0 (1)
2 (6 10 ) 0 (2)
5 6 5 8 0 (constraint)
(1) (2) (2 16 )( ) 0 18 or (3)Substituting Eq (3) into Eqs (1) and (2) l
L x x xxL x x xxL x x x x
x x x x
λ
λ
λλ λ
part = + + =partpart = + + =partpart = + + minus =part+ + + = rArr = minus = minus
1 2 1 2eads to
and which corresponds to 14Thus minimummaximum distance 12x x x x J= = minus =
=
MingsianMingsian R R BaiBai13
22 2
1
1 1
2 2
The result above is identical to the previously obtained result in Linear Algebr
ellipse
45 rotatio
a
1 ( )4
1 11 1
n12
xx
x xx x
deg⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠
prime+ =prime
primeminus= rArr
prime2x prime
2x
1x prime
1x
2
MingsianMingsian R R BaiBai14
Curl 旋度
curl curl In Cartesian coordinates
spatial operator indep
ˆˆ ˆ
ˆˆ ˆ( )
ende
(
nt of coordi
) (
D
nates
ef
)
x y z
y x x xz z
i j k
vx y z
v v v
v v v vv vi j ky z x z x y
v v
part part partnablatimes =part part part
part part part partpart part= minus minus minus + minuspart
times
part part part part part
nabla
rarr
r
r r
0vnablatimes =rr 0vnablatimes ne
rr
MingsianMingsian R R BaiBai15
circulaPhysica
tion o spin ol interpretati
f fluidr or vortion
y
citΓ
( )0 0x dx y dy+ +
( )0 0x y ( )0 0x dx y+
Γ
1
2
3
4
( )0 0x y dy+
MingsianMingsian R R BaiBai16
yy
vv dy
ypart
+part
Γdyminus
dyyvv x
x partpart
+
dy
dx
dxminus
y yy
v vv dx dy
x ypart part
+ +part part
x xx
v vv dx dyx y
part part+ +part part
yy
vv dx
xpart
+part
xx
vv dxx
part+part
xv
yv
vv
dλ paddle weel
MingsianMingsian R R BaiBai17
contour
1 2 3 4
circulation
x x y y x x y y
x
v d
v d v d v d v d
v
λ
λ λ λ λrarr
Γ sdot
= + + +
asymp
intint int int int
vv
12
xv dxx
part+part ydx v
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
+ 12
y yv vdx dy
x ypart part
+ +part part
x
dy
v
⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦
+
12
xv dxx
part+part
( )xy
v dy dx vy
⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦
part+ minus +part
12
yvdy
ypart
+part
( )
( )
Similar process can be applied to the circulation in the and directionsThus the total circulation per unit area i curl s
y xz
dy
v v dxdy v dxdyx y
xv v
y
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎝ ⎠
minus
part part= minus = nablatimespart part
=nablatimes
v
v v
MingsianMingsian R R BaiBai18
Identities of differential operators ( p380 Greenberg )
( )
( ) ( )( )( )
( ) ( )( ) is required
Example of deriv
ˆ ˆˆ ˆ ˆ ˆ
ation
x y z
x y z
v v vx y z
uv u v u v
uv i j k uv i uv j uv kx y z
uv u v u v
u v v u u v
u v u vv v u u
+ +part part partpart part part
⎛ ⎞⎜ ⎟⎝ ⎠
nablasdot = nabla sdot + nablasdot
part part partnablasdot = + + sdot + + =part part part
nablatimes =nabla times + nablatimes
nablasdot times = sdotnablatimes minus sdotnablatimes
nablatimes times = nabla minus + minusnablanablasdot sdotsdot678 678
v
v v v
vL
v v v
v v v v v v
v v v v v v v v( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )2 cf
x y zu u ux y z
A B C A
u v
v w v w w v v w w v
v v v C B A B C
+ +part part partpart part part
sdotnabla
nabla sdot = sdotnabla + sdotnabla + times nablatimes + times
times
nablatimes
nablatimes nablatimes =nabla nabla times = sdot minussdot sdotminusnabla
678v
v v v
v v v v v vv v
v v v v v v
v v
v
v v
ˆ ˆˆ ˆ ˆ ˆ( ) ( ) ( )x y z x y zv i v j v k i j k u v v v ux y z x y zpart part part part part part+ + sdot + + = + +part part part part part part
v v
MingsianMingsian R R BaiBai19
2 2 22
2 2 2
0 (dc direct current) solenoidal0 (cg center of gravity) conservative potent
Laplacia
Zero identitie
n
s
ial
x y z
vu
⎧⎪⎨⎪⎩
part part partΔ =nabla nablasdotnabla = + +part part part
nablasdotnablatimes = rarr
nablatimesnabla = rarr
v
v
These differential operators will take more complicated and irregular form for the other coordinate systems
MingsianMingsian R R BaiBai20
2 2 3
2 2
ˆˆ ˆEx 2
ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ 2 0 2
2 0 0 2yx z
u x y v x yi z j k
u i j k u xyi x j k xyi x jx y z
vv vv xy xyx y z
⎛ ⎞⎜ ⎟⎝ ⎠
= = minus +
part part partnabla = + + = + + = +part part part
partpart partnablasdot = + + = + minus =part part part
v
v
MingsianMingsian R R BaiBai21
( ) ( ) ( )
( ) ( )
2 2 2 2
2 2 22
2 2 2
2
ˆˆ ˆ
ˆˆ ˆ
ˆ ˆˆ ˆ ˆ 0 3 0 0 0 3
2 0 0 2
ˆˆ ˆ
y yx xz z
x y z
i j kv vv vv vv i j k
x y z y z x z x yv v v
z i j x k z i x k
u u uu y yx y zv v v
i j kx y z
⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠ ⎝ ⎠
⎛
⎝
part partpart partpart partpart part partnablatimes = = minus minus minus + minuspart part part part part part part part part
= + minus minus + minus = minus
part part partnabla = + + = + + =part part part
nablatimes nablatimes =nabla nablasdot minusnabla
part part part= + +part part part
v
v v v
( ) ( )( ) ( ) ( ) ( )
( )
2 2 22 3
2 2 2ˆˆ ˆ2 2
ˆ ˆˆ ˆ ˆ ˆ 2 2 0 2 0 0 0 0 6 0 0 0
ˆ 2 6
xy x yi z j kx y z
yi xj k y i z j k
x z j
⎛ ⎞⎞⎜ ⎟⎜ ⎟
⎠ ⎝ ⎠⎡ ⎤⎣ ⎦
part part partminus + + minus +part part part
= + + minus + + + + minus + + minus
= +
MingsianMingsian R R BaiBai22
( )v w r w r r wnablatimes =nablatimes times = nablasdot minus nablasdotv v v v v v v ( )r w+ sdotnablav v ( ) constant w
w rminus sdotnablav
v v
Q
Ex Rigid body rotation
MingsianMingsian R R BaiBai23
( )( ) ( )
( ) ( )
ˆˆ ˆ
ˆ ˆˆ ˆ ˆ ˆ 1 1 1 3
ˆˆ ˆ
ˆˆ ˆ
Thus 3 2 is linked directly
x y z
x y z
r xi yj zk
r i j k xi yj zkx y z
w r w w w xi yj zkx y z
w i w j w k w
w r w r w r w w wv
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎝ ⎠
= + +
part part partnablasdot = + + + + = + + =part part part
part part partsdotnabla = + + + +part part part
= + + =
nablatimes times == nablasdot minus sdotnabla = minus =rArrnablatimes
v
v
v v
v
v v v v v v v v v
v spinn to ing
MingsianMingsian R R BaiBai24
Line surface and volume integrals
x
yC
B
A
isΔ
( )ii ηξ
Piecewise smooth curve
Line integral
MingsianMingsian R R BaiBai25
0 1lim ( ) ( ) or ( )
Intepretation of a line integral as an area ( )
i
Bn
i i isA
C
i C
F s F x y ds F x y d
x
s
F y ds
ξ ηΔ rarr =
Δ =sum int
int
int
x
y
z
C
( )z F x y=
MingsianMingsian R R BaiBai26
the area is always on the left of an obs
Positive dierver trave
rection of irsing on the
ntegrationcurve
RC
CCW CW
If is a regular closed curve piecewise smo
( ) ( )
(contou
oth simple ( )
r integral)
x y zC C
x y zC
F dR F F F dx dy dz
F dx F dy
C
F dz
sdot = + + + +
= + +
lowast
int int
int
i j k i j kv v
無交叉
work dW F dR= sdotv v
MingsianMingsian R R BaiBai27
0 1
Surface integral
lim ( ) ( )i
n
i i i iS i S
F S F x y z dSξ η ζΔ rarr =
Δ =sum intint( ) i i iξ η ζ
iSΔ
0 1
mass density
lim ( ) ( )
Ex
Volume integral
( )
i
n
i i i iV i V
V
F V F x y z dV
M x y z dV
ξ η ζ
ρ
Δ rarr =Δ =
=
sum intintint
intintint( ) i i iξ η ζ
iVΔ
MingsianMingsian R R BaiBai28
Ref (1)Hildebrand advanced Calculus for application Prentice-Hall 1976 p306-313(2) Greenberg p383-389 (3) Arfken ch2
1 2 3
1 2 3
1 2 3
1 2 3
(
Orthogonal coordinate systems base vectors are not constant in s
) ( )( )( )( )
ˆˆ ˆ
pacep x y z p u u ux x u u uy y u u uz z u u u
r xi yj zk
primerarr
= +
rarr
===
+v
1uv
2uv3uv
1u
2u
3uP coord curve
xy
z
O
X Curvilinear Coordinates
MingsianMingsian R R BaiBai29
1
11
1 1 1
1
1
1 1 1
11 1
1 1
2 2 2 3 3 3
Tangent vector to the at
where is the arc length
1
where unit vector
Similarly
curv
e
u psr rU
u s us
rs
U h uds rh udu sU h u U h u
partpart part= =part part part
part =part
there4 =part= =part
= =
v vv
vQ
v v
vv
v vv v
1u cood curve
1uv
1du 1dS
drv
P
MingsianMingsian R R BaiBai30
3 31 2 1 21 2 3
1 2 32
curv
In summary scale factor
If is the arc length along a in any directione
k k k
kk k
k k
k k k
U h u
ds rh Udu u
ds h dus
du dudu du du dudr r r r U U Uds u ds u ds u ds ds ds ds
dr drds ds
=
part= = =part
=
part part part= + + = + +part part part
= sdot
v v
v v
v v v v v v v
v v 3 3
1 1
23 3 32 2
21 1 1
2 2 2 2 2 2 2 2 2 21 1 2 2 3 3 1 2 3
1 (for orthogonal coord)
jii j
i j
ji ii j i
i j i
dududr U Uds ds ds
dudu duU U hds ds ds
ds h du h du h du ds ds ds
= =
= = =
⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
= sdot
rarr = sdot =
rarr = + + = + +
sum sum
sumsum sum
v v v
v v
MingsianMingsian R R BaiBai31
1 1 2 2 3 3
1 2 3
are mutually vectors having with arc length
k k k k kds h du U du
U du U du U duds ds ds
= =
rArr perp
v
v v v
1uv
2uv
3uv
1dS
2dS
3dS
dS
MingsianMingsian R R BaiBai32
( ) ( )
( ) ( )
1 1 2 2 3 3 1 2 3 1 2 3 1 2 3
1 2 3 1 2 3
1
1 2 2 3 3 2 2 2 3 3 3 2 3 2 3 1
Differential volume
Differential On const coord surface
area
Gradi t
en
d U du U du U du u u u h h h du du du
d h h h du du du
ud U du U du h u du h u du h h du du u
df
τ
τ
σ
= times sdot = times sdot
=
=
= times = times =
= nabla
v v v v v v
v vv v v v
1 2 31 2 3
1 2 3 1 1 2 2 3 31 2 3
1 1 1 2 2 2 3 3 3
f f ff dr df du du duu u u
r r rdr du du du U du U du U duu u u
h u du h u du h u du
part part partsdot rarr = + +part part part
part part part= + + = + +part part part
= + +
v
v v v v v vv
v v v
=1 (right hand rule)
MingsianMingsian R R BaiBai33
Rectangular Cartesian Coordinates
ij
k
MingsianMingsian R R BaiBai34
Circular Cylindrical Coordinatescossin
xyz z
ρ ϕρ ϕ
===
Ref Arfken
( )P zρ ϕ
MingsianMingsian R R BaiBai35
Polar Spherical Coordinatessin cossin sincos
x ry rz r
θ ϕθ ϕθ
===
Ref Arfken
0
0
0
sin cos sin sin coscos cos cos sin sin
sin cos
θ ϕ θ ϕ θθ ϕ θ ϕ θϕ ϕ
= + +
= + minus= minus +
r i j kθ i j kφ i j
2
2
sinsin
dS r d ddV r dr d d
θ θ ϕ
θ θ ϕ
=
=( )P r θ ϕ
MingsianMingsian R R BaiBai36
Integral TheoremsHeart of this chapter
Interchange of dimensionality Div Curl Grad in integrals High-dimensional integration
rarr
1
Thm1 Divergence theorem Let be a closed region bounded by piecewise smooth orientable surface and let (con
I Divergence theor
tinuous derivatives up to the first order
em (Gauss theorem)
)
VS v Cisin
r
ˆ If denotes the outward unit normal on
the ˆn V S
v dV n v dS
n S
nablasdot = sdotint intr r
MingsianMingsian R R BaiBai37
jS
jP
jV
S
0
ˆBy definition ( ) lim
ˆ( )
ˆ( )
j
j
j
j
Sj V
j
j j S
j j Sj j
n v dSv P
V
v P V n v dS
v P V n v dS
Δ rarr
⎧ ⎫sdot⎪ ⎪nabla sdot ⎨ ⎬Δ⎪ ⎪⎩ ⎭
nabla sdot Δ asymp sdot
nabla sdot Δ = sdot
int
intsum sumint
rr
r r
r r
Pf
MingsianMingsian R R BaiBai38
0
After interior surfaces cancel out only exterior surface (Why Opposite normal vectors)
ˆ( )
volumes exterior surfaces
ˆlim
QE
survive
j
j
j j Sj j
V SV
v p V n v dS
v dV n v dSΔ rarr
nabla sdot Δ = sdot
rArr nablasdot = sdot
sum sumint
int int
r r
r r
D
MingsianMingsian R R BaiBai39
( )
scalar variable constant vectorvector variable
Alternative formsLet be a an arbitrary and
a
ˆ
1 Le
(
ˆt
V
V S
S
V
v ap
v d
v dV vn d
V n v dS
v va
va dV v a v a
S
nablasdot = sdot
= rArr
nablasdot = nabla sdot + nabla
=
sdot
nablaint intint int
int
r
v
r r
r r
r r rQ ( )
( )
ˆ)
2 Let ˆV S
V
V S
dV n va dS
v a p
a p
p dV n p d
d p
S
V a
⎛ ⎞= sdot⎜ ⎟
⎝ ⎠
= times rArr
nablasdot times = sdot nabla
times
times
nabla = times
int int
int int
int
r
r r r
r r v
r
vQ
r
( ) ( ) ( )
( ) ( )
ˆ
ˆV S
V S
a p dV n a p dS
a p dV a n p dS
⎛ ⎞⎡ ⎤minus sdot nablatimes = sdot times⎣ ⎦⎜ ⎟⎜ ⎟⎜ ⎟rArr minus sdot nablatimes = minus sdot times⎜ ⎟⎝ ⎠
int int
int int
r rv v
r rv v
MingsianMingsian R R BaiBai40
( )
( )
From Leibnitzs rule
(1-D) ( )
( ) ( )
B t
A t
d f x t dxdt
f x t dx B ttpart prime= +part
int( )
( )( ( ) ) ( )
B t
A tf B t t A tprimeminusint ( ( ) )
(assume ( ) ( ) )
f A t t
A t B t const= =
ContinuityEx The Equation
rate of increase of mass
mass densityV
dM d dVdt dt
ρ
ρ
= int
flow velocityvv
MingsianMingsian R R BaiBai41
ˆ ˆ ( )
(
Conservation
) ( Divergence theorem)
of mass
( ) 0
V S S
V
V
dV n v dS n v dSt
v dV
v dVt
ρ ρ ρ
ρ
ρ ρ⎡ ⎤⎢ ⎥⎣ ⎦
part = minus sdot = minus sdotpart
= minus nablasdot
partrArr +nablasdot =part
int int int
int
int
v v
vQ
v
ˆOn the other hand rate influof mass through n ox i t
Extended to 3-D for time-independent fixed boundary
(3-D) ( ) ( )
Thus
S
V V
V
S V n v dS
Sd f x y z t dV f x y z t dVdt tdM dVdt t
ρ
ρ
= minus sdot
part=part
part=part
int
int int
intv
Fixed control volume (Eulerian coordinates)Otherwise Raynoldrsquos transport theoremMoving control volume Lagrangian coordinates
MingsianMingsian R R BaiBai42
But the control volime is only arbitrary
In particular for flow
( ) 0
con
(continuity equati
stant
on)
Divergence the
0 incompress
Note is t
ible
orem h
V
v
vt
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
e mother of the other integral theorems
MingsianMingsian R R BaiBai43
Ex Archimedes principle ( ) The buoyant force ( ) on a body with volume immersed in a fluid is equal to the weight of the fluid
ˆ displaced by the object ie In the followin
B
V
gVρ= minusf k
阿基米德原理
浮力
ˆg figure the ˆ with and being the fluid density gravitational
acceleration depth and outward normal unit vector to the surface
fhydraulic pressure g z
g z
S
ρ
ρ
= minusp n
n
1
the modified divergence theorem
ˆ
Hint
scalar fuctionV S
vdV v dS v Cnabla = isinint int n
MingsianMingsian R R BaiBai44
Levelsurfacefluid
V
S
airyx
z
ˆf g zρ= minusp n
ˆPf ( ) ( )
ˆ ˆ ˆ ˆ ˆ (0 0 ) v v
B fS S V
V V
g z g zdS dS dV
g dV g dV gV
ρ ρ
ρ ρ ρ
= = minus = minus nabla
= minus + + = minus = minus
int int int
int int
f p n
i j k k k
k
MingsianMingsian R R BaiBai45
Note This can also be thought of as simple balance of forces (resultant force that acts on and weight of filled with fluid)S V
2
~ a little history about George Green Recall the diveregence theorem
ˆ
Let
II th
e Green
are arbitrary fun
s identiti
ctions
es
V S
v dV n v dS
v u vu v C
nablasdot = sdot
= nablaisin
int intv v
v
ˆ ( ) ( )
ˆ ( )V S
V S
u v dV n u v dS
u v u v dV un v dS
nablasdot nabla = sdot nabla
nabla sdotnabla + nablasdotnabla = sdotnabla
int int
int int
MingsianMingsian R R BaiBai46
2 -( ) -(1)V S
vu v u v dV u dSnpart
nabla sdotnabla + nabla =partint int firs(Greens identt ity)
2
If
( ) --(2)
(1) (2) V S
u vuv u v u dV v dSn
harrpart
nabla sdotnabla + nabla =part
minus
int int
( )2 2
V S
dv duu v v u dV u v dSdn dn
⎛ ⎞nabla minus nabla = minus⎜ ⎟⎝ ⎠int int
second(Greens identity)
MingsianMingsian R R BaiBai47
( )
( )
( ) ( )
1st id ( )
Greens identities in the 1D case integration by parts
b b bb ba a
a a ab
ba
ab
ba
a
v udx v u v u dx u v uv dx uv
u v u v vu dx vu
uv vu dx uv vu
primeprime prime prime prime prime prime primeprime prime= minus rArr + =
prime prime primeprime primeharr + =
primeprime primeprime prime primerArr minus = minus
int int int
int
2
2nd id ( )
Good for any adjoint operators in addition to Good also for and integration by parts Important for theory amp reciprocity
From th
Gree
e 1s
2-
t
ns functi
identity
D 1-
tak 1
D
e
on
u
nablararr
=
int
2
V S
vv dV dSnpartnabla =part
rArr int int2Ex 0 st 0 on the boundary 0 ( 1st id u u unabla = = rArr equiv 以 証)
MingsianMingsian R R BaiBai48
1
Thm1 Stokes theorem Let be defined in Let be a piecewise-smooth orientable surface bounded by a piec
III Stokes theorem and
ewise-smooth simple clos
Greens
ed curve Then
theore
m
v C R S
C
isinv
( )
ˆwhere is a unit normal to according t right hand rulo the e
ˆS C
n S
n v dS v dRsdot nablatimes = sdotint intvv v
(not necessarily planar)S
Cn vnablatimes v vv
MingsianMingsian R R BaiBai49
Pf
( )( )
circulatioWe have shown in deriving thatthe in -direction
ˆor
ˆ
n
z
z
j j j j j
j j j j jj j
vz v dx dy
v dR n v S
v dR n v S
nablatimes
=nablatimes
sdot = sdot nablatimes Δ
sdot = sdot nablatimes Δ
intsum sumint
v
v
vv v
vv v
C
S
jS
jC
jP
ˆ jn
jSΔ
MingsianMingsian R R BaiBai50
( )
( )ext surface
After the interior boundaries cancel out only the exterior boundaries (Why)
ˆ
ˆ0 QED
Alternative forms scalar
survive
va
j
j j j j jC
jC S
v dR n v S
S v dR n v dS
φ
sdot = sdot nablatimes Δ
Δ rarr rArr sdot = sdot nablatimes
sum sumint
int int
vv v
vv v
( )
riable constant vector vector variable
1 Let
2 Let
ˆ
ˆ S C
S C
a p
v n dR dR
n p dS dR p
a
v a p
φ φφ= rArr timesnabla =
timesnabla times= rArr timestimes =
int int
int int
v v
v v
v
v v
vv vv v
MingsianMingsian R R BaiBai51
( ) ( )
( )
( )
( ) ( )
1
Thm2 Greens theorem ( planar version of Stokes theorem)ˆ ˆ Let
pf By Stokes theoremˆ
ˆ ˆ ˆ ˆ ˆ ˆ
S C
S C
S C
v P x y i Q x y j C
n v dS v dR
k Pi Qj dS Pi Qj dx i dy
Q P dS Pdx Qdyx y
⎛ ⎞part partminus = +⎜ ⎟
= + isin
sdot nablatimes = sdot
⎡ ⎤sdot nablatimes + =
part part⎝ ⎠
+ sdot +⎣ ⎦
int int
int
int
int
int
v
vv v
( )
( )
( )
ˆ
ˆˆ ˆ
ˆˆ ˆ ˆ ˆ( ) ( )
0
QEDS C
j
i j kQ PPi Qj i j k
x y z x yP Q
Q P dS Pdx Qdyx y
⎛ ⎞part part part part partnablatimes + = = + + minus⎜ ⎟part part part part part⎝ ⎠
⎛ ⎞part partrArr minus = +⎜ ⎟part part⎝ ⎠int int
L L
S
C
i
j
x
y
MingsianMingsian R R BaiBai52
Potential TheorypotentiGeorge Green coined the term (l
a )
rarr 位能 勢能
例子 重力場 電磁場 流場 聲場 彈性力學
1
2
Thm1 If is in a simply connected region then the following statements are equivalent(i) There exists a scala
Conservative Field
potential functior such that
(ii
n
) 0
v
C
C
v
v R
φφ=nabla
nablatimes
isin
isin
=r
r
r
r in (Irrotational field curl-free)
cf 0 st solenoidal field divergence-free
0
(iii) in (Conservative field)C
Rv w v w
v R Rd
nablasdot = exist nabla
sdot =
= times
int
r r r
r
r
r
MingsianMingsian R R BaiBai53
(iv) is connecting and
independent o
Ex Think
f the pa
of dynamics is the force
thB
A
v dR A B
v
sdotintrr
r
1 2
1
0
Pf
(i) (ii) ( ) 0 (zero identity cg rule) irrotational
ˆ(ii) (iii) ( ) 0 (Stokes theorem)
(iii) (iv) ( ) 0
C S
C C C
C
v
v dR n v dS
v dR v dR
v dR
φ
=
rarr nablatimes = nablatimes nabla =
rarr sdot = sdot nablatimes =
rarr sdot = minus sdot =
rArr sdot =
int int
int int int
int
r
rr
rr r
r rr r
rr r
2
independent of pathC
v dRsdotintr
A
B
1C
2C
MingsianMingsian R R BaiBai54
1 2
(iv) (i) constant ( ) ( )
--(1)
But (2)
(1) (2) QED
Note on application
Check
B
C C A
v dR v dR B A d
v dR d
d dx dy dz dRx y z
v dR dR v R
φ φ φ
φφ φ φφ φ
φ φ
rarr sdot = sdot = = minus =
rArr sdot =part part part
= + + = nabla sdot minus minus minus minus minuspart part part
sdot = nabla sdot rArr = nabla forall
int int intr rr r
rr
r
r r rr r
use simpler path (iv)if 0
find (i)v
φ⎧
nablatimes = ⎨⎩
rr
A
B
x
y
Cz
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai5
0 2 2
0 2 2
0
Similarly
ˆˆ ˆ ˆˆ ( ) ( )2
for some point ( ) on the - 2
Thus the net outflow per unit volume in -direction
ˆlim
x y z xx back x back
x face
V
xn v dS i v i v j v k dS v x y z y z
xx y z x back
x
n v dS
V
minus minus
minus
rarr
Δsdot = minus sdot + + = minus minus Δ Δ
Δminus
⎧ ⎫sdot⎪⎨ ⎬⎪⎩
int int
int
r
r0 0
0
( ) ( )2 2lim
x x
x y z
x xv x y z v x y z y z
Δ Δ Δ rarr
Δ Δ⎡ ⎤+ minus minus Δ Δ⎢ ⎥⎪ ⎣ ⎦=⎪⎭
x y zΔ Δ Δ
0 0
0
( ) ( )2 2 lim
Hence the net outflow per unit volume in the 3 directi
div
ons sums up to
(space operat ) or
x xx
x
yx zvv
x xv x y z v x y z vx x
vv vx y z
Δ rarr
Δ Δ+ minus minus part
=
partpart partnabla = + +
part part part
=Δ part
r r
MingsianMingsian R R BaiBai6
Gradient 梯度
g
R
r
ec
ad
all
ient
the divergence operator div
ˆˆ ˆIn Cartesian coordinates
Define the operator
a spatial differential operator
ˆˆ ˆgr
independent
ad
of coo
v v
i j kx
u u uu u i j kx y
y
z
z
part part part
=nablasdotpart part part
nabla + +part part part
rarr
nabla = + +part part part
r r
rdinates chosen ie is invariant wrt coordinates
directional derivative ( )ˆˆ ˆLet ( )
Physical interpretation
(position vector)
ˆ( ) (
u
u u x y z r xi yj zk
u u u u udu r dx dy dz ix y z x
nabla
= = + +
part part part part part= + + = +part part part part part
r
r
方向導數
ˆˆ ˆ
ˆ ˆˆ ˆ
( ) (
ˆ
)
) ( )
( )
dr r xi yj zk
uj k idx jdy kdzy z
du x u ddu r xu dr
= + +
=
part+ sdot + +part
primerarrsdot =nabla
r r
r r可視為 的推廣
MingsianMingsian R R BaiBai7
ˆ rate of change of in the direction Di
ˆ
rectional De
( )
ˆw
rivative
here
u sdu drr u u sdr dr
drsdr
=nabla sdot = nabla sdotr
r
r
s0
ˆ( ) ( )limh
du u r hs u rdr hrarr
+ minus=r r
drv
r dr+r r
rr
oNote
ˆ
ˆ )
ˆ
ˆ(
dudr
u s
s u
d sdr
s u
=nabla sdot
= sdotnabla
rArr
nabla
nabla
sdot=
sdotθ
s
s
( )0 yxx
y
u
unabla
drdu
duslopedr
=
( )yxfu =
MingsianMingsian R R BaiBai8
ˆEx Normal gradient outward normal
Note is often written as in the PDE literature
ˆˆ ˆˆPartial derivatives If then
ˆ
Physical interpretati
n
du udn n
du u u us i j kdr
du n ud
z
n
x y
partpart
part part part= rarrpart part part
sdotnabla
max
maximum rate of increase in a direction
ˆ ˆ ˆ ˆcos ( ) cos ( )
ˆ is independent of
ˆmax occurs when ( ) 0 and
points in the direction of ma
o
n
x a
du u s u s u s u u sdr
u sdu duu s udr dr
duudr
θ θ
θ ⎛ ⎞⎜ ⎟⎝ ⎠
=nabla sdot = nabla nabla = nabla nabla
nabla
there4 nabla = = nabla
rArrnabla
Q
( )max
(interpretation 1nd
gradient search steepest desce
)
nt
duudr
⎛ ⎞⎜ ⎟⎝ ⎠
nabla =
rArr minusnabla最佳化方法 的基礎
MingsianMingsian R R BaiBai9
(interpretatDirection of ion 2 normal to the level surface ( )
On the 0 ( )
ie
level surfac
o
e
n
)udu u dr
u cdu u dr
u dr u c
nabla= nabla sdot
== = nabla sdotnabla perp =
r
r
r
unabla
dr
unabla
Q
P
1cu=
2u c=
山
x
y
( )u x y
x
y
Hill-climbing search in optimization
MingsianMingsian R R BaiBai10
Ex Plane equationˆˆ ˆ( ) Normal vector
Ex 0 is the necessary condition for searching extrema (from multivariate Tylor expansion)
Ex Revisit of the exampl
u x y z ax by cz d u ai bj ck
u
= + + = rArr nabla = + +
nabla =v
多變數函數求極值
2 21 1 2 2
e in Linear Algebra using method of What is the maximum and minimum distance of the origin to the following curve
Lagrange multipli s
6 8
er
5 5x x x x+ + =
MingsianMingsian R R BaiBai11
1 2
2 21 2( )
2 21 1 2 2
Sol
This problem can be posed as a problemmin max ( )
constrained optimization
5 6 5 8 0 ( )which can
objectivecost function
constraint
equatioals
nx x
J x x
st g x x x x
= +
= + + minus =
2 2 2 21 2 1 1 2 2
o be converted into an unconstrained optimization problem by defining the ( ) (5 6 5 8)where is referred to as the thatLa
L
g ora
agr
nge ften h multi
angi
plie as
an
r ph
L x x x x x xλλ
= + + + + minus
ysical meaning in certain problemsThus the extreme values of will occur wh en or L J gJ nabla = nabla nabla0
MingsianMingsian R R BaiBai12
1 1 21
2 1 22
2 21 1 2 2
1 2 1 2
2 (10 6 ) 0 (1)
2 (6 10 ) 0 (2)
5 6 5 8 0 (constraint)
(1) (2) (2 16 )( ) 0 18 or (3)Substituting Eq (3) into Eqs (1) and (2) l
L x x xxL x x xxL x x x x
x x x x
λ
λ
λλ λ
part = + + =partpart = + + =partpart = + + minus =part+ + + = rArr = minus = minus
1 2 1 2eads to
and which corresponds to 14Thus minimummaximum distance 12x x x x J= = minus =
=
MingsianMingsian R R BaiBai13
22 2
1
1 1
2 2
The result above is identical to the previously obtained result in Linear Algebr
ellipse
45 rotatio
a
1 ( )4
1 11 1
n12
xx
x xx x
deg⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠
prime+ =prime
primeminus= rArr
prime2x prime
2x
1x prime
1x
2
MingsianMingsian R R BaiBai14
Curl 旋度
curl curl In Cartesian coordinates
spatial operator indep
ˆˆ ˆ
ˆˆ ˆ( )
ende
(
nt of coordi
) (
D
nates
ef
)
x y z
y x x xz z
i j k
vx y z
v v v
v v v vv vi j ky z x z x y
v v
part part partnablatimes =part part part
part part part partpart part= minus minus minus + minuspart
times
part part part part part
nabla
rarr
r
r r
0vnablatimes =rr 0vnablatimes ne
rr
MingsianMingsian R R BaiBai15
circulaPhysica
tion o spin ol interpretati
f fluidr or vortion
y
citΓ
( )0 0x dx y dy+ +
( )0 0x y ( )0 0x dx y+
Γ
1
2
3
4
( )0 0x y dy+
MingsianMingsian R R BaiBai16
yy
vv dy
ypart
+part
Γdyminus
dyyvv x
x partpart
+
dy
dx
dxminus
y yy
v vv dx dy
x ypart part
+ +part part
x xx
v vv dx dyx y
part part+ +part part
yy
vv dx
xpart
+part
xx
vv dxx
part+part
xv
yv
vv
dλ paddle weel
MingsianMingsian R R BaiBai17
contour
1 2 3 4
circulation
x x y y x x y y
x
v d
v d v d v d v d
v
λ
λ λ λ λrarr
Γ sdot
= + + +
asymp
intint int int int
vv
12
xv dxx
part+part ydx v
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
+ 12
y yv vdx dy
x ypart part
+ +part part
x
dy
v
⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦
+
12
xv dxx
part+part
( )xy
v dy dx vy
⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦
part+ minus +part
12
yvdy
ypart
+part
( )
( )
Similar process can be applied to the circulation in the and directionsThus the total circulation per unit area i curl s
y xz
dy
v v dxdy v dxdyx y
xv v
y
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎝ ⎠
minus
part part= minus = nablatimespart part
=nablatimes
v
v v
MingsianMingsian R R BaiBai18
Identities of differential operators ( p380 Greenberg )
( )
( ) ( )( )( )
( ) ( )( ) is required
Example of deriv
ˆ ˆˆ ˆ ˆ ˆ
ation
x y z
x y z
v v vx y z
uv u v u v
uv i j k uv i uv j uv kx y z
uv u v u v
u v v u u v
u v u vv v u u
+ +part part partpart part part
⎛ ⎞⎜ ⎟⎝ ⎠
nablasdot = nabla sdot + nablasdot
part part partnablasdot = + + sdot + + =part part part
nablatimes =nabla times + nablatimes
nablasdot times = sdotnablatimes minus sdotnablatimes
nablatimes times = nabla minus + minusnablanablasdot sdotsdot678 678
v
v v v
vL
v v v
v v v v v v
v v v v v v v v( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )2 cf
x y zu u ux y z
A B C A
u v
v w v w w v v w w v
v v v C B A B C
+ +part part partpart part part
sdotnabla
nabla sdot = sdotnabla + sdotnabla + times nablatimes + times
times
nablatimes
nablatimes nablatimes =nabla nabla times = sdot minussdot sdotminusnabla
678v
v v v
v v v v v vv v
v v v v v v
v v
v
v v
ˆ ˆˆ ˆ ˆ ˆ( ) ( ) ( )x y z x y zv i v j v k i j k u v v v ux y z x y zpart part part part part part+ + sdot + + = + +part part part part part part
v v
MingsianMingsian R R BaiBai19
2 2 22
2 2 2
0 (dc direct current) solenoidal0 (cg center of gravity) conservative potent
Laplacia
Zero identitie
n
s
ial
x y z
vu
⎧⎪⎨⎪⎩
part part partΔ =nabla nablasdotnabla = + +part part part
nablasdotnablatimes = rarr
nablatimesnabla = rarr
v
v
These differential operators will take more complicated and irregular form for the other coordinate systems
MingsianMingsian R R BaiBai20
2 2 3
2 2
ˆˆ ˆEx 2
ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ 2 0 2
2 0 0 2yx z
u x y v x yi z j k
u i j k u xyi x j k xyi x jx y z
vv vv xy xyx y z
⎛ ⎞⎜ ⎟⎝ ⎠
= = minus +
part part partnabla = + + = + + = +part part part
partpart partnablasdot = + + = + minus =part part part
v
v
MingsianMingsian R R BaiBai21
( ) ( ) ( )
( ) ( )
2 2 2 2
2 2 22
2 2 2
2
ˆˆ ˆ
ˆˆ ˆ
ˆ ˆˆ ˆ ˆ 0 3 0 0 0 3
2 0 0 2
ˆˆ ˆ
y yx xz z
x y z
i j kv vv vv vv i j k
x y z y z x z x yv v v
z i j x k z i x k
u u uu y yx y zv v v
i j kx y z
⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠ ⎝ ⎠
⎛
⎝
part partpart partpart partpart part partnablatimes = = minus minus minus + minuspart part part part part part part part part
= + minus minus + minus = minus
part part partnabla = + + = + + =part part part
nablatimes nablatimes =nabla nablasdot minusnabla
part part part= + +part part part
v
v v v
( ) ( )( ) ( ) ( ) ( )
( )
2 2 22 3
2 2 2ˆˆ ˆ2 2
ˆ ˆˆ ˆ ˆ ˆ 2 2 0 2 0 0 0 0 6 0 0 0
ˆ 2 6
xy x yi z j kx y z
yi xj k y i z j k
x z j
⎛ ⎞⎞⎜ ⎟⎜ ⎟
⎠ ⎝ ⎠⎡ ⎤⎣ ⎦
part part partminus + + minus +part part part
= + + minus + + + + minus + + minus
= +
MingsianMingsian R R BaiBai22
( )v w r w r r wnablatimes =nablatimes times = nablasdot minus nablasdotv v v v v v v ( )r w+ sdotnablav v ( ) constant w
w rminus sdotnablav
v v
Q
Ex Rigid body rotation
MingsianMingsian R R BaiBai23
( )( ) ( )
( ) ( )
ˆˆ ˆ
ˆ ˆˆ ˆ ˆ ˆ 1 1 1 3
ˆˆ ˆ
ˆˆ ˆ
Thus 3 2 is linked directly
x y z
x y z
r xi yj zk
r i j k xi yj zkx y z
w r w w w xi yj zkx y z
w i w j w k w
w r w r w r w w wv
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎝ ⎠
= + +
part part partnablasdot = + + + + = + + =part part part
part part partsdotnabla = + + + +part part part
= + + =
nablatimes times == nablasdot minus sdotnabla = minus =rArrnablatimes
v
v
v v
v
v v v v v v v v v
v spinn to ing
MingsianMingsian R R BaiBai24
Line surface and volume integrals
x
yC
B
A
isΔ
( )ii ηξ
Piecewise smooth curve
Line integral
MingsianMingsian R R BaiBai25
0 1lim ( ) ( ) or ( )
Intepretation of a line integral as an area ( )
i
Bn
i i isA
C
i C
F s F x y ds F x y d
x
s
F y ds
ξ ηΔ rarr =
Δ =sum int
int
int
x
y
z
C
( )z F x y=
MingsianMingsian R R BaiBai26
the area is always on the left of an obs
Positive dierver trave
rection of irsing on the
ntegrationcurve
RC
CCW CW
If is a regular closed curve piecewise smo
( ) ( )
(contou
oth simple ( )
r integral)
x y zC C
x y zC
F dR F F F dx dy dz
F dx F dy
C
F dz
sdot = + + + +
= + +
lowast
int int
int
i j k i j kv v
無交叉
work dW F dR= sdotv v
MingsianMingsian R R BaiBai27
0 1
Surface integral
lim ( ) ( )i
n
i i i iS i S
F S F x y z dSξ η ζΔ rarr =
Δ =sum intint( ) i i iξ η ζ
iSΔ
0 1
mass density
lim ( ) ( )
Ex
Volume integral
( )
i
n
i i i iV i V
V
F V F x y z dV
M x y z dV
ξ η ζ
ρ
Δ rarr =Δ =
=
sum intintint
intintint( ) i i iξ η ζ
iVΔ
MingsianMingsian R R BaiBai28
Ref (1)Hildebrand advanced Calculus for application Prentice-Hall 1976 p306-313(2) Greenberg p383-389 (3) Arfken ch2
1 2 3
1 2 3
1 2 3
1 2 3
(
Orthogonal coordinate systems base vectors are not constant in s
) ( )( )( )( )
ˆˆ ˆ
pacep x y z p u u ux x u u uy y u u uz z u u u
r xi yj zk
primerarr
= +
rarr
===
+v
1uv
2uv3uv
1u
2u
3uP coord curve
xy
z
O
X Curvilinear Coordinates
MingsianMingsian R R BaiBai29
1
11
1 1 1
1
1
1 1 1
11 1
1 1
2 2 2 3 3 3
Tangent vector to the at
where is the arc length
1
where unit vector
Similarly
curv
e
u psr rU
u s us
rs
U h uds rh udu sU h u U h u
partpart part= =part part part
part =part
there4 =part= =part
= =
v vv
vQ
v v
vv
v vv v
1u cood curve
1uv
1du 1dS
drv
P
MingsianMingsian R R BaiBai30
3 31 2 1 21 2 3
1 2 32
curv
In summary scale factor
If is the arc length along a in any directione
k k k
kk k
k k
k k k
U h u
ds rh Udu u
ds h dus
du dudu du du dudr r r r U U Uds u ds u ds u ds ds ds ds
dr drds ds
=
part= = =part
=
part part part= + + = + +part part part
= sdot
v v
v v
v v v v v v v
v v 3 3
1 1
23 3 32 2
21 1 1
2 2 2 2 2 2 2 2 2 21 1 2 2 3 3 1 2 3
1 (for orthogonal coord)
jii j
i j
ji ii j i
i j i
dududr U Uds ds ds
dudu duU U hds ds ds
ds h du h du h du ds ds ds
= =
= = =
⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
= sdot
rarr = sdot =
rarr = + + = + +
sum sum
sumsum sum
v v v
v v
MingsianMingsian R R BaiBai31
1 1 2 2 3 3
1 2 3
are mutually vectors having with arc length
k k k k kds h du U du
U du U du U duds ds ds
= =
rArr perp
v
v v v
1uv
2uv
3uv
1dS
2dS
3dS
dS
MingsianMingsian R R BaiBai32
( ) ( )
( ) ( )
1 1 2 2 3 3 1 2 3 1 2 3 1 2 3
1 2 3 1 2 3
1
1 2 2 3 3 2 2 2 3 3 3 2 3 2 3 1
Differential volume
Differential On const coord surface
area
Gradi t
en
d U du U du U du u u u h h h du du du
d h h h du du du
ud U du U du h u du h u du h h du du u
df
τ
τ
σ
= times sdot = times sdot
=
=
= times = times =
= nabla
v v v v v v
v vv v v v
1 2 31 2 3
1 2 3 1 1 2 2 3 31 2 3
1 1 1 2 2 2 3 3 3
f f ff dr df du du duu u u
r r rdr du du du U du U du U duu u u
h u du h u du h u du
part part partsdot rarr = + +part part part
part part part= + + = + +part part part
= + +
v
v v v v v vv
v v v
=1 (right hand rule)
MingsianMingsian R R BaiBai33
Rectangular Cartesian Coordinates
ij
k
MingsianMingsian R R BaiBai34
Circular Cylindrical Coordinatescossin
xyz z
ρ ϕρ ϕ
===
Ref Arfken
( )P zρ ϕ
MingsianMingsian R R BaiBai35
Polar Spherical Coordinatessin cossin sincos
x ry rz r
θ ϕθ ϕθ
===
Ref Arfken
0
0
0
sin cos sin sin coscos cos cos sin sin
sin cos
θ ϕ θ ϕ θθ ϕ θ ϕ θϕ ϕ
= + +
= + minus= minus +
r i j kθ i j kφ i j
2
2
sinsin
dS r d ddV r dr d d
θ θ ϕ
θ θ ϕ
=
=( )P r θ ϕ
MingsianMingsian R R BaiBai36
Integral TheoremsHeart of this chapter
Interchange of dimensionality Div Curl Grad in integrals High-dimensional integration
rarr
1
Thm1 Divergence theorem Let be a closed region bounded by piecewise smooth orientable surface and let (con
I Divergence theor
tinuous derivatives up to the first order
em (Gauss theorem)
)
VS v Cisin
r
ˆ If denotes the outward unit normal on
the ˆn V S
v dV n v dS
n S
nablasdot = sdotint intr r
MingsianMingsian R R BaiBai37
jS
jP
jV
S
0
ˆBy definition ( ) lim
ˆ( )
ˆ( )
j
j
j
j
Sj V
j
j j S
j j Sj j
n v dSv P
V
v P V n v dS
v P V n v dS
Δ rarr
⎧ ⎫sdot⎪ ⎪nabla sdot ⎨ ⎬Δ⎪ ⎪⎩ ⎭
nabla sdot Δ asymp sdot
nabla sdot Δ = sdot
int
intsum sumint
rr
r r
r r
Pf
MingsianMingsian R R BaiBai38
0
After interior surfaces cancel out only exterior surface (Why Opposite normal vectors)
ˆ( )
volumes exterior surfaces
ˆlim
QE
survive
j
j
j j Sj j
V SV
v p V n v dS
v dV n v dSΔ rarr
nabla sdot Δ = sdot
rArr nablasdot = sdot
sum sumint
int int
r r
r r
D
MingsianMingsian R R BaiBai39
( )
scalar variable constant vectorvector variable
Alternative formsLet be a an arbitrary and
a
ˆ
1 Le
(
ˆt
V
V S
S
V
v ap
v d
v dV vn d
V n v dS
v va
va dV v a v a
S
nablasdot = sdot
= rArr
nablasdot = nabla sdot + nabla
=
sdot
nablaint intint int
int
r
v
r r
r r
r r rQ ( )
( )
ˆ)
2 Let ˆV S
V
V S
dV n va dS
v a p
a p
p dV n p d
d p
S
V a
⎛ ⎞= sdot⎜ ⎟
⎝ ⎠
= times rArr
nablasdot times = sdot nabla
times
times
nabla = times
int int
int int
int
r
r r r
r r v
r
vQ
r
( ) ( ) ( )
( ) ( )
ˆ
ˆV S
V S
a p dV n a p dS
a p dV a n p dS
⎛ ⎞⎡ ⎤minus sdot nablatimes = sdot times⎣ ⎦⎜ ⎟⎜ ⎟⎜ ⎟rArr minus sdot nablatimes = minus sdot times⎜ ⎟⎝ ⎠
int int
int int
r rv v
r rv v
MingsianMingsian R R BaiBai40
( )
( )
From Leibnitzs rule
(1-D) ( )
( ) ( )
B t
A t
d f x t dxdt
f x t dx B ttpart prime= +part
int( )
( )( ( ) ) ( )
B t
A tf B t t A tprimeminusint ( ( ) )
(assume ( ) ( ) )
f A t t
A t B t const= =
ContinuityEx The Equation
rate of increase of mass
mass densityV
dM d dVdt dt
ρ
ρ
= int
flow velocityvv
MingsianMingsian R R BaiBai41
ˆ ˆ ( )
(
Conservation
) ( Divergence theorem)
of mass
( ) 0
V S S
V
V
dV n v dS n v dSt
v dV
v dVt
ρ ρ ρ
ρ
ρ ρ⎡ ⎤⎢ ⎥⎣ ⎦
part = minus sdot = minus sdotpart
= minus nablasdot
partrArr +nablasdot =part
int int int
int
int
v v
vQ
v
ˆOn the other hand rate influof mass through n ox i t
Extended to 3-D for time-independent fixed boundary
(3-D) ( ) ( )
Thus
S
V V
V
S V n v dS
Sd f x y z t dV f x y z t dVdt tdM dVdt t
ρ
ρ
= minus sdot
part=part
part=part
int
int int
intv
Fixed control volume (Eulerian coordinates)Otherwise Raynoldrsquos transport theoremMoving control volume Lagrangian coordinates
MingsianMingsian R R BaiBai42
But the control volime is only arbitrary
In particular for flow
( ) 0
con
(continuity equati
stant
on)
Divergence the
0 incompress
Note is t
ible
orem h
V
v
vt
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
e mother of the other integral theorems
MingsianMingsian R R BaiBai43
Ex Archimedes principle ( ) The buoyant force ( ) on a body with volume immersed in a fluid is equal to the weight of the fluid
ˆ displaced by the object ie In the followin
B
V
gVρ= minusf k
阿基米德原理
浮力
ˆg figure the ˆ with and being the fluid density gravitational
acceleration depth and outward normal unit vector to the surface
fhydraulic pressure g z
g z
S
ρ
ρ
= minusp n
n
1
the modified divergence theorem
ˆ
Hint
scalar fuctionV S
vdV v dS v Cnabla = isinint int n
MingsianMingsian R R BaiBai44
Levelsurfacefluid
V
S
airyx
z
ˆf g zρ= minusp n
ˆPf ( ) ( )
ˆ ˆ ˆ ˆ ˆ (0 0 ) v v
B fS S V
V V
g z g zdS dS dV
g dV g dV gV
ρ ρ
ρ ρ ρ
= = minus = minus nabla
= minus + + = minus = minus
int int int
int int
f p n
i j k k k
k
MingsianMingsian R R BaiBai45
Note This can also be thought of as simple balance of forces (resultant force that acts on and weight of filled with fluid)S V
2
~ a little history about George Green Recall the diveregence theorem
ˆ
Let
II th
e Green
are arbitrary fun
s identiti
ctions
es
V S
v dV n v dS
v u vu v C
nablasdot = sdot
= nablaisin
int intv v
v
ˆ ( ) ( )
ˆ ( )V S
V S
u v dV n u v dS
u v u v dV un v dS
nablasdot nabla = sdot nabla
nabla sdotnabla + nablasdotnabla = sdotnabla
int int
int int
MingsianMingsian R R BaiBai46
2 -( ) -(1)V S
vu v u v dV u dSnpart
nabla sdotnabla + nabla =partint int firs(Greens identt ity)
2
If
( ) --(2)
(1) (2) V S
u vuv u v u dV v dSn
harrpart
nabla sdotnabla + nabla =part
minus
int int
( )2 2
V S
dv duu v v u dV u v dSdn dn
⎛ ⎞nabla minus nabla = minus⎜ ⎟⎝ ⎠int int
second(Greens identity)
MingsianMingsian R R BaiBai47
( )
( )
( ) ( )
1st id ( )
Greens identities in the 1D case integration by parts
b b bb ba a
a a ab
ba
ab
ba
a
v udx v u v u dx u v uv dx uv
u v u v vu dx vu
uv vu dx uv vu
primeprime prime prime prime prime prime primeprime prime= minus rArr + =
prime prime primeprime primeharr + =
primeprime primeprime prime primerArr minus = minus
int int int
int
2
2nd id ( )
Good for any adjoint operators in addition to Good also for and integration by parts Important for theory amp reciprocity
From th
Gree
e 1s
2-
t
ns functi
identity
D 1-
tak 1
D
e
on
u
nablararr
=
int
2
V S
vv dV dSnpartnabla =part
rArr int int2Ex 0 st 0 on the boundary 0 ( 1st id u u unabla = = rArr equiv 以 証)
MingsianMingsian R R BaiBai48
1
Thm1 Stokes theorem Let be defined in Let be a piecewise-smooth orientable surface bounded by a piec
III Stokes theorem and
ewise-smooth simple clos
Greens
ed curve Then
theore
m
v C R S
C
isinv
( )
ˆwhere is a unit normal to according t right hand rulo the e
ˆS C
n S
n v dS v dRsdot nablatimes = sdotint intvv v
(not necessarily planar)S
Cn vnablatimes v vv
MingsianMingsian R R BaiBai49
Pf
( )( )
circulatioWe have shown in deriving thatthe in -direction
ˆor
ˆ
n
z
z
j j j j j
j j j j jj j
vz v dx dy
v dR n v S
v dR n v S
nablatimes
=nablatimes
sdot = sdot nablatimes Δ
sdot = sdot nablatimes Δ
intsum sumint
v
v
vv v
vv v
C
S
jS
jC
jP
ˆ jn
jSΔ
MingsianMingsian R R BaiBai50
( )
( )ext surface
After the interior boundaries cancel out only the exterior boundaries (Why)
ˆ
ˆ0 QED
Alternative forms scalar
survive
va
j
j j j j jC
jC S
v dR n v S
S v dR n v dS
φ
sdot = sdot nablatimes Δ
Δ rarr rArr sdot = sdot nablatimes
sum sumint
int int
vv v
vv v
( )
riable constant vector vector variable
1 Let
2 Let
ˆ
ˆ S C
S C
a p
v n dR dR
n p dS dR p
a
v a p
φ φφ= rArr timesnabla =
timesnabla times= rArr timestimes =
int int
int int
v v
v v
v
v v
vv vv v
MingsianMingsian R R BaiBai51
( ) ( )
( )
( )
( ) ( )
1
Thm2 Greens theorem ( planar version of Stokes theorem)ˆ ˆ Let
pf By Stokes theoremˆ
ˆ ˆ ˆ ˆ ˆ ˆ
S C
S C
S C
v P x y i Q x y j C
n v dS v dR
k Pi Qj dS Pi Qj dx i dy
Q P dS Pdx Qdyx y
⎛ ⎞part partminus = +⎜ ⎟
= + isin
sdot nablatimes = sdot
⎡ ⎤sdot nablatimes + =
part part⎝ ⎠
+ sdot +⎣ ⎦
int int
int
int
int
int
v
vv v
( )
( )
( )
ˆ
ˆˆ ˆ
ˆˆ ˆ ˆ ˆ( ) ( )
0
QEDS C
j
i j kQ PPi Qj i j k
x y z x yP Q
Q P dS Pdx Qdyx y
⎛ ⎞part part part part partnablatimes + = = + + minus⎜ ⎟part part part part part⎝ ⎠
⎛ ⎞part partrArr minus = +⎜ ⎟part part⎝ ⎠int int
L L
S
C
i
j
x
y
MingsianMingsian R R BaiBai52
Potential TheorypotentiGeorge Green coined the term (l
a )
rarr 位能 勢能
例子 重力場 電磁場 流場 聲場 彈性力學
1
2
Thm1 If is in a simply connected region then the following statements are equivalent(i) There exists a scala
Conservative Field
potential functior such that
(ii
n
) 0
v
C
C
v
v R
φφ=nabla
nablatimes
isin
isin
=r
r
r
r in (Irrotational field curl-free)
cf 0 st solenoidal field divergence-free
0
(iii) in (Conservative field)C
Rv w v w
v R Rd
nablasdot = exist nabla
sdot =
= times
int
r r r
r
r
r
MingsianMingsian R R BaiBai53
(iv) is connecting and
independent o
Ex Think
f the pa
of dynamics is the force
thB
A
v dR A B
v
sdotintrr
r
1 2
1
0
Pf
(i) (ii) ( ) 0 (zero identity cg rule) irrotational
ˆ(ii) (iii) ( ) 0 (Stokes theorem)
(iii) (iv) ( ) 0
C S
C C C
C
v
v dR n v dS
v dR v dR
v dR
φ
=
rarr nablatimes = nablatimes nabla =
rarr sdot = sdot nablatimes =
rarr sdot = minus sdot =
rArr sdot =
int int
int int int
int
r
rr
rr r
r rr r
rr r
2
independent of pathC
v dRsdotintr
A
B
1C
2C
MingsianMingsian R R BaiBai54
1 2
(iv) (i) constant ( ) ( )
--(1)
But (2)
(1) (2) QED
Note on application
Check
B
C C A
v dR v dR B A d
v dR d
d dx dy dz dRx y z
v dR dR v R
φ φ φ
φφ φ φφ φ
φ φ
rarr sdot = sdot = = minus =
rArr sdot =part part part
= + + = nabla sdot minus minus minus minus minuspart part part
sdot = nabla sdot rArr = nabla forall
int int intr rr r
rr
r
r r rr r
use simpler path (iv)if 0
find (i)v
φ⎧
nablatimes = ⎨⎩
rr
A
B
x
y
Cz
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai6
Gradient 梯度
g
R
r
ec
ad
all
ient
the divergence operator div
ˆˆ ˆIn Cartesian coordinates
Define the operator
a spatial differential operator
ˆˆ ˆgr
independent
ad
of coo
v v
i j kx
u u uu u i j kx y
y
z
z
part part part
=nablasdotpart part part
nabla + +part part part
rarr
nabla = + +part part part
r r
rdinates chosen ie is invariant wrt coordinates
directional derivative ( )ˆˆ ˆLet ( )
Physical interpretation
(position vector)
ˆ( ) (
u
u u x y z r xi yj zk
u u u u udu r dx dy dz ix y z x
nabla
= = + +
part part part part part= + + = +part part part part part
r
r
方向導數
ˆˆ ˆ
ˆ ˆˆ ˆ
( ) (
ˆ
)
) ( )
( )
dr r xi yj zk
uj k idx jdy kdzy z
du x u ddu r xu dr
= + +
=
part+ sdot + +part
primerarrsdot =nabla
r r
r r可視為 的推廣
MingsianMingsian R R BaiBai7
ˆ rate of change of in the direction Di
ˆ
rectional De
( )
ˆw
rivative
here
u sdu drr u u sdr dr
drsdr
=nabla sdot = nabla sdotr
r
r
s0
ˆ( ) ( )limh
du u r hs u rdr hrarr
+ minus=r r
drv
r dr+r r
rr
oNote
ˆ
ˆ )
ˆ
ˆ(
dudr
u s
s u
d sdr
s u
=nabla sdot
= sdotnabla
rArr
nabla
nabla
sdot=
sdotθ
s
s
( )0 yxx
y
u
unabla
drdu
duslopedr
=
( )yxfu =
MingsianMingsian R R BaiBai8
ˆEx Normal gradient outward normal
Note is often written as in the PDE literature
ˆˆ ˆˆPartial derivatives If then
ˆ
Physical interpretati
n
du udn n
du u u us i j kdr
du n ud
z
n
x y
partpart
part part part= rarrpart part part
sdotnabla
max
maximum rate of increase in a direction
ˆ ˆ ˆ ˆcos ( ) cos ( )
ˆ is independent of
ˆmax occurs when ( ) 0 and
points in the direction of ma
o
n
x a
du u s u s u s u u sdr
u sdu duu s udr dr
duudr
θ θ
θ ⎛ ⎞⎜ ⎟⎝ ⎠
=nabla sdot = nabla nabla = nabla nabla
nabla
there4 nabla = = nabla
rArrnabla
Q
( )max
(interpretation 1nd
gradient search steepest desce
)
nt
duudr
⎛ ⎞⎜ ⎟⎝ ⎠
nabla =
rArr minusnabla最佳化方法 的基礎
MingsianMingsian R R BaiBai9
(interpretatDirection of ion 2 normal to the level surface ( )
On the 0 ( )
ie
level surfac
o
e
n
)udu u dr
u cdu u dr
u dr u c
nabla= nabla sdot
== = nabla sdotnabla perp =
r
r
r
unabla
dr
unabla
Q
P
1cu=
2u c=
山
x
y
( )u x y
x
y
Hill-climbing search in optimization
MingsianMingsian R R BaiBai10
Ex Plane equationˆˆ ˆ( ) Normal vector
Ex 0 is the necessary condition for searching extrema (from multivariate Tylor expansion)
Ex Revisit of the exampl
u x y z ax by cz d u ai bj ck
u
= + + = rArr nabla = + +
nabla =v
多變數函數求極值
2 21 1 2 2
e in Linear Algebra using method of What is the maximum and minimum distance of the origin to the following curve
Lagrange multipli s
6 8
er
5 5x x x x+ + =
MingsianMingsian R R BaiBai11
1 2
2 21 2( )
2 21 1 2 2
Sol
This problem can be posed as a problemmin max ( )
constrained optimization
5 6 5 8 0 ( )which can
objectivecost function
constraint
equatioals
nx x
J x x
st g x x x x
= +
= + + minus =
2 2 2 21 2 1 1 2 2
o be converted into an unconstrained optimization problem by defining the ( ) (5 6 5 8)where is referred to as the thatLa
L
g ora
agr
nge ften h multi
angi
plie as
an
r ph
L x x x x x xλλ
= + + + + minus
ysical meaning in certain problemsThus the extreme values of will occur wh en or L J gJ nabla = nabla nabla0
MingsianMingsian R R BaiBai12
1 1 21
2 1 22
2 21 1 2 2
1 2 1 2
2 (10 6 ) 0 (1)
2 (6 10 ) 0 (2)
5 6 5 8 0 (constraint)
(1) (2) (2 16 )( ) 0 18 or (3)Substituting Eq (3) into Eqs (1) and (2) l
L x x xxL x x xxL x x x x
x x x x
λ
λ
λλ λ
part = + + =partpart = + + =partpart = + + minus =part+ + + = rArr = minus = minus
1 2 1 2eads to
and which corresponds to 14Thus minimummaximum distance 12x x x x J= = minus =
=
MingsianMingsian R R BaiBai13
22 2
1
1 1
2 2
The result above is identical to the previously obtained result in Linear Algebr
ellipse
45 rotatio
a
1 ( )4
1 11 1
n12
xx
x xx x
deg⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠
prime+ =prime
primeminus= rArr
prime2x prime
2x
1x prime
1x
2
MingsianMingsian R R BaiBai14
Curl 旋度
curl curl In Cartesian coordinates
spatial operator indep
ˆˆ ˆ
ˆˆ ˆ( )
ende
(
nt of coordi
) (
D
nates
ef
)
x y z
y x x xz z
i j k
vx y z
v v v
v v v vv vi j ky z x z x y
v v
part part partnablatimes =part part part
part part part partpart part= minus minus minus + minuspart
times
part part part part part
nabla
rarr
r
r r
0vnablatimes =rr 0vnablatimes ne
rr
MingsianMingsian R R BaiBai15
circulaPhysica
tion o spin ol interpretati
f fluidr or vortion
y
citΓ
( )0 0x dx y dy+ +
( )0 0x y ( )0 0x dx y+
Γ
1
2
3
4
( )0 0x y dy+
MingsianMingsian R R BaiBai16
yy
vv dy
ypart
+part
Γdyminus
dyyvv x
x partpart
+
dy
dx
dxminus
y yy
v vv dx dy
x ypart part
+ +part part
x xx
v vv dx dyx y
part part+ +part part
yy
vv dx
xpart
+part
xx
vv dxx
part+part
xv
yv
vv
dλ paddle weel
MingsianMingsian R R BaiBai17
contour
1 2 3 4
circulation
x x y y x x y y
x
v d
v d v d v d v d
v
λ
λ λ λ λrarr
Γ sdot
= + + +
asymp
intint int int int
vv
12
xv dxx
part+part ydx v
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
+ 12
y yv vdx dy
x ypart part
+ +part part
x
dy
v
⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦
+
12
xv dxx
part+part
( )xy
v dy dx vy
⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦
part+ minus +part
12
yvdy
ypart
+part
( )
( )
Similar process can be applied to the circulation in the and directionsThus the total circulation per unit area i curl s
y xz
dy
v v dxdy v dxdyx y
xv v
y
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎝ ⎠
minus
part part= minus = nablatimespart part
=nablatimes
v
v v
MingsianMingsian R R BaiBai18
Identities of differential operators ( p380 Greenberg )
( )
( ) ( )( )( )
( ) ( )( ) is required
Example of deriv
ˆ ˆˆ ˆ ˆ ˆ
ation
x y z
x y z
v v vx y z
uv u v u v
uv i j k uv i uv j uv kx y z
uv u v u v
u v v u u v
u v u vv v u u
+ +part part partpart part part
⎛ ⎞⎜ ⎟⎝ ⎠
nablasdot = nabla sdot + nablasdot
part part partnablasdot = + + sdot + + =part part part
nablatimes =nabla times + nablatimes
nablasdot times = sdotnablatimes minus sdotnablatimes
nablatimes times = nabla minus + minusnablanablasdot sdotsdot678 678
v
v v v
vL
v v v
v v v v v v
v v v v v v v v( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )2 cf
x y zu u ux y z
A B C A
u v
v w v w w v v w w v
v v v C B A B C
+ +part part partpart part part
sdotnabla
nabla sdot = sdotnabla + sdotnabla + times nablatimes + times
times
nablatimes
nablatimes nablatimes =nabla nabla times = sdot minussdot sdotminusnabla
678v
v v v
v v v v v vv v
v v v v v v
v v
v
v v
ˆ ˆˆ ˆ ˆ ˆ( ) ( ) ( )x y z x y zv i v j v k i j k u v v v ux y z x y zpart part part part part part+ + sdot + + = + +part part part part part part
v v
MingsianMingsian R R BaiBai19
2 2 22
2 2 2
0 (dc direct current) solenoidal0 (cg center of gravity) conservative potent
Laplacia
Zero identitie
n
s
ial
x y z
vu
⎧⎪⎨⎪⎩
part part partΔ =nabla nablasdotnabla = + +part part part
nablasdotnablatimes = rarr
nablatimesnabla = rarr
v
v
These differential operators will take more complicated and irregular form for the other coordinate systems
MingsianMingsian R R BaiBai20
2 2 3
2 2
ˆˆ ˆEx 2
ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ 2 0 2
2 0 0 2yx z
u x y v x yi z j k
u i j k u xyi x j k xyi x jx y z
vv vv xy xyx y z
⎛ ⎞⎜ ⎟⎝ ⎠
= = minus +
part part partnabla = + + = + + = +part part part
partpart partnablasdot = + + = + minus =part part part
v
v
MingsianMingsian R R BaiBai21
( ) ( ) ( )
( ) ( )
2 2 2 2
2 2 22
2 2 2
2
ˆˆ ˆ
ˆˆ ˆ
ˆ ˆˆ ˆ ˆ 0 3 0 0 0 3
2 0 0 2
ˆˆ ˆ
y yx xz z
x y z
i j kv vv vv vv i j k
x y z y z x z x yv v v
z i j x k z i x k
u u uu y yx y zv v v
i j kx y z
⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠ ⎝ ⎠
⎛
⎝
part partpart partpart partpart part partnablatimes = = minus minus minus + minuspart part part part part part part part part
= + minus minus + minus = minus
part part partnabla = + + = + + =part part part
nablatimes nablatimes =nabla nablasdot minusnabla
part part part= + +part part part
v
v v v
( ) ( )( ) ( ) ( ) ( )
( )
2 2 22 3
2 2 2ˆˆ ˆ2 2
ˆ ˆˆ ˆ ˆ ˆ 2 2 0 2 0 0 0 0 6 0 0 0
ˆ 2 6
xy x yi z j kx y z
yi xj k y i z j k
x z j
⎛ ⎞⎞⎜ ⎟⎜ ⎟
⎠ ⎝ ⎠⎡ ⎤⎣ ⎦
part part partminus + + minus +part part part
= + + minus + + + + minus + + minus
= +
MingsianMingsian R R BaiBai22
( )v w r w r r wnablatimes =nablatimes times = nablasdot minus nablasdotv v v v v v v ( )r w+ sdotnablav v ( ) constant w
w rminus sdotnablav
v v
Q
Ex Rigid body rotation
MingsianMingsian R R BaiBai23
( )( ) ( )
( ) ( )
ˆˆ ˆ
ˆ ˆˆ ˆ ˆ ˆ 1 1 1 3
ˆˆ ˆ
ˆˆ ˆ
Thus 3 2 is linked directly
x y z
x y z
r xi yj zk
r i j k xi yj zkx y z
w r w w w xi yj zkx y z
w i w j w k w
w r w r w r w w wv
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎝ ⎠
= + +
part part partnablasdot = + + + + = + + =part part part
part part partsdotnabla = + + + +part part part
= + + =
nablatimes times == nablasdot minus sdotnabla = minus =rArrnablatimes
v
v
v v
v
v v v v v v v v v
v spinn to ing
MingsianMingsian R R BaiBai24
Line surface and volume integrals
x
yC
B
A
isΔ
( )ii ηξ
Piecewise smooth curve
Line integral
MingsianMingsian R R BaiBai25
0 1lim ( ) ( ) or ( )
Intepretation of a line integral as an area ( )
i
Bn
i i isA
C
i C
F s F x y ds F x y d
x
s
F y ds
ξ ηΔ rarr =
Δ =sum int
int
int
x
y
z
C
( )z F x y=
MingsianMingsian R R BaiBai26
the area is always on the left of an obs
Positive dierver trave
rection of irsing on the
ntegrationcurve
RC
CCW CW
If is a regular closed curve piecewise smo
( ) ( )
(contou
oth simple ( )
r integral)
x y zC C
x y zC
F dR F F F dx dy dz
F dx F dy
C
F dz
sdot = + + + +
= + +
lowast
int int
int
i j k i j kv v
無交叉
work dW F dR= sdotv v
MingsianMingsian R R BaiBai27
0 1
Surface integral
lim ( ) ( )i
n
i i i iS i S
F S F x y z dSξ η ζΔ rarr =
Δ =sum intint( ) i i iξ η ζ
iSΔ
0 1
mass density
lim ( ) ( )
Ex
Volume integral
( )
i
n
i i i iV i V
V
F V F x y z dV
M x y z dV
ξ η ζ
ρ
Δ rarr =Δ =
=
sum intintint
intintint( ) i i iξ η ζ
iVΔ
MingsianMingsian R R BaiBai28
Ref (1)Hildebrand advanced Calculus for application Prentice-Hall 1976 p306-313(2) Greenberg p383-389 (3) Arfken ch2
1 2 3
1 2 3
1 2 3
1 2 3
(
Orthogonal coordinate systems base vectors are not constant in s
) ( )( )( )( )
ˆˆ ˆ
pacep x y z p u u ux x u u uy y u u uz z u u u
r xi yj zk
primerarr
= +
rarr
===
+v
1uv
2uv3uv
1u
2u
3uP coord curve
xy
z
O
X Curvilinear Coordinates
MingsianMingsian R R BaiBai29
1
11
1 1 1
1
1
1 1 1
11 1
1 1
2 2 2 3 3 3
Tangent vector to the at
where is the arc length
1
where unit vector
Similarly
curv
e
u psr rU
u s us
rs
U h uds rh udu sU h u U h u
partpart part= =part part part
part =part
there4 =part= =part
= =
v vv
vQ
v v
vv
v vv v
1u cood curve
1uv
1du 1dS
drv
P
MingsianMingsian R R BaiBai30
3 31 2 1 21 2 3
1 2 32
curv
In summary scale factor
If is the arc length along a in any directione
k k k
kk k
k k
k k k
U h u
ds rh Udu u
ds h dus
du dudu du du dudr r r r U U Uds u ds u ds u ds ds ds ds
dr drds ds
=
part= = =part
=
part part part= + + = + +part part part
= sdot
v v
v v
v v v v v v v
v v 3 3
1 1
23 3 32 2
21 1 1
2 2 2 2 2 2 2 2 2 21 1 2 2 3 3 1 2 3
1 (for orthogonal coord)
jii j
i j
ji ii j i
i j i
dududr U Uds ds ds
dudu duU U hds ds ds
ds h du h du h du ds ds ds
= =
= = =
⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
= sdot
rarr = sdot =
rarr = + + = + +
sum sum
sumsum sum
v v v
v v
MingsianMingsian R R BaiBai31
1 1 2 2 3 3
1 2 3
are mutually vectors having with arc length
k k k k kds h du U du
U du U du U duds ds ds
= =
rArr perp
v
v v v
1uv
2uv
3uv
1dS
2dS
3dS
dS
MingsianMingsian R R BaiBai32
( ) ( )
( ) ( )
1 1 2 2 3 3 1 2 3 1 2 3 1 2 3
1 2 3 1 2 3
1
1 2 2 3 3 2 2 2 3 3 3 2 3 2 3 1
Differential volume
Differential On const coord surface
area
Gradi t
en
d U du U du U du u u u h h h du du du
d h h h du du du
ud U du U du h u du h u du h h du du u
df
τ
τ
σ
= times sdot = times sdot
=
=
= times = times =
= nabla
v v v v v v
v vv v v v
1 2 31 2 3
1 2 3 1 1 2 2 3 31 2 3
1 1 1 2 2 2 3 3 3
f f ff dr df du du duu u u
r r rdr du du du U du U du U duu u u
h u du h u du h u du
part part partsdot rarr = + +part part part
part part part= + + = + +part part part
= + +
v
v v v v v vv
v v v
=1 (right hand rule)
MingsianMingsian R R BaiBai33
Rectangular Cartesian Coordinates
ij
k
MingsianMingsian R R BaiBai34
Circular Cylindrical Coordinatescossin
xyz z
ρ ϕρ ϕ
===
Ref Arfken
( )P zρ ϕ
MingsianMingsian R R BaiBai35
Polar Spherical Coordinatessin cossin sincos
x ry rz r
θ ϕθ ϕθ
===
Ref Arfken
0
0
0
sin cos sin sin coscos cos cos sin sin
sin cos
θ ϕ θ ϕ θθ ϕ θ ϕ θϕ ϕ
= + +
= + minus= minus +
r i j kθ i j kφ i j
2
2
sinsin
dS r d ddV r dr d d
θ θ ϕ
θ θ ϕ
=
=( )P r θ ϕ
MingsianMingsian R R BaiBai36
Integral TheoremsHeart of this chapter
Interchange of dimensionality Div Curl Grad in integrals High-dimensional integration
rarr
1
Thm1 Divergence theorem Let be a closed region bounded by piecewise smooth orientable surface and let (con
I Divergence theor
tinuous derivatives up to the first order
em (Gauss theorem)
)
VS v Cisin
r
ˆ If denotes the outward unit normal on
the ˆn V S
v dV n v dS
n S
nablasdot = sdotint intr r
MingsianMingsian R R BaiBai37
jS
jP
jV
S
0
ˆBy definition ( ) lim
ˆ( )
ˆ( )
j
j
j
j
Sj V
j
j j S
j j Sj j
n v dSv P
V
v P V n v dS
v P V n v dS
Δ rarr
⎧ ⎫sdot⎪ ⎪nabla sdot ⎨ ⎬Δ⎪ ⎪⎩ ⎭
nabla sdot Δ asymp sdot
nabla sdot Δ = sdot
int
intsum sumint
rr
r r
r r
Pf
MingsianMingsian R R BaiBai38
0
After interior surfaces cancel out only exterior surface (Why Opposite normal vectors)
ˆ( )
volumes exterior surfaces
ˆlim
QE
survive
j
j
j j Sj j
V SV
v p V n v dS
v dV n v dSΔ rarr
nabla sdot Δ = sdot
rArr nablasdot = sdot
sum sumint
int int
r r
r r
D
MingsianMingsian R R BaiBai39
( )
scalar variable constant vectorvector variable
Alternative formsLet be a an arbitrary and
a
ˆ
1 Le
(
ˆt
V
V S
S
V
v ap
v d
v dV vn d
V n v dS
v va
va dV v a v a
S
nablasdot = sdot
= rArr
nablasdot = nabla sdot + nabla
=
sdot
nablaint intint int
int
r
v
r r
r r
r r rQ ( )
( )
ˆ)
2 Let ˆV S
V
V S
dV n va dS
v a p
a p
p dV n p d
d p
S
V a
⎛ ⎞= sdot⎜ ⎟
⎝ ⎠
= times rArr
nablasdot times = sdot nabla
times
times
nabla = times
int int
int int
int
r
r r r
r r v
r
vQ
r
( ) ( ) ( )
( ) ( )
ˆ
ˆV S
V S
a p dV n a p dS
a p dV a n p dS
⎛ ⎞⎡ ⎤minus sdot nablatimes = sdot times⎣ ⎦⎜ ⎟⎜ ⎟⎜ ⎟rArr minus sdot nablatimes = minus sdot times⎜ ⎟⎝ ⎠
int int
int int
r rv v
r rv v
MingsianMingsian R R BaiBai40
( )
( )
From Leibnitzs rule
(1-D) ( )
( ) ( )
B t
A t
d f x t dxdt
f x t dx B ttpart prime= +part
int( )
( )( ( ) ) ( )
B t
A tf B t t A tprimeminusint ( ( ) )
(assume ( ) ( ) )
f A t t
A t B t const= =
ContinuityEx The Equation
rate of increase of mass
mass densityV
dM d dVdt dt
ρ
ρ
= int
flow velocityvv
MingsianMingsian R R BaiBai41
ˆ ˆ ( )
(
Conservation
) ( Divergence theorem)
of mass
( ) 0
V S S
V
V
dV n v dS n v dSt
v dV
v dVt
ρ ρ ρ
ρ
ρ ρ⎡ ⎤⎢ ⎥⎣ ⎦
part = minus sdot = minus sdotpart
= minus nablasdot
partrArr +nablasdot =part
int int int
int
int
v v
vQ
v
ˆOn the other hand rate influof mass through n ox i t
Extended to 3-D for time-independent fixed boundary
(3-D) ( ) ( )
Thus
S
V V
V
S V n v dS
Sd f x y z t dV f x y z t dVdt tdM dVdt t
ρ
ρ
= minus sdot
part=part
part=part
int
int int
intv
Fixed control volume (Eulerian coordinates)Otherwise Raynoldrsquos transport theoremMoving control volume Lagrangian coordinates
MingsianMingsian R R BaiBai42
But the control volime is only arbitrary
In particular for flow
( ) 0
con
(continuity equati
stant
on)
Divergence the
0 incompress
Note is t
ible
orem h
V
v
vt
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
e mother of the other integral theorems
MingsianMingsian R R BaiBai43
Ex Archimedes principle ( ) The buoyant force ( ) on a body with volume immersed in a fluid is equal to the weight of the fluid
ˆ displaced by the object ie In the followin
B
V
gVρ= minusf k
阿基米德原理
浮力
ˆg figure the ˆ with and being the fluid density gravitational
acceleration depth and outward normal unit vector to the surface
fhydraulic pressure g z
g z
S
ρ
ρ
= minusp n
n
1
the modified divergence theorem
ˆ
Hint
scalar fuctionV S
vdV v dS v Cnabla = isinint int n
MingsianMingsian R R BaiBai44
Levelsurfacefluid
V
S
airyx
z
ˆf g zρ= minusp n
ˆPf ( ) ( )
ˆ ˆ ˆ ˆ ˆ (0 0 ) v v
B fS S V
V V
g z g zdS dS dV
g dV g dV gV
ρ ρ
ρ ρ ρ
= = minus = minus nabla
= minus + + = minus = minus
int int int
int int
f p n
i j k k k
k
MingsianMingsian R R BaiBai45
Note This can also be thought of as simple balance of forces (resultant force that acts on and weight of filled with fluid)S V
2
~ a little history about George Green Recall the diveregence theorem
ˆ
Let
II th
e Green
are arbitrary fun
s identiti
ctions
es
V S
v dV n v dS
v u vu v C
nablasdot = sdot
= nablaisin
int intv v
v
ˆ ( ) ( )
ˆ ( )V S
V S
u v dV n u v dS
u v u v dV un v dS
nablasdot nabla = sdot nabla
nabla sdotnabla + nablasdotnabla = sdotnabla
int int
int int
MingsianMingsian R R BaiBai46
2 -( ) -(1)V S
vu v u v dV u dSnpart
nabla sdotnabla + nabla =partint int firs(Greens identt ity)
2
If
( ) --(2)
(1) (2) V S
u vuv u v u dV v dSn
harrpart
nabla sdotnabla + nabla =part
minus
int int
( )2 2
V S
dv duu v v u dV u v dSdn dn
⎛ ⎞nabla minus nabla = minus⎜ ⎟⎝ ⎠int int
second(Greens identity)
MingsianMingsian R R BaiBai47
( )
( )
( ) ( )
1st id ( )
Greens identities in the 1D case integration by parts
b b bb ba a
a a ab
ba
ab
ba
a
v udx v u v u dx u v uv dx uv
u v u v vu dx vu
uv vu dx uv vu
primeprime prime prime prime prime prime primeprime prime= minus rArr + =
prime prime primeprime primeharr + =
primeprime primeprime prime primerArr minus = minus
int int int
int
2
2nd id ( )
Good for any adjoint operators in addition to Good also for and integration by parts Important for theory amp reciprocity
From th
Gree
e 1s
2-
t
ns functi
identity
D 1-
tak 1
D
e
on
u
nablararr
=
int
2
V S
vv dV dSnpartnabla =part
rArr int int2Ex 0 st 0 on the boundary 0 ( 1st id u u unabla = = rArr equiv 以 証)
MingsianMingsian R R BaiBai48
1
Thm1 Stokes theorem Let be defined in Let be a piecewise-smooth orientable surface bounded by a piec
III Stokes theorem and
ewise-smooth simple clos
Greens
ed curve Then
theore
m
v C R S
C
isinv
( )
ˆwhere is a unit normal to according t right hand rulo the e
ˆS C
n S
n v dS v dRsdot nablatimes = sdotint intvv v
(not necessarily planar)S
Cn vnablatimes v vv
MingsianMingsian R R BaiBai49
Pf
( )( )
circulatioWe have shown in deriving thatthe in -direction
ˆor
ˆ
n
z
z
j j j j j
j j j j jj j
vz v dx dy
v dR n v S
v dR n v S
nablatimes
=nablatimes
sdot = sdot nablatimes Δ
sdot = sdot nablatimes Δ
intsum sumint
v
v
vv v
vv v
C
S
jS
jC
jP
ˆ jn
jSΔ
MingsianMingsian R R BaiBai50
( )
( )ext surface
After the interior boundaries cancel out only the exterior boundaries (Why)
ˆ
ˆ0 QED
Alternative forms scalar
survive
va
j
j j j j jC
jC S
v dR n v S
S v dR n v dS
φ
sdot = sdot nablatimes Δ
Δ rarr rArr sdot = sdot nablatimes
sum sumint
int int
vv v
vv v
( )
riable constant vector vector variable
1 Let
2 Let
ˆ
ˆ S C
S C
a p
v n dR dR
n p dS dR p
a
v a p
φ φφ= rArr timesnabla =
timesnabla times= rArr timestimes =
int int
int int
v v
v v
v
v v
vv vv v
MingsianMingsian R R BaiBai51
( ) ( )
( )
( )
( ) ( )
1
Thm2 Greens theorem ( planar version of Stokes theorem)ˆ ˆ Let
pf By Stokes theoremˆ
ˆ ˆ ˆ ˆ ˆ ˆ
S C
S C
S C
v P x y i Q x y j C
n v dS v dR
k Pi Qj dS Pi Qj dx i dy
Q P dS Pdx Qdyx y
⎛ ⎞part partminus = +⎜ ⎟
= + isin
sdot nablatimes = sdot
⎡ ⎤sdot nablatimes + =
part part⎝ ⎠
+ sdot +⎣ ⎦
int int
int
int
int
int
v
vv v
( )
( )
( )
ˆ
ˆˆ ˆ
ˆˆ ˆ ˆ ˆ( ) ( )
0
QEDS C
j
i j kQ PPi Qj i j k
x y z x yP Q
Q P dS Pdx Qdyx y
⎛ ⎞part part part part partnablatimes + = = + + minus⎜ ⎟part part part part part⎝ ⎠
⎛ ⎞part partrArr minus = +⎜ ⎟part part⎝ ⎠int int
L L
S
C
i
j
x
y
MingsianMingsian R R BaiBai52
Potential TheorypotentiGeorge Green coined the term (l
a )
rarr 位能 勢能
例子 重力場 電磁場 流場 聲場 彈性力學
1
2
Thm1 If is in a simply connected region then the following statements are equivalent(i) There exists a scala
Conservative Field
potential functior such that
(ii
n
) 0
v
C
C
v
v R
φφ=nabla
nablatimes
isin
isin
=r
r
r
r in (Irrotational field curl-free)
cf 0 st solenoidal field divergence-free
0
(iii) in (Conservative field)C
Rv w v w
v R Rd
nablasdot = exist nabla
sdot =
= times
int
r r r
r
r
r
MingsianMingsian R R BaiBai53
(iv) is connecting and
independent o
Ex Think
f the pa
of dynamics is the force
thB
A
v dR A B
v
sdotintrr
r
1 2
1
0
Pf
(i) (ii) ( ) 0 (zero identity cg rule) irrotational
ˆ(ii) (iii) ( ) 0 (Stokes theorem)
(iii) (iv) ( ) 0
C S
C C C
C
v
v dR n v dS
v dR v dR
v dR
φ
=
rarr nablatimes = nablatimes nabla =
rarr sdot = sdot nablatimes =
rarr sdot = minus sdot =
rArr sdot =
int int
int int int
int
r
rr
rr r
r rr r
rr r
2
independent of pathC
v dRsdotintr
A
B
1C
2C
MingsianMingsian R R BaiBai54
1 2
(iv) (i) constant ( ) ( )
--(1)
But (2)
(1) (2) QED
Note on application
Check
B
C C A
v dR v dR B A d
v dR d
d dx dy dz dRx y z
v dR dR v R
φ φ φ
φφ φ φφ φ
φ φ
rarr sdot = sdot = = minus =
rArr sdot =part part part
= + + = nabla sdot minus minus minus minus minuspart part part
sdot = nabla sdot rArr = nabla forall
int int intr rr r
rr
r
r r rr r
use simpler path (iv)if 0
find (i)v
φ⎧
nablatimes = ⎨⎩
rr
A
B
x
y
Cz
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai7
ˆ rate of change of in the direction Di
ˆ
rectional De
( )
ˆw
rivative
here
u sdu drr u u sdr dr
drsdr
=nabla sdot = nabla sdotr
r
r
s0
ˆ( ) ( )limh
du u r hs u rdr hrarr
+ minus=r r
drv
r dr+r r
rr
oNote
ˆ
ˆ )
ˆ
ˆ(
dudr
u s
s u
d sdr
s u
=nabla sdot
= sdotnabla
rArr
nabla
nabla
sdot=
sdotθ
s
s
( )0 yxx
y
u
unabla
drdu
duslopedr
=
( )yxfu =
MingsianMingsian R R BaiBai8
ˆEx Normal gradient outward normal
Note is often written as in the PDE literature
ˆˆ ˆˆPartial derivatives If then
ˆ
Physical interpretati
n
du udn n
du u u us i j kdr
du n ud
z
n
x y
partpart
part part part= rarrpart part part
sdotnabla
max
maximum rate of increase in a direction
ˆ ˆ ˆ ˆcos ( ) cos ( )
ˆ is independent of
ˆmax occurs when ( ) 0 and
points in the direction of ma
o
n
x a
du u s u s u s u u sdr
u sdu duu s udr dr
duudr
θ θ
θ ⎛ ⎞⎜ ⎟⎝ ⎠
=nabla sdot = nabla nabla = nabla nabla
nabla
there4 nabla = = nabla
rArrnabla
Q
( )max
(interpretation 1nd
gradient search steepest desce
)
nt
duudr
⎛ ⎞⎜ ⎟⎝ ⎠
nabla =
rArr minusnabla最佳化方法 的基礎
MingsianMingsian R R BaiBai9
(interpretatDirection of ion 2 normal to the level surface ( )
On the 0 ( )
ie
level surfac
o
e
n
)udu u dr
u cdu u dr
u dr u c
nabla= nabla sdot
== = nabla sdotnabla perp =
r
r
r
unabla
dr
unabla
Q
P
1cu=
2u c=
山
x
y
( )u x y
x
y
Hill-climbing search in optimization
MingsianMingsian R R BaiBai10
Ex Plane equationˆˆ ˆ( ) Normal vector
Ex 0 is the necessary condition for searching extrema (from multivariate Tylor expansion)
Ex Revisit of the exampl
u x y z ax by cz d u ai bj ck
u
= + + = rArr nabla = + +
nabla =v
多變數函數求極值
2 21 1 2 2
e in Linear Algebra using method of What is the maximum and minimum distance of the origin to the following curve
Lagrange multipli s
6 8
er
5 5x x x x+ + =
MingsianMingsian R R BaiBai11
1 2
2 21 2( )
2 21 1 2 2
Sol
This problem can be posed as a problemmin max ( )
constrained optimization
5 6 5 8 0 ( )which can
objectivecost function
constraint
equatioals
nx x
J x x
st g x x x x
= +
= + + minus =
2 2 2 21 2 1 1 2 2
o be converted into an unconstrained optimization problem by defining the ( ) (5 6 5 8)where is referred to as the thatLa
L
g ora
agr
nge ften h multi
angi
plie as
an
r ph
L x x x x x xλλ
= + + + + minus
ysical meaning in certain problemsThus the extreme values of will occur wh en or L J gJ nabla = nabla nabla0
MingsianMingsian R R BaiBai12
1 1 21
2 1 22
2 21 1 2 2
1 2 1 2
2 (10 6 ) 0 (1)
2 (6 10 ) 0 (2)
5 6 5 8 0 (constraint)
(1) (2) (2 16 )( ) 0 18 or (3)Substituting Eq (3) into Eqs (1) and (2) l
L x x xxL x x xxL x x x x
x x x x
λ
λ
λλ λ
part = + + =partpart = + + =partpart = + + minus =part+ + + = rArr = minus = minus
1 2 1 2eads to
and which corresponds to 14Thus minimummaximum distance 12x x x x J= = minus =
=
MingsianMingsian R R BaiBai13
22 2
1
1 1
2 2
The result above is identical to the previously obtained result in Linear Algebr
ellipse
45 rotatio
a
1 ( )4
1 11 1
n12
xx
x xx x
deg⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠
prime+ =prime
primeminus= rArr
prime2x prime
2x
1x prime
1x
2
MingsianMingsian R R BaiBai14
Curl 旋度
curl curl In Cartesian coordinates
spatial operator indep
ˆˆ ˆ
ˆˆ ˆ( )
ende
(
nt of coordi
) (
D
nates
ef
)
x y z
y x x xz z
i j k
vx y z
v v v
v v v vv vi j ky z x z x y
v v
part part partnablatimes =part part part
part part part partpart part= minus minus minus + minuspart
times
part part part part part
nabla
rarr
r
r r
0vnablatimes =rr 0vnablatimes ne
rr
MingsianMingsian R R BaiBai15
circulaPhysica
tion o spin ol interpretati
f fluidr or vortion
y
citΓ
( )0 0x dx y dy+ +
( )0 0x y ( )0 0x dx y+
Γ
1
2
3
4
( )0 0x y dy+
MingsianMingsian R R BaiBai16
yy
vv dy
ypart
+part
Γdyminus
dyyvv x
x partpart
+
dy
dx
dxminus
y yy
v vv dx dy
x ypart part
+ +part part
x xx
v vv dx dyx y
part part+ +part part
yy
vv dx
xpart
+part
xx
vv dxx
part+part
xv
yv
vv
dλ paddle weel
MingsianMingsian R R BaiBai17
contour
1 2 3 4
circulation
x x y y x x y y
x
v d
v d v d v d v d
v
λ
λ λ λ λrarr
Γ sdot
= + + +
asymp
intint int int int
vv
12
xv dxx
part+part ydx v
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
+ 12
y yv vdx dy
x ypart part
+ +part part
x
dy
v
⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦
+
12
xv dxx
part+part
( )xy
v dy dx vy
⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦
part+ minus +part
12
yvdy
ypart
+part
( )
( )
Similar process can be applied to the circulation in the and directionsThus the total circulation per unit area i curl s
y xz
dy
v v dxdy v dxdyx y
xv v
y
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎝ ⎠
minus
part part= minus = nablatimespart part
=nablatimes
v
v v
MingsianMingsian R R BaiBai18
Identities of differential operators ( p380 Greenberg )
( )
( ) ( )( )( )
( ) ( )( ) is required
Example of deriv
ˆ ˆˆ ˆ ˆ ˆ
ation
x y z
x y z
v v vx y z
uv u v u v
uv i j k uv i uv j uv kx y z
uv u v u v
u v v u u v
u v u vv v u u
+ +part part partpart part part
⎛ ⎞⎜ ⎟⎝ ⎠
nablasdot = nabla sdot + nablasdot
part part partnablasdot = + + sdot + + =part part part
nablatimes =nabla times + nablatimes
nablasdot times = sdotnablatimes minus sdotnablatimes
nablatimes times = nabla minus + minusnablanablasdot sdotsdot678 678
v
v v v
vL
v v v
v v v v v v
v v v v v v v v( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )2 cf
x y zu u ux y z
A B C A
u v
v w v w w v v w w v
v v v C B A B C
+ +part part partpart part part
sdotnabla
nabla sdot = sdotnabla + sdotnabla + times nablatimes + times
times
nablatimes
nablatimes nablatimes =nabla nabla times = sdot minussdot sdotminusnabla
678v
v v v
v v v v v vv v
v v v v v v
v v
v
v v
ˆ ˆˆ ˆ ˆ ˆ( ) ( ) ( )x y z x y zv i v j v k i j k u v v v ux y z x y zpart part part part part part+ + sdot + + = + +part part part part part part
v v
MingsianMingsian R R BaiBai19
2 2 22
2 2 2
0 (dc direct current) solenoidal0 (cg center of gravity) conservative potent
Laplacia
Zero identitie
n
s
ial
x y z
vu
⎧⎪⎨⎪⎩
part part partΔ =nabla nablasdotnabla = + +part part part
nablasdotnablatimes = rarr
nablatimesnabla = rarr
v
v
These differential operators will take more complicated and irregular form for the other coordinate systems
MingsianMingsian R R BaiBai20
2 2 3
2 2
ˆˆ ˆEx 2
ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ 2 0 2
2 0 0 2yx z
u x y v x yi z j k
u i j k u xyi x j k xyi x jx y z
vv vv xy xyx y z
⎛ ⎞⎜ ⎟⎝ ⎠
= = minus +
part part partnabla = + + = + + = +part part part
partpart partnablasdot = + + = + minus =part part part
v
v
MingsianMingsian R R BaiBai21
( ) ( ) ( )
( ) ( )
2 2 2 2
2 2 22
2 2 2
2
ˆˆ ˆ
ˆˆ ˆ
ˆ ˆˆ ˆ ˆ 0 3 0 0 0 3
2 0 0 2
ˆˆ ˆ
y yx xz z
x y z
i j kv vv vv vv i j k
x y z y z x z x yv v v
z i j x k z i x k
u u uu y yx y zv v v
i j kx y z
⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠ ⎝ ⎠
⎛
⎝
part partpart partpart partpart part partnablatimes = = minus minus minus + minuspart part part part part part part part part
= + minus minus + minus = minus
part part partnabla = + + = + + =part part part
nablatimes nablatimes =nabla nablasdot minusnabla
part part part= + +part part part
v
v v v
( ) ( )( ) ( ) ( ) ( )
( )
2 2 22 3
2 2 2ˆˆ ˆ2 2
ˆ ˆˆ ˆ ˆ ˆ 2 2 0 2 0 0 0 0 6 0 0 0
ˆ 2 6
xy x yi z j kx y z
yi xj k y i z j k
x z j
⎛ ⎞⎞⎜ ⎟⎜ ⎟
⎠ ⎝ ⎠⎡ ⎤⎣ ⎦
part part partminus + + minus +part part part
= + + minus + + + + minus + + minus
= +
MingsianMingsian R R BaiBai22
( )v w r w r r wnablatimes =nablatimes times = nablasdot minus nablasdotv v v v v v v ( )r w+ sdotnablav v ( ) constant w
w rminus sdotnablav
v v
Q
Ex Rigid body rotation
MingsianMingsian R R BaiBai23
( )( ) ( )
( ) ( )
ˆˆ ˆ
ˆ ˆˆ ˆ ˆ ˆ 1 1 1 3
ˆˆ ˆ
ˆˆ ˆ
Thus 3 2 is linked directly
x y z
x y z
r xi yj zk
r i j k xi yj zkx y z
w r w w w xi yj zkx y z
w i w j w k w
w r w r w r w w wv
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎝ ⎠
= + +
part part partnablasdot = + + + + = + + =part part part
part part partsdotnabla = + + + +part part part
= + + =
nablatimes times == nablasdot minus sdotnabla = minus =rArrnablatimes
v
v
v v
v
v v v v v v v v v
v spinn to ing
MingsianMingsian R R BaiBai24
Line surface and volume integrals
x
yC
B
A
isΔ
( )ii ηξ
Piecewise smooth curve
Line integral
MingsianMingsian R R BaiBai25
0 1lim ( ) ( ) or ( )
Intepretation of a line integral as an area ( )
i
Bn
i i isA
C
i C
F s F x y ds F x y d
x
s
F y ds
ξ ηΔ rarr =
Δ =sum int
int
int
x
y
z
C
( )z F x y=
MingsianMingsian R R BaiBai26
the area is always on the left of an obs
Positive dierver trave
rection of irsing on the
ntegrationcurve
RC
CCW CW
If is a regular closed curve piecewise smo
( ) ( )
(contou
oth simple ( )
r integral)
x y zC C
x y zC
F dR F F F dx dy dz
F dx F dy
C
F dz
sdot = + + + +
= + +
lowast
int int
int
i j k i j kv v
無交叉
work dW F dR= sdotv v
MingsianMingsian R R BaiBai27
0 1
Surface integral
lim ( ) ( )i
n
i i i iS i S
F S F x y z dSξ η ζΔ rarr =
Δ =sum intint( ) i i iξ η ζ
iSΔ
0 1
mass density
lim ( ) ( )
Ex
Volume integral
( )
i
n
i i i iV i V
V
F V F x y z dV
M x y z dV
ξ η ζ
ρ
Δ rarr =Δ =
=
sum intintint
intintint( ) i i iξ η ζ
iVΔ
MingsianMingsian R R BaiBai28
Ref (1)Hildebrand advanced Calculus for application Prentice-Hall 1976 p306-313(2) Greenberg p383-389 (3) Arfken ch2
1 2 3
1 2 3
1 2 3
1 2 3
(
Orthogonal coordinate systems base vectors are not constant in s
) ( )( )( )( )
ˆˆ ˆ
pacep x y z p u u ux x u u uy y u u uz z u u u
r xi yj zk
primerarr
= +
rarr
===
+v
1uv
2uv3uv
1u
2u
3uP coord curve
xy
z
O
X Curvilinear Coordinates
MingsianMingsian R R BaiBai29
1
11
1 1 1
1
1
1 1 1
11 1
1 1
2 2 2 3 3 3
Tangent vector to the at
where is the arc length
1
where unit vector
Similarly
curv
e
u psr rU
u s us
rs
U h uds rh udu sU h u U h u
partpart part= =part part part
part =part
there4 =part= =part
= =
v vv
vQ
v v
vv
v vv v
1u cood curve
1uv
1du 1dS
drv
P
MingsianMingsian R R BaiBai30
3 31 2 1 21 2 3
1 2 32
curv
In summary scale factor
If is the arc length along a in any directione
k k k
kk k
k k
k k k
U h u
ds rh Udu u
ds h dus
du dudu du du dudr r r r U U Uds u ds u ds u ds ds ds ds
dr drds ds
=
part= = =part
=
part part part= + + = + +part part part
= sdot
v v
v v
v v v v v v v
v v 3 3
1 1
23 3 32 2
21 1 1
2 2 2 2 2 2 2 2 2 21 1 2 2 3 3 1 2 3
1 (for orthogonal coord)
jii j
i j
ji ii j i
i j i
dududr U Uds ds ds
dudu duU U hds ds ds
ds h du h du h du ds ds ds
= =
= = =
⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
= sdot
rarr = sdot =
rarr = + + = + +
sum sum
sumsum sum
v v v
v v
MingsianMingsian R R BaiBai31
1 1 2 2 3 3
1 2 3
are mutually vectors having with arc length
k k k k kds h du U du
U du U du U duds ds ds
= =
rArr perp
v
v v v
1uv
2uv
3uv
1dS
2dS
3dS
dS
MingsianMingsian R R BaiBai32
( ) ( )
( ) ( )
1 1 2 2 3 3 1 2 3 1 2 3 1 2 3
1 2 3 1 2 3
1
1 2 2 3 3 2 2 2 3 3 3 2 3 2 3 1
Differential volume
Differential On const coord surface
area
Gradi t
en
d U du U du U du u u u h h h du du du
d h h h du du du
ud U du U du h u du h u du h h du du u
df
τ
τ
σ
= times sdot = times sdot
=
=
= times = times =
= nabla
v v v v v v
v vv v v v
1 2 31 2 3
1 2 3 1 1 2 2 3 31 2 3
1 1 1 2 2 2 3 3 3
f f ff dr df du du duu u u
r r rdr du du du U du U du U duu u u
h u du h u du h u du
part part partsdot rarr = + +part part part
part part part= + + = + +part part part
= + +
v
v v v v v vv
v v v
=1 (right hand rule)
MingsianMingsian R R BaiBai33
Rectangular Cartesian Coordinates
ij
k
MingsianMingsian R R BaiBai34
Circular Cylindrical Coordinatescossin
xyz z
ρ ϕρ ϕ
===
Ref Arfken
( )P zρ ϕ
MingsianMingsian R R BaiBai35
Polar Spherical Coordinatessin cossin sincos
x ry rz r
θ ϕθ ϕθ
===
Ref Arfken
0
0
0
sin cos sin sin coscos cos cos sin sin
sin cos
θ ϕ θ ϕ θθ ϕ θ ϕ θϕ ϕ
= + +
= + minus= minus +
r i j kθ i j kφ i j
2
2
sinsin
dS r d ddV r dr d d
θ θ ϕ
θ θ ϕ
=
=( )P r θ ϕ
MingsianMingsian R R BaiBai36
Integral TheoremsHeart of this chapter
Interchange of dimensionality Div Curl Grad in integrals High-dimensional integration
rarr
1
Thm1 Divergence theorem Let be a closed region bounded by piecewise smooth orientable surface and let (con
I Divergence theor
tinuous derivatives up to the first order
em (Gauss theorem)
)
VS v Cisin
r
ˆ If denotes the outward unit normal on
the ˆn V S
v dV n v dS
n S
nablasdot = sdotint intr r
MingsianMingsian R R BaiBai37
jS
jP
jV
S
0
ˆBy definition ( ) lim
ˆ( )
ˆ( )
j
j
j
j
Sj V
j
j j S
j j Sj j
n v dSv P
V
v P V n v dS
v P V n v dS
Δ rarr
⎧ ⎫sdot⎪ ⎪nabla sdot ⎨ ⎬Δ⎪ ⎪⎩ ⎭
nabla sdot Δ asymp sdot
nabla sdot Δ = sdot
int
intsum sumint
rr
r r
r r
Pf
MingsianMingsian R R BaiBai38
0
After interior surfaces cancel out only exterior surface (Why Opposite normal vectors)
ˆ( )
volumes exterior surfaces
ˆlim
QE
survive
j
j
j j Sj j
V SV
v p V n v dS
v dV n v dSΔ rarr
nabla sdot Δ = sdot
rArr nablasdot = sdot
sum sumint
int int
r r
r r
D
MingsianMingsian R R BaiBai39
( )
scalar variable constant vectorvector variable
Alternative formsLet be a an arbitrary and
a
ˆ
1 Le
(
ˆt
V
V S
S
V
v ap
v d
v dV vn d
V n v dS
v va
va dV v a v a
S
nablasdot = sdot
= rArr
nablasdot = nabla sdot + nabla
=
sdot
nablaint intint int
int
r
v
r r
r r
r r rQ ( )
( )
ˆ)
2 Let ˆV S
V
V S
dV n va dS
v a p
a p
p dV n p d
d p
S
V a
⎛ ⎞= sdot⎜ ⎟
⎝ ⎠
= times rArr
nablasdot times = sdot nabla
times
times
nabla = times
int int
int int
int
r
r r r
r r v
r
vQ
r
( ) ( ) ( )
( ) ( )
ˆ
ˆV S
V S
a p dV n a p dS
a p dV a n p dS
⎛ ⎞⎡ ⎤minus sdot nablatimes = sdot times⎣ ⎦⎜ ⎟⎜ ⎟⎜ ⎟rArr minus sdot nablatimes = minus sdot times⎜ ⎟⎝ ⎠
int int
int int
r rv v
r rv v
MingsianMingsian R R BaiBai40
( )
( )
From Leibnitzs rule
(1-D) ( )
( ) ( )
B t
A t
d f x t dxdt
f x t dx B ttpart prime= +part
int( )
( )( ( ) ) ( )
B t
A tf B t t A tprimeminusint ( ( ) )
(assume ( ) ( ) )
f A t t
A t B t const= =
ContinuityEx The Equation
rate of increase of mass
mass densityV
dM d dVdt dt
ρ
ρ
= int
flow velocityvv
MingsianMingsian R R BaiBai41
ˆ ˆ ( )
(
Conservation
) ( Divergence theorem)
of mass
( ) 0
V S S
V
V
dV n v dS n v dSt
v dV
v dVt
ρ ρ ρ
ρ
ρ ρ⎡ ⎤⎢ ⎥⎣ ⎦
part = minus sdot = minus sdotpart
= minus nablasdot
partrArr +nablasdot =part
int int int
int
int
v v
vQ
v
ˆOn the other hand rate influof mass through n ox i t
Extended to 3-D for time-independent fixed boundary
(3-D) ( ) ( )
Thus
S
V V
V
S V n v dS
Sd f x y z t dV f x y z t dVdt tdM dVdt t
ρ
ρ
= minus sdot
part=part
part=part
int
int int
intv
Fixed control volume (Eulerian coordinates)Otherwise Raynoldrsquos transport theoremMoving control volume Lagrangian coordinates
MingsianMingsian R R BaiBai42
But the control volime is only arbitrary
In particular for flow
( ) 0
con
(continuity equati
stant
on)
Divergence the
0 incompress
Note is t
ible
orem h
V
v
vt
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
e mother of the other integral theorems
MingsianMingsian R R BaiBai43
Ex Archimedes principle ( ) The buoyant force ( ) on a body with volume immersed in a fluid is equal to the weight of the fluid
ˆ displaced by the object ie In the followin
B
V
gVρ= minusf k
阿基米德原理
浮力
ˆg figure the ˆ with and being the fluid density gravitational
acceleration depth and outward normal unit vector to the surface
fhydraulic pressure g z
g z
S
ρ
ρ
= minusp n
n
1
the modified divergence theorem
ˆ
Hint
scalar fuctionV S
vdV v dS v Cnabla = isinint int n
MingsianMingsian R R BaiBai44
Levelsurfacefluid
V
S
airyx
z
ˆf g zρ= minusp n
ˆPf ( ) ( )
ˆ ˆ ˆ ˆ ˆ (0 0 ) v v
B fS S V
V V
g z g zdS dS dV
g dV g dV gV
ρ ρ
ρ ρ ρ
= = minus = minus nabla
= minus + + = minus = minus
int int int
int int
f p n
i j k k k
k
MingsianMingsian R R BaiBai45
Note This can also be thought of as simple balance of forces (resultant force that acts on and weight of filled with fluid)S V
2
~ a little history about George Green Recall the diveregence theorem
ˆ
Let
II th
e Green
are arbitrary fun
s identiti
ctions
es
V S
v dV n v dS
v u vu v C
nablasdot = sdot
= nablaisin
int intv v
v
ˆ ( ) ( )
ˆ ( )V S
V S
u v dV n u v dS
u v u v dV un v dS
nablasdot nabla = sdot nabla
nabla sdotnabla + nablasdotnabla = sdotnabla
int int
int int
MingsianMingsian R R BaiBai46
2 -( ) -(1)V S
vu v u v dV u dSnpart
nabla sdotnabla + nabla =partint int firs(Greens identt ity)
2
If
( ) --(2)
(1) (2) V S
u vuv u v u dV v dSn
harrpart
nabla sdotnabla + nabla =part
minus
int int
( )2 2
V S
dv duu v v u dV u v dSdn dn
⎛ ⎞nabla minus nabla = minus⎜ ⎟⎝ ⎠int int
second(Greens identity)
MingsianMingsian R R BaiBai47
( )
( )
( ) ( )
1st id ( )
Greens identities in the 1D case integration by parts
b b bb ba a
a a ab
ba
ab
ba
a
v udx v u v u dx u v uv dx uv
u v u v vu dx vu
uv vu dx uv vu
primeprime prime prime prime prime prime primeprime prime= minus rArr + =
prime prime primeprime primeharr + =
primeprime primeprime prime primerArr minus = minus
int int int
int
2
2nd id ( )
Good for any adjoint operators in addition to Good also for and integration by parts Important for theory amp reciprocity
From th
Gree
e 1s
2-
t
ns functi
identity
D 1-
tak 1
D
e
on
u
nablararr
=
int
2
V S
vv dV dSnpartnabla =part
rArr int int2Ex 0 st 0 on the boundary 0 ( 1st id u u unabla = = rArr equiv 以 証)
MingsianMingsian R R BaiBai48
1
Thm1 Stokes theorem Let be defined in Let be a piecewise-smooth orientable surface bounded by a piec
III Stokes theorem and
ewise-smooth simple clos
Greens
ed curve Then
theore
m
v C R S
C
isinv
( )
ˆwhere is a unit normal to according t right hand rulo the e
ˆS C
n S
n v dS v dRsdot nablatimes = sdotint intvv v
(not necessarily planar)S
Cn vnablatimes v vv
MingsianMingsian R R BaiBai49
Pf
( )( )
circulatioWe have shown in deriving thatthe in -direction
ˆor
ˆ
n
z
z
j j j j j
j j j j jj j
vz v dx dy
v dR n v S
v dR n v S
nablatimes
=nablatimes
sdot = sdot nablatimes Δ
sdot = sdot nablatimes Δ
intsum sumint
v
v
vv v
vv v
C
S
jS
jC
jP
ˆ jn
jSΔ
MingsianMingsian R R BaiBai50
( )
( )ext surface
After the interior boundaries cancel out only the exterior boundaries (Why)
ˆ
ˆ0 QED
Alternative forms scalar
survive
va
j
j j j j jC
jC S
v dR n v S
S v dR n v dS
φ
sdot = sdot nablatimes Δ
Δ rarr rArr sdot = sdot nablatimes
sum sumint
int int
vv v
vv v
( )
riable constant vector vector variable
1 Let
2 Let
ˆ
ˆ S C
S C
a p
v n dR dR
n p dS dR p
a
v a p
φ φφ= rArr timesnabla =
timesnabla times= rArr timestimes =
int int
int int
v v
v v
v
v v
vv vv v
MingsianMingsian R R BaiBai51
( ) ( )
( )
( )
( ) ( )
1
Thm2 Greens theorem ( planar version of Stokes theorem)ˆ ˆ Let
pf By Stokes theoremˆ
ˆ ˆ ˆ ˆ ˆ ˆ
S C
S C
S C
v P x y i Q x y j C
n v dS v dR
k Pi Qj dS Pi Qj dx i dy
Q P dS Pdx Qdyx y
⎛ ⎞part partminus = +⎜ ⎟
= + isin
sdot nablatimes = sdot
⎡ ⎤sdot nablatimes + =
part part⎝ ⎠
+ sdot +⎣ ⎦
int int
int
int
int
int
v
vv v
( )
( )
( )
ˆ
ˆˆ ˆ
ˆˆ ˆ ˆ ˆ( ) ( )
0
QEDS C
j
i j kQ PPi Qj i j k
x y z x yP Q
Q P dS Pdx Qdyx y
⎛ ⎞part part part part partnablatimes + = = + + minus⎜ ⎟part part part part part⎝ ⎠
⎛ ⎞part partrArr minus = +⎜ ⎟part part⎝ ⎠int int
L L
S
C
i
j
x
y
MingsianMingsian R R BaiBai52
Potential TheorypotentiGeorge Green coined the term (l
a )
rarr 位能 勢能
例子 重力場 電磁場 流場 聲場 彈性力學
1
2
Thm1 If is in a simply connected region then the following statements are equivalent(i) There exists a scala
Conservative Field
potential functior such that
(ii
n
) 0
v
C
C
v
v R
φφ=nabla
nablatimes
isin
isin
=r
r
r
r in (Irrotational field curl-free)
cf 0 st solenoidal field divergence-free
0
(iii) in (Conservative field)C
Rv w v w
v R Rd
nablasdot = exist nabla
sdot =
= times
int
r r r
r
r
r
MingsianMingsian R R BaiBai53
(iv) is connecting and
independent o
Ex Think
f the pa
of dynamics is the force
thB
A
v dR A B
v
sdotintrr
r
1 2
1
0
Pf
(i) (ii) ( ) 0 (zero identity cg rule) irrotational
ˆ(ii) (iii) ( ) 0 (Stokes theorem)
(iii) (iv) ( ) 0
C S
C C C
C
v
v dR n v dS
v dR v dR
v dR
φ
=
rarr nablatimes = nablatimes nabla =
rarr sdot = sdot nablatimes =
rarr sdot = minus sdot =
rArr sdot =
int int
int int int
int
r
rr
rr r
r rr r
rr r
2
independent of pathC
v dRsdotintr
A
B
1C
2C
MingsianMingsian R R BaiBai54
1 2
(iv) (i) constant ( ) ( )
--(1)
But (2)
(1) (2) QED
Note on application
Check
B
C C A
v dR v dR B A d
v dR d
d dx dy dz dRx y z
v dR dR v R
φ φ φ
φφ φ φφ φ
φ φ
rarr sdot = sdot = = minus =
rArr sdot =part part part
= + + = nabla sdot minus minus minus minus minuspart part part
sdot = nabla sdot rArr = nabla forall
int int intr rr r
rr
r
r r rr r
use simpler path (iv)if 0
find (i)v
φ⎧
nablatimes = ⎨⎩
rr
A
B
x
y
Cz
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai8
ˆEx Normal gradient outward normal
Note is often written as in the PDE literature
ˆˆ ˆˆPartial derivatives If then
ˆ
Physical interpretati
n
du udn n
du u u us i j kdr
du n ud
z
n
x y
partpart
part part part= rarrpart part part
sdotnabla
max
maximum rate of increase in a direction
ˆ ˆ ˆ ˆcos ( ) cos ( )
ˆ is independent of
ˆmax occurs when ( ) 0 and
points in the direction of ma
o
n
x a
du u s u s u s u u sdr
u sdu duu s udr dr
duudr
θ θ
θ ⎛ ⎞⎜ ⎟⎝ ⎠
=nabla sdot = nabla nabla = nabla nabla
nabla
there4 nabla = = nabla
rArrnabla
Q
( )max
(interpretation 1nd
gradient search steepest desce
)
nt
duudr
⎛ ⎞⎜ ⎟⎝ ⎠
nabla =
rArr minusnabla最佳化方法 的基礎
MingsianMingsian R R BaiBai9
(interpretatDirection of ion 2 normal to the level surface ( )
On the 0 ( )
ie
level surfac
o
e
n
)udu u dr
u cdu u dr
u dr u c
nabla= nabla sdot
== = nabla sdotnabla perp =
r
r
r
unabla
dr
unabla
Q
P
1cu=
2u c=
山
x
y
( )u x y
x
y
Hill-climbing search in optimization
MingsianMingsian R R BaiBai10
Ex Plane equationˆˆ ˆ( ) Normal vector
Ex 0 is the necessary condition for searching extrema (from multivariate Tylor expansion)
Ex Revisit of the exampl
u x y z ax by cz d u ai bj ck
u
= + + = rArr nabla = + +
nabla =v
多變數函數求極值
2 21 1 2 2
e in Linear Algebra using method of What is the maximum and minimum distance of the origin to the following curve
Lagrange multipli s
6 8
er
5 5x x x x+ + =
MingsianMingsian R R BaiBai11
1 2
2 21 2( )
2 21 1 2 2
Sol
This problem can be posed as a problemmin max ( )
constrained optimization
5 6 5 8 0 ( )which can
objectivecost function
constraint
equatioals
nx x
J x x
st g x x x x
= +
= + + minus =
2 2 2 21 2 1 1 2 2
o be converted into an unconstrained optimization problem by defining the ( ) (5 6 5 8)where is referred to as the thatLa
L
g ora
agr
nge ften h multi
angi
plie as
an
r ph
L x x x x x xλλ
= + + + + minus
ysical meaning in certain problemsThus the extreme values of will occur wh en or L J gJ nabla = nabla nabla0
MingsianMingsian R R BaiBai12
1 1 21
2 1 22
2 21 1 2 2
1 2 1 2
2 (10 6 ) 0 (1)
2 (6 10 ) 0 (2)
5 6 5 8 0 (constraint)
(1) (2) (2 16 )( ) 0 18 or (3)Substituting Eq (3) into Eqs (1) and (2) l
L x x xxL x x xxL x x x x
x x x x
λ
λ
λλ λ
part = + + =partpart = + + =partpart = + + minus =part+ + + = rArr = minus = minus
1 2 1 2eads to
and which corresponds to 14Thus minimummaximum distance 12x x x x J= = minus =
=
MingsianMingsian R R BaiBai13
22 2
1
1 1
2 2
The result above is identical to the previously obtained result in Linear Algebr
ellipse
45 rotatio
a
1 ( )4
1 11 1
n12
xx
x xx x
deg⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠
prime+ =prime
primeminus= rArr
prime2x prime
2x
1x prime
1x
2
MingsianMingsian R R BaiBai14
Curl 旋度
curl curl In Cartesian coordinates
spatial operator indep
ˆˆ ˆ
ˆˆ ˆ( )
ende
(
nt of coordi
) (
D
nates
ef
)
x y z
y x x xz z
i j k
vx y z
v v v
v v v vv vi j ky z x z x y
v v
part part partnablatimes =part part part
part part part partpart part= minus minus minus + minuspart
times
part part part part part
nabla
rarr
r
r r
0vnablatimes =rr 0vnablatimes ne
rr
MingsianMingsian R R BaiBai15
circulaPhysica
tion o spin ol interpretati
f fluidr or vortion
y
citΓ
( )0 0x dx y dy+ +
( )0 0x y ( )0 0x dx y+
Γ
1
2
3
4
( )0 0x y dy+
MingsianMingsian R R BaiBai16
yy
vv dy
ypart
+part
Γdyminus
dyyvv x
x partpart
+
dy
dx
dxminus
y yy
v vv dx dy
x ypart part
+ +part part
x xx
v vv dx dyx y
part part+ +part part
yy
vv dx
xpart
+part
xx
vv dxx
part+part
xv
yv
vv
dλ paddle weel
MingsianMingsian R R BaiBai17
contour
1 2 3 4
circulation
x x y y x x y y
x
v d
v d v d v d v d
v
λ
λ λ λ λrarr
Γ sdot
= + + +
asymp
intint int int int
vv
12
xv dxx
part+part ydx v
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
+ 12
y yv vdx dy
x ypart part
+ +part part
x
dy
v
⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦
+
12
xv dxx
part+part
( )xy
v dy dx vy
⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦
part+ minus +part
12
yvdy
ypart
+part
( )
( )
Similar process can be applied to the circulation in the and directionsThus the total circulation per unit area i curl s
y xz
dy
v v dxdy v dxdyx y
xv v
y
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎝ ⎠
minus
part part= minus = nablatimespart part
=nablatimes
v
v v
MingsianMingsian R R BaiBai18
Identities of differential operators ( p380 Greenberg )
( )
( ) ( )( )( )
( ) ( )( ) is required
Example of deriv
ˆ ˆˆ ˆ ˆ ˆ
ation
x y z
x y z
v v vx y z
uv u v u v
uv i j k uv i uv j uv kx y z
uv u v u v
u v v u u v
u v u vv v u u
+ +part part partpart part part
⎛ ⎞⎜ ⎟⎝ ⎠
nablasdot = nabla sdot + nablasdot
part part partnablasdot = + + sdot + + =part part part
nablatimes =nabla times + nablatimes
nablasdot times = sdotnablatimes minus sdotnablatimes
nablatimes times = nabla minus + minusnablanablasdot sdotsdot678 678
v
v v v
vL
v v v
v v v v v v
v v v v v v v v( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )2 cf
x y zu u ux y z
A B C A
u v
v w v w w v v w w v
v v v C B A B C
+ +part part partpart part part
sdotnabla
nabla sdot = sdotnabla + sdotnabla + times nablatimes + times
times
nablatimes
nablatimes nablatimes =nabla nabla times = sdot minussdot sdotminusnabla
678v
v v v
v v v v v vv v
v v v v v v
v v
v
v v
ˆ ˆˆ ˆ ˆ ˆ( ) ( ) ( )x y z x y zv i v j v k i j k u v v v ux y z x y zpart part part part part part+ + sdot + + = + +part part part part part part
v v
MingsianMingsian R R BaiBai19
2 2 22
2 2 2
0 (dc direct current) solenoidal0 (cg center of gravity) conservative potent
Laplacia
Zero identitie
n
s
ial
x y z
vu
⎧⎪⎨⎪⎩
part part partΔ =nabla nablasdotnabla = + +part part part
nablasdotnablatimes = rarr
nablatimesnabla = rarr
v
v
These differential operators will take more complicated and irregular form for the other coordinate systems
MingsianMingsian R R BaiBai20
2 2 3
2 2
ˆˆ ˆEx 2
ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ 2 0 2
2 0 0 2yx z
u x y v x yi z j k
u i j k u xyi x j k xyi x jx y z
vv vv xy xyx y z
⎛ ⎞⎜ ⎟⎝ ⎠
= = minus +
part part partnabla = + + = + + = +part part part
partpart partnablasdot = + + = + minus =part part part
v
v
MingsianMingsian R R BaiBai21
( ) ( ) ( )
( ) ( )
2 2 2 2
2 2 22
2 2 2
2
ˆˆ ˆ
ˆˆ ˆ
ˆ ˆˆ ˆ ˆ 0 3 0 0 0 3
2 0 0 2
ˆˆ ˆ
y yx xz z
x y z
i j kv vv vv vv i j k
x y z y z x z x yv v v
z i j x k z i x k
u u uu y yx y zv v v
i j kx y z
⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠ ⎝ ⎠
⎛
⎝
part partpart partpart partpart part partnablatimes = = minus minus minus + minuspart part part part part part part part part
= + minus minus + minus = minus
part part partnabla = + + = + + =part part part
nablatimes nablatimes =nabla nablasdot minusnabla
part part part= + +part part part
v
v v v
( ) ( )( ) ( ) ( ) ( )
( )
2 2 22 3
2 2 2ˆˆ ˆ2 2
ˆ ˆˆ ˆ ˆ ˆ 2 2 0 2 0 0 0 0 6 0 0 0
ˆ 2 6
xy x yi z j kx y z
yi xj k y i z j k
x z j
⎛ ⎞⎞⎜ ⎟⎜ ⎟
⎠ ⎝ ⎠⎡ ⎤⎣ ⎦
part part partminus + + minus +part part part
= + + minus + + + + minus + + minus
= +
MingsianMingsian R R BaiBai22
( )v w r w r r wnablatimes =nablatimes times = nablasdot minus nablasdotv v v v v v v ( )r w+ sdotnablav v ( ) constant w
w rminus sdotnablav
v v
Q
Ex Rigid body rotation
MingsianMingsian R R BaiBai23
( )( ) ( )
( ) ( )
ˆˆ ˆ
ˆ ˆˆ ˆ ˆ ˆ 1 1 1 3
ˆˆ ˆ
ˆˆ ˆ
Thus 3 2 is linked directly
x y z
x y z
r xi yj zk
r i j k xi yj zkx y z
w r w w w xi yj zkx y z
w i w j w k w
w r w r w r w w wv
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎝ ⎠
= + +
part part partnablasdot = + + + + = + + =part part part
part part partsdotnabla = + + + +part part part
= + + =
nablatimes times == nablasdot minus sdotnabla = minus =rArrnablatimes
v
v
v v
v
v v v v v v v v v
v spinn to ing
MingsianMingsian R R BaiBai24
Line surface and volume integrals
x
yC
B
A
isΔ
( )ii ηξ
Piecewise smooth curve
Line integral
MingsianMingsian R R BaiBai25
0 1lim ( ) ( ) or ( )
Intepretation of a line integral as an area ( )
i
Bn
i i isA
C
i C
F s F x y ds F x y d
x
s
F y ds
ξ ηΔ rarr =
Δ =sum int
int
int
x
y
z
C
( )z F x y=
MingsianMingsian R R BaiBai26
the area is always on the left of an obs
Positive dierver trave
rection of irsing on the
ntegrationcurve
RC
CCW CW
If is a regular closed curve piecewise smo
( ) ( )
(contou
oth simple ( )
r integral)
x y zC C
x y zC
F dR F F F dx dy dz
F dx F dy
C
F dz
sdot = + + + +
= + +
lowast
int int
int
i j k i j kv v
無交叉
work dW F dR= sdotv v
MingsianMingsian R R BaiBai27
0 1
Surface integral
lim ( ) ( )i
n
i i i iS i S
F S F x y z dSξ η ζΔ rarr =
Δ =sum intint( ) i i iξ η ζ
iSΔ
0 1
mass density
lim ( ) ( )
Ex
Volume integral
( )
i
n
i i i iV i V
V
F V F x y z dV
M x y z dV
ξ η ζ
ρ
Δ rarr =Δ =
=
sum intintint
intintint( ) i i iξ η ζ
iVΔ
MingsianMingsian R R BaiBai28
Ref (1)Hildebrand advanced Calculus for application Prentice-Hall 1976 p306-313(2) Greenberg p383-389 (3) Arfken ch2
1 2 3
1 2 3
1 2 3
1 2 3
(
Orthogonal coordinate systems base vectors are not constant in s
) ( )( )( )( )
ˆˆ ˆ
pacep x y z p u u ux x u u uy y u u uz z u u u
r xi yj zk
primerarr
= +
rarr
===
+v
1uv
2uv3uv
1u
2u
3uP coord curve
xy
z
O
X Curvilinear Coordinates
MingsianMingsian R R BaiBai29
1
11
1 1 1
1
1
1 1 1
11 1
1 1
2 2 2 3 3 3
Tangent vector to the at
where is the arc length
1
where unit vector
Similarly
curv
e
u psr rU
u s us
rs
U h uds rh udu sU h u U h u
partpart part= =part part part
part =part
there4 =part= =part
= =
v vv
vQ
v v
vv
v vv v
1u cood curve
1uv
1du 1dS
drv
P
MingsianMingsian R R BaiBai30
3 31 2 1 21 2 3
1 2 32
curv
In summary scale factor
If is the arc length along a in any directione
k k k
kk k
k k
k k k
U h u
ds rh Udu u
ds h dus
du dudu du du dudr r r r U U Uds u ds u ds u ds ds ds ds
dr drds ds
=
part= = =part
=
part part part= + + = + +part part part
= sdot
v v
v v
v v v v v v v
v v 3 3
1 1
23 3 32 2
21 1 1
2 2 2 2 2 2 2 2 2 21 1 2 2 3 3 1 2 3
1 (for orthogonal coord)
jii j
i j
ji ii j i
i j i
dududr U Uds ds ds
dudu duU U hds ds ds
ds h du h du h du ds ds ds
= =
= = =
⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
= sdot
rarr = sdot =
rarr = + + = + +
sum sum
sumsum sum
v v v
v v
MingsianMingsian R R BaiBai31
1 1 2 2 3 3
1 2 3
are mutually vectors having with arc length
k k k k kds h du U du
U du U du U duds ds ds
= =
rArr perp
v
v v v
1uv
2uv
3uv
1dS
2dS
3dS
dS
MingsianMingsian R R BaiBai32
( ) ( )
( ) ( )
1 1 2 2 3 3 1 2 3 1 2 3 1 2 3
1 2 3 1 2 3
1
1 2 2 3 3 2 2 2 3 3 3 2 3 2 3 1
Differential volume
Differential On const coord surface
area
Gradi t
en
d U du U du U du u u u h h h du du du
d h h h du du du
ud U du U du h u du h u du h h du du u
df
τ
τ
σ
= times sdot = times sdot
=
=
= times = times =
= nabla
v v v v v v
v vv v v v
1 2 31 2 3
1 2 3 1 1 2 2 3 31 2 3
1 1 1 2 2 2 3 3 3
f f ff dr df du du duu u u
r r rdr du du du U du U du U duu u u
h u du h u du h u du
part part partsdot rarr = + +part part part
part part part= + + = + +part part part
= + +
v
v v v v v vv
v v v
=1 (right hand rule)
MingsianMingsian R R BaiBai33
Rectangular Cartesian Coordinates
ij
k
MingsianMingsian R R BaiBai34
Circular Cylindrical Coordinatescossin
xyz z
ρ ϕρ ϕ
===
Ref Arfken
( )P zρ ϕ
MingsianMingsian R R BaiBai35
Polar Spherical Coordinatessin cossin sincos
x ry rz r
θ ϕθ ϕθ
===
Ref Arfken
0
0
0
sin cos sin sin coscos cos cos sin sin
sin cos
θ ϕ θ ϕ θθ ϕ θ ϕ θϕ ϕ
= + +
= + minus= minus +
r i j kθ i j kφ i j
2
2
sinsin
dS r d ddV r dr d d
θ θ ϕ
θ θ ϕ
=
=( )P r θ ϕ
MingsianMingsian R R BaiBai36
Integral TheoremsHeart of this chapter
Interchange of dimensionality Div Curl Grad in integrals High-dimensional integration
rarr
1
Thm1 Divergence theorem Let be a closed region bounded by piecewise smooth orientable surface and let (con
I Divergence theor
tinuous derivatives up to the first order
em (Gauss theorem)
)
VS v Cisin
r
ˆ If denotes the outward unit normal on
the ˆn V S
v dV n v dS
n S
nablasdot = sdotint intr r
MingsianMingsian R R BaiBai37
jS
jP
jV
S
0
ˆBy definition ( ) lim
ˆ( )
ˆ( )
j
j
j
j
Sj V
j
j j S
j j Sj j
n v dSv P
V
v P V n v dS
v P V n v dS
Δ rarr
⎧ ⎫sdot⎪ ⎪nabla sdot ⎨ ⎬Δ⎪ ⎪⎩ ⎭
nabla sdot Δ asymp sdot
nabla sdot Δ = sdot
int
intsum sumint
rr
r r
r r
Pf
MingsianMingsian R R BaiBai38
0
After interior surfaces cancel out only exterior surface (Why Opposite normal vectors)
ˆ( )
volumes exterior surfaces
ˆlim
QE
survive
j
j
j j Sj j
V SV
v p V n v dS
v dV n v dSΔ rarr
nabla sdot Δ = sdot
rArr nablasdot = sdot
sum sumint
int int
r r
r r
D
MingsianMingsian R R BaiBai39
( )
scalar variable constant vectorvector variable
Alternative formsLet be a an arbitrary and
a
ˆ
1 Le
(
ˆt
V
V S
S
V
v ap
v d
v dV vn d
V n v dS
v va
va dV v a v a
S
nablasdot = sdot
= rArr
nablasdot = nabla sdot + nabla
=
sdot
nablaint intint int
int
r
v
r r
r r
r r rQ ( )
( )
ˆ)
2 Let ˆV S
V
V S
dV n va dS
v a p
a p
p dV n p d
d p
S
V a
⎛ ⎞= sdot⎜ ⎟
⎝ ⎠
= times rArr
nablasdot times = sdot nabla
times
times
nabla = times
int int
int int
int
r
r r r
r r v
r
vQ
r
( ) ( ) ( )
( ) ( )
ˆ
ˆV S
V S
a p dV n a p dS
a p dV a n p dS
⎛ ⎞⎡ ⎤minus sdot nablatimes = sdot times⎣ ⎦⎜ ⎟⎜ ⎟⎜ ⎟rArr minus sdot nablatimes = minus sdot times⎜ ⎟⎝ ⎠
int int
int int
r rv v
r rv v
MingsianMingsian R R BaiBai40
( )
( )
From Leibnitzs rule
(1-D) ( )
( ) ( )
B t
A t
d f x t dxdt
f x t dx B ttpart prime= +part
int( )
( )( ( ) ) ( )
B t
A tf B t t A tprimeminusint ( ( ) )
(assume ( ) ( ) )
f A t t
A t B t const= =
ContinuityEx The Equation
rate of increase of mass
mass densityV
dM d dVdt dt
ρ
ρ
= int
flow velocityvv
MingsianMingsian R R BaiBai41
ˆ ˆ ( )
(
Conservation
) ( Divergence theorem)
of mass
( ) 0
V S S
V
V
dV n v dS n v dSt
v dV
v dVt
ρ ρ ρ
ρ
ρ ρ⎡ ⎤⎢ ⎥⎣ ⎦
part = minus sdot = minus sdotpart
= minus nablasdot
partrArr +nablasdot =part
int int int
int
int
v v
vQ
v
ˆOn the other hand rate influof mass through n ox i t
Extended to 3-D for time-independent fixed boundary
(3-D) ( ) ( )
Thus
S
V V
V
S V n v dS
Sd f x y z t dV f x y z t dVdt tdM dVdt t
ρ
ρ
= minus sdot
part=part
part=part
int
int int
intv
Fixed control volume (Eulerian coordinates)Otherwise Raynoldrsquos transport theoremMoving control volume Lagrangian coordinates
MingsianMingsian R R BaiBai42
But the control volime is only arbitrary
In particular for flow
( ) 0
con
(continuity equati
stant
on)
Divergence the
0 incompress
Note is t
ible
orem h
V
v
vt
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
e mother of the other integral theorems
MingsianMingsian R R BaiBai43
Ex Archimedes principle ( ) The buoyant force ( ) on a body with volume immersed in a fluid is equal to the weight of the fluid
ˆ displaced by the object ie In the followin
B
V
gVρ= minusf k
阿基米德原理
浮力
ˆg figure the ˆ with and being the fluid density gravitational
acceleration depth and outward normal unit vector to the surface
fhydraulic pressure g z
g z
S
ρ
ρ
= minusp n
n
1
the modified divergence theorem
ˆ
Hint
scalar fuctionV S
vdV v dS v Cnabla = isinint int n
MingsianMingsian R R BaiBai44
Levelsurfacefluid
V
S
airyx
z
ˆf g zρ= minusp n
ˆPf ( ) ( )
ˆ ˆ ˆ ˆ ˆ (0 0 ) v v
B fS S V
V V
g z g zdS dS dV
g dV g dV gV
ρ ρ
ρ ρ ρ
= = minus = minus nabla
= minus + + = minus = minus
int int int
int int
f p n
i j k k k
k
MingsianMingsian R R BaiBai45
Note This can also be thought of as simple balance of forces (resultant force that acts on and weight of filled with fluid)S V
2
~ a little history about George Green Recall the diveregence theorem
ˆ
Let
II th
e Green
are arbitrary fun
s identiti
ctions
es
V S
v dV n v dS
v u vu v C
nablasdot = sdot
= nablaisin
int intv v
v
ˆ ( ) ( )
ˆ ( )V S
V S
u v dV n u v dS
u v u v dV un v dS
nablasdot nabla = sdot nabla
nabla sdotnabla + nablasdotnabla = sdotnabla
int int
int int
MingsianMingsian R R BaiBai46
2 -( ) -(1)V S
vu v u v dV u dSnpart
nabla sdotnabla + nabla =partint int firs(Greens identt ity)
2
If
( ) --(2)
(1) (2) V S
u vuv u v u dV v dSn
harrpart
nabla sdotnabla + nabla =part
minus
int int
( )2 2
V S
dv duu v v u dV u v dSdn dn
⎛ ⎞nabla minus nabla = minus⎜ ⎟⎝ ⎠int int
second(Greens identity)
MingsianMingsian R R BaiBai47
( )
( )
( ) ( )
1st id ( )
Greens identities in the 1D case integration by parts
b b bb ba a
a a ab
ba
ab
ba
a
v udx v u v u dx u v uv dx uv
u v u v vu dx vu
uv vu dx uv vu
primeprime prime prime prime prime prime primeprime prime= minus rArr + =
prime prime primeprime primeharr + =
primeprime primeprime prime primerArr minus = minus
int int int
int
2
2nd id ( )
Good for any adjoint operators in addition to Good also for and integration by parts Important for theory amp reciprocity
From th
Gree
e 1s
2-
t
ns functi
identity
D 1-
tak 1
D
e
on
u
nablararr
=
int
2
V S
vv dV dSnpartnabla =part
rArr int int2Ex 0 st 0 on the boundary 0 ( 1st id u u unabla = = rArr equiv 以 証)
MingsianMingsian R R BaiBai48
1
Thm1 Stokes theorem Let be defined in Let be a piecewise-smooth orientable surface bounded by a piec
III Stokes theorem and
ewise-smooth simple clos
Greens
ed curve Then
theore
m
v C R S
C
isinv
( )
ˆwhere is a unit normal to according t right hand rulo the e
ˆS C
n S
n v dS v dRsdot nablatimes = sdotint intvv v
(not necessarily planar)S
Cn vnablatimes v vv
MingsianMingsian R R BaiBai49
Pf
( )( )
circulatioWe have shown in deriving thatthe in -direction
ˆor
ˆ
n
z
z
j j j j j
j j j j jj j
vz v dx dy
v dR n v S
v dR n v S
nablatimes
=nablatimes
sdot = sdot nablatimes Δ
sdot = sdot nablatimes Δ
intsum sumint
v
v
vv v
vv v
C
S
jS
jC
jP
ˆ jn
jSΔ
MingsianMingsian R R BaiBai50
( )
( )ext surface
After the interior boundaries cancel out only the exterior boundaries (Why)
ˆ
ˆ0 QED
Alternative forms scalar
survive
va
j
j j j j jC
jC S
v dR n v S
S v dR n v dS
φ
sdot = sdot nablatimes Δ
Δ rarr rArr sdot = sdot nablatimes
sum sumint
int int
vv v
vv v
( )
riable constant vector vector variable
1 Let
2 Let
ˆ
ˆ S C
S C
a p
v n dR dR
n p dS dR p
a
v a p
φ φφ= rArr timesnabla =
timesnabla times= rArr timestimes =
int int
int int
v v
v v
v
v v
vv vv v
MingsianMingsian R R BaiBai51
( ) ( )
( )
( )
( ) ( )
1
Thm2 Greens theorem ( planar version of Stokes theorem)ˆ ˆ Let
pf By Stokes theoremˆ
ˆ ˆ ˆ ˆ ˆ ˆ
S C
S C
S C
v P x y i Q x y j C
n v dS v dR
k Pi Qj dS Pi Qj dx i dy
Q P dS Pdx Qdyx y
⎛ ⎞part partminus = +⎜ ⎟
= + isin
sdot nablatimes = sdot
⎡ ⎤sdot nablatimes + =
part part⎝ ⎠
+ sdot +⎣ ⎦
int int
int
int
int
int
v
vv v
( )
( )
( )
ˆ
ˆˆ ˆ
ˆˆ ˆ ˆ ˆ( ) ( )
0
QEDS C
j
i j kQ PPi Qj i j k
x y z x yP Q
Q P dS Pdx Qdyx y
⎛ ⎞part part part part partnablatimes + = = + + minus⎜ ⎟part part part part part⎝ ⎠
⎛ ⎞part partrArr minus = +⎜ ⎟part part⎝ ⎠int int
L L
S
C
i
j
x
y
MingsianMingsian R R BaiBai52
Potential TheorypotentiGeorge Green coined the term (l
a )
rarr 位能 勢能
例子 重力場 電磁場 流場 聲場 彈性力學
1
2
Thm1 If is in a simply connected region then the following statements are equivalent(i) There exists a scala
Conservative Field
potential functior such that
(ii
n
) 0
v
C
C
v
v R
φφ=nabla
nablatimes
isin
isin
=r
r
r
r in (Irrotational field curl-free)
cf 0 st solenoidal field divergence-free
0
(iii) in (Conservative field)C
Rv w v w
v R Rd
nablasdot = exist nabla
sdot =
= times
int
r r r
r
r
r
MingsianMingsian R R BaiBai53
(iv) is connecting and
independent o
Ex Think
f the pa
of dynamics is the force
thB
A
v dR A B
v
sdotintrr
r
1 2
1
0
Pf
(i) (ii) ( ) 0 (zero identity cg rule) irrotational
ˆ(ii) (iii) ( ) 0 (Stokes theorem)
(iii) (iv) ( ) 0
C S
C C C
C
v
v dR n v dS
v dR v dR
v dR
φ
=
rarr nablatimes = nablatimes nabla =
rarr sdot = sdot nablatimes =
rarr sdot = minus sdot =
rArr sdot =
int int
int int int
int
r
rr
rr r
r rr r
rr r
2
independent of pathC
v dRsdotintr
A
B
1C
2C
MingsianMingsian R R BaiBai54
1 2
(iv) (i) constant ( ) ( )
--(1)
But (2)
(1) (2) QED
Note on application
Check
B
C C A
v dR v dR B A d
v dR d
d dx dy dz dRx y z
v dR dR v R
φ φ φ
φφ φ φφ φ
φ φ
rarr sdot = sdot = = minus =
rArr sdot =part part part
= + + = nabla sdot minus minus minus minus minuspart part part
sdot = nabla sdot rArr = nabla forall
int int intr rr r
rr
r
r r rr r
use simpler path (iv)if 0
find (i)v
φ⎧
nablatimes = ⎨⎩
rr
A
B
x
y
Cz
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai9
(interpretatDirection of ion 2 normal to the level surface ( )
On the 0 ( )
ie
level surfac
o
e
n
)udu u dr
u cdu u dr
u dr u c
nabla= nabla sdot
== = nabla sdotnabla perp =
r
r
r
unabla
dr
unabla
Q
P
1cu=
2u c=
山
x
y
( )u x y
x
y
Hill-climbing search in optimization
MingsianMingsian R R BaiBai10
Ex Plane equationˆˆ ˆ( ) Normal vector
Ex 0 is the necessary condition for searching extrema (from multivariate Tylor expansion)
Ex Revisit of the exampl
u x y z ax by cz d u ai bj ck
u
= + + = rArr nabla = + +
nabla =v
多變數函數求極值
2 21 1 2 2
e in Linear Algebra using method of What is the maximum and minimum distance of the origin to the following curve
Lagrange multipli s
6 8
er
5 5x x x x+ + =
MingsianMingsian R R BaiBai11
1 2
2 21 2( )
2 21 1 2 2
Sol
This problem can be posed as a problemmin max ( )
constrained optimization
5 6 5 8 0 ( )which can
objectivecost function
constraint
equatioals
nx x
J x x
st g x x x x
= +
= + + minus =
2 2 2 21 2 1 1 2 2
o be converted into an unconstrained optimization problem by defining the ( ) (5 6 5 8)where is referred to as the thatLa
L
g ora
agr
nge ften h multi
angi
plie as
an
r ph
L x x x x x xλλ
= + + + + minus
ysical meaning in certain problemsThus the extreme values of will occur wh en or L J gJ nabla = nabla nabla0
MingsianMingsian R R BaiBai12
1 1 21
2 1 22
2 21 1 2 2
1 2 1 2
2 (10 6 ) 0 (1)
2 (6 10 ) 0 (2)
5 6 5 8 0 (constraint)
(1) (2) (2 16 )( ) 0 18 or (3)Substituting Eq (3) into Eqs (1) and (2) l
L x x xxL x x xxL x x x x
x x x x
λ
λ
λλ λ
part = + + =partpart = + + =partpart = + + minus =part+ + + = rArr = minus = minus
1 2 1 2eads to
and which corresponds to 14Thus minimummaximum distance 12x x x x J= = minus =
=
MingsianMingsian R R BaiBai13
22 2
1
1 1
2 2
The result above is identical to the previously obtained result in Linear Algebr
ellipse
45 rotatio
a
1 ( )4
1 11 1
n12
xx
x xx x
deg⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠
prime+ =prime
primeminus= rArr
prime2x prime
2x
1x prime
1x
2
MingsianMingsian R R BaiBai14
Curl 旋度
curl curl In Cartesian coordinates
spatial operator indep
ˆˆ ˆ
ˆˆ ˆ( )
ende
(
nt of coordi
) (
D
nates
ef
)
x y z
y x x xz z
i j k
vx y z
v v v
v v v vv vi j ky z x z x y
v v
part part partnablatimes =part part part
part part part partpart part= minus minus minus + minuspart
times
part part part part part
nabla
rarr
r
r r
0vnablatimes =rr 0vnablatimes ne
rr
MingsianMingsian R R BaiBai15
circulaPhysica
tion o spin ol interpretati
f fluidr or vortion
y
citΓ
( )0 0x dx y dy+ +
( )0 0x y ( )0 0x dx y+
Γ
1
2
3
4
( )0 0x y dy+
MingsianMingsian R R BaiBai16
yy
vv dy
ypart
+part
Γdyminus
dyyvv x
x partpart
+
dy
dx
dxminus
y yy
v vv dx dy
x ypart part
+ +part part
x xx
v vv dx dyx y
part part+ +part part
yy
vv dx
xpart
+part
xx
vv dxx
part+part
xv
yv
vv
dλ paddle weel
MingsianMingsian R R BaiBai17
contour
1 2 3 4
circulation
x x y y x x y y
x
v d
v d v d v d v d
v
λ
λ λ λ λrarr
Γ sdot
= + + +
asymp
intint int int int
vv
12
xv dxx
part+part ydx v
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
+ 12
y yv vdx dy
x ypart part
+ +part part
x
dy
v
⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦
+
12
xv dxx
part+part
( )xy
v dy dx vy
⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦
part+ minus +part
12
yvdy
ypart
+part
( )
( )
Similar process can be applied to the circulation in the and directionsThus the total circulation per unit area i curl s
y xz
dy
v v dxdy v dxdyx y
xv v
y
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎝ ⎠
minus
part part= minus = nablatimespart part
=nablatimes
v
v v
MingsianMingsian R R BaiBai18
Identities of differential operators ( p380 Greenberg )
( )
( ) ( )( )( )
( ) ( )( ) is required
Example of deriv
ˆ ˆˆ ˆ ˆ ˆ
ation
x y z
x y z
v v vx y z
uv u v u v
uv i j k uv i uv j uv kx y z
uv u v u v
u v v u u v
u v u vv v u u
+ +part part partpart part part
⎛ ⎞⎜ ⎟⎝ ⎠
nablasdot = nabla sdot + nablasdot
part part partnablasdot = + + sdot + + =part part part
nablatimes =nabla times + nablatimes
nablasdot times = sdotnablatimes minus sdotnablatimes
nablatimes times = nabla minus + minusnablanablasdot sdotsdot678 678
v
v v v
vL
v v v
v v v v v v
v v v v v v v v( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )2 cf
x y zu u ux y z
A B C A
u v
v w v w w v v w w v
v v v C B A B C
+ +part part partpart part part
sdotnabla
nabla sdot = sdotnabla + sdotnabla + times nablatimes + times
times
nablatimes
nablatimes nablatimes =nabla nabla times = sdot minussdot sdotminusnabla
678v
v v v
v v v v v vv v
v v v v v v
v v
v
v v
ˆ ˆˆ ˆ ˆ ˆ( ) ( ) ( )x y z x y zv i v j v k i j k u v v v ux y z x y zpart part part part part part+ + sdot + + = + +part part part part part part
v v
MingsianMingsian R R BaiBai19
2 2 22
2 2 2
0 (dc direct current) solenoidal0 (cg center of gravity) conservative potent
Laplacia
Zero identitie
n
s
ial
x y z
vu
⎧⎪⎨⎪⎩
part part partΔ =nabla nablasdotnabla = + +part part part
nablasdotnablatimes = rarr
nablatimesnabla = rarr
v
v
These differential operators will take more complicated and irregular form for the other coordinate systems
MingsianMingsian R R BaiBai20
2 2 3
2 2
ˆˆ ˆEx 2
ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ 2 0 2
2 0 0 2yx z
u x y v x yi z j k
u i j k u xyi x j k xyi x jx y z
vv vv xy xyx y z
⎛ ⎞⎜ ⎟⎝ ⎠
= = minus +
part part partnabla = + + = + + = +part part part
partpart partnablasdot = + + = + minus =part part part
v
v
MingsianMingsian R R BaiBai21
( ) ( ) ( )
( ) ( )
2 2 2 2
2 2 22
2 2 2
2
ˆˆ ˆ
ˆˆ ˆ
ˆ ˆˆ ˆ ˆ 0 3 0 0 0 3
2 0 0 2
ˆˆ ˆ
y yx xz z
x y z
i j kv vv vv vv i j k
x y z y z x z x yv v v
z i j x k z i x k
u u uu y yx y zv v v
i j kx y z
⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠ ⎝ ⎠
⎛
⎝
part partpart partpart partpart part partnablatimes = = minus minus minus + minuspart part part part part part part part part
= + minus minus + minus = minus
part part partnabla = + + = + + =part part part
nablatimes nablatimes =nabla nablasdot minusnabla
part part part= + +part part part
v
v v v
( ) ( )( ) ( ) ( ) ( )
( )
2 2 22 3
2 2 2ˆˆ ˆ2 2
ˆ ˆˆ ˆ ˆ ˆ 2 2 0 2 0 0 0 0 6 0 0 0
ˆ 2 6
xy x yi z j kx y z
yi xj k y i z j k
x z j
⎛ ⎞⎞⎜ ⎟⎜ ⎟
⎠ ⎝ ⎠⎡ ⎤⎣ ⎦
part part partminus + + minus +part part part
= + + minus + + + + minus + + minus
= +
MingsianMingsian R R BaiBai22
( )v w r w r r wnablatimes =nablatimes times = nablasdot minus nablasdotv v v v v v v ( )r w+ sdotnablav v ( ) constant w
w rminus sdotnablav
v v
Q
Ex Rigid body rotation
MingsianMingsian R R BaiBai23
( )( ) ( )
( ) ( )
ˆˆ ˆ
ˆ ˆˆ ˆ ˆ ˆ 1 1 1 3
ˆˆ ˆ
ˆˆ ˆ
Thus 3 2 is linked directly
x y z
x y z
r xi yj zk
r i j k xi yj zkx y z
w r w w w xi yj zkx y z
w i w j w k w
w r w r w r w w wv
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎝ ⎠
= + +
part part partnablasdot = + + + + = + + =part part part
part part partsdotnabla = + + + +part part part
= + + =
nablatimes times == nablasdot minus sdotnabla = minus =rArrnablatimes
v
v
v v
v
v v v v v v v v v
v spinn to ing
MingsianMingsian R R BaiBai24
Line surface and volume integrals
x
yC
B
A
isΔ
( )ii ηξ
Piecewise smooth curve
Line integral
MingsianMingsian R R BaiBai25
0 1lim ( ) ( ) or ( )
Intepretation of a line integral as an area ( )
i
Bn
i i isA
C
i C
F s F x y ds F x y d
x
s
F y ds
ξ ηΔ rarr =
Δ =sum int
int
int
x
y
z
C
( )z F x y=
MingsianMingsian R R BaiBai26
the area is always on the left of an obs
Positive dierver trave
rection of irsing on the
ntegrationcurve
RC
CCW CW
If is a regular closed curve piecewise smo
( ) ( )
(contou
oth simple ( )
r integral)
x y zC C
x y zC
F dR F F F dx dy dz
F dx F dy
C
F dz
sdot = + + + +
= + +
lowast
int int
int
i j k i j kv v
無交叉
work dW F dR= sdotv v
MingsianMingsian R R BaiBai27
0 1
Surface integral
lim ( ) ( )i
n
i i i iS i S
F S F x y z dSξ η ζΔ rarr =
Δ =sum intint( ) i i iξ η ζ
iSΔ
0 1
mass density
lim ( ) ( )
Ex
Volume integral
( )
i
n
i i i iV i V
V
F V F x y z dV
M x y z dV
ξ η ζ
ρ
Δ rarr =Δ =
=
sum intintint
intintint( ) i i iξ η ζ
iVΔ
MingsianMingsian R R BaiBai28
Ref (1)Hildebrand advanced Calculus for application Prentice-Hall 1976 p306-313(2) Greenberg p383-389 (3) Arfken ch2
1 2 3
1 2 3
1 2 3
1 2 3
(
Orthogonal coordinate systems base vectors are not constant in s
) ( )( )( )( )
ˆˆ ˆ
pacep x y z p u u ux x u u uy y u u uz z u u u
r xi yj zk
primerarr
= +
rarr
===
+v
1uv
2uv3uv
1u
2u
3uP coord curve
xy
z
O
X Curvilinear Coordinates
MingsianMingsian R R BaiBai29
1
11
1 1 1
1
1
1 1 1
11 1
1 1
2 2 2 3 3 3
Tangent vector to the at
where is the arc length
1
where unit vector
Similarly
curv
e
u psr rU
u s us
rs
U h uds rh udu sU h u U h u
partpart part= =part part part
part =part
there4 =part= =part
= =
v vv
vQ
v v
vv
v vv v
1u cood curve
1uv
1du 1dS
drv
P
MingsianMingsian R R BaiBai30
3 31 2 1 21 2 3
1 2 32
curv
In summary scale factor
If is the arc length along a in any directione
k k k
kk k
k k
k k k
U h u
ds rh Udu u
ds h dus
du dudu du du dudr r r r U U Uds u ds u ds u ds ds ds ds
dr drds ds
=
part= = =part
=
part part part= + + = + +part part part
= sdot
v v
v v
v v v v v v v
v v 3 3
1 1
23 3 32 2
21 1 1
2 2 2 2 2 2 2 2 2 21 1 2 2 3 3 1 2 3
1 (for orthogonal coord)
jii j
i j
ji ii j i
i j i
dududr U Uds ds ds
dudu duU U hds ds ds
ds h du h du h du ds ds ds
= =
= = =
⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
= sdot
rarr = sdot =
rarr = + + = + +
sum sum
sumsum sum
v v v
v v
MingsianMingsian R R BaiBai31
1 1 2 2 3 3
1 2 3
are mutually vectors having with arc length
k k k k kds h du U du
U du U du U duds ds ds
= =
rArr perp
v
v v v
1uv
2uv
3uv
1dS
2dS
3dS
dS
MingsianMingsian R R BaiBai32
( ) ( )
( ) ( )
1 1 2 2 3 3 1 2 3 1 2 3 1 2 3
1 2 3 1 2 3
1
1 2 2 3 3 2 2 2 3 3 3 2 3 2 3 1
Differential volume
Differential On const coord surface
area
Gradi t
en
d U du U du U du u u u h h h du du du
d h h h du du du
ud U du U du h u du h u du h h du du u
df
τ
τ
σ
= times sdot = times sdot
=
=
= times = times =
= nabla
v v v v v v
v vv v v v
1 2 31 2 3
1 2 3 1 1 2 2 3 31 2 3
1 1 1 2 2 2 3 3 3
f f ff dr df du du duu u u
r r rdr du du du U du U du U duu u u
h u du h u du h u du
part part partsdot rarr = + +part part part
part part part= + + = + +part part part
= + +
v
v v v v v vv
v v v
=1 (right hand rule)
MingsianMingsian R R BaiBai33
Rectangular Cartesian Coordinates
ij
k
MingsianMingsian R R BaiBai34
Circular Cylindrical Coordinatescossin
xyz z
ρ ϕρ ϕ
===
Ref Arfken
( )P zρ ϕ
MingsianMingsian R R BaiBai35
Polar Spherical Coordinatessin cossin sincos
x ry rz r
θ ϕθ ϕθ
===
Ref Arfken
0
0
0
sin cos sin sin coscos cos cos sin sin
sin cos
θ ϕ θ ϕ θθ ϕ θ ϕ θϕ ϕ
= + +
= + minus= minus +
r i j kθ i j kφ i j
2
2
sinsin
dS r d ddV r dr d d
θ θ ϕ
θ θ ϕ
=
=( )P r θ ϕ
MingsianMingsian R R BaiBai36
Integral TheoremsHeart of this chapter
Interchange of dimensionality Div Curl Grad in integrals High-dimensional integration
rarr
1
Thm1 Divergence theorem Let be a closed region bounded by piecewise smooth orientable surface and let (con
I Divergence theor
tinuous derivatives up to the first order
em (Gauss theorem)
)
VS v Cisin
r
ˆ If denotes the outward unit normal on
the ˆn V S
v dV n v dS
n S
nablasdot = sdotint intr r
MingsianMingsian R R BaiBai37
jS
jP
jV
S
0
ˆBy definition ( ) lim
ˆ( )
ˆ( )
j
j
j
j
Sj V
j
j j S
j j Sj j
n v dSv P
V
v P V n v dS
v P V n v dS
Δ rarr
⎧ ⎫sdot⎪ ⎪nabla sdot ⎨ ⎬Δ⎪ ⎪⎩ ⎭
nabla sdot Δ asymp sdot
nabla sdot Δ = sdot
int
intsum sumint
rr
r r
r r
Pf
MingsianMingsian R R BaiBai38
0
After interior surfaces cancel out only exterior surface (Why Opposite normal vectors)
ˆ( )
volumes exterior surfaces
ˆlim
QE
survive
j
j
j j Sj j
V SV
v p V n v dS
v dV n v dSΔ rarr
nabla sdot Δ = sdot
rArr nablasdot = sdot
sum sumint
int int
r r
r r
D
MingsianMingsian R R BaiBai39
( )
scalar variable constant vectorvector variable
Alternative formsLet be a an arbitrary and
a
ˆ
1 Le
(
ˆt
V
V S
S
V
v ap
v d
v dV vn d
V n v dS
v va
va dV v a v a
S
nablasdot = sdot
= rArr
nablasdot = nabla sdot + nabla
=
sdot
nablaint intint int
int
r
v
r r
r r
r r rQ ( )
( )
ˆ)
2 Let ˆV S
V
V S
dV n va dS
v a p
a p
p dV n p d
d p
S
V a
⎛ ⎞= sdot⎜ ⎟
⎝ ⎠
= times rArr
nablasdot times = sdot nabla
times
times
nabla = times
int int
int int
int
r
r r r
r r v
r
vQ
r
( ) ( ) ( )
( ) ( )
ˆ
ˆV S
V S
a p dV n a p dS
a p dV a n p dS
⎛ ⎞⎡ ⎤minus sdot nablatimes = sdot times⎣ ⎦⎜ ⎟⎜ ⎟⎜ ⎟rArr minus sdot nablatimes = minus sdot times⎜ ⎟⎝ ⎠
int int
int int
r rv v
r rv v
MingsianMingsian R R BaiBai40
( )
( )
From Leibnitzs rule
(1-D) ( )
( ) ( )
B t
A t
d f x t dxdt
f x t dx B ttpart prime= +part
int( )
( )( ( ) ) ( )
B t
A tf B t t A tprimeminusint ( ( ) )
(assume ( ) ( ) )
f A t t
A t B t const= =
ContinuityEx The Equation
rate of increase of mass
mass densityV
dM d dVdt dt
ρ
ρ
= int
flow velocityvv
MingsianMingsian R R BaiBai41
ˆ ˆ ( )
(
Conservation
) ( Divergence theorem)
of mass
( ) 0
V S S
V
V
dV n v dS n v dSt
v dV
v dVt
ρ ρ ρ
ρ
ρ ρ⎡ ⎤⎢ ⎥⎣ ⎦
part = minus sdot = minus sdotpart
= minus nablasdot
partrArr +nablasdot =part
int int int
int
int
v v
vQ
v
ˆOn the other hand rate influof mass through n ox i t
Extended to 3-D for time-independent fixed boundary
(3-D) ( ) ( )
Thus
S
V V
V
S V n v dS
Sd f x y z t dV f x y z t dVdt tdM dVdt t
ρ
ρ
= minus sdot
part=part
part=part
int
int int
intv
Fixed control volume (Eulerian coordinates)Otherwise Raynoldrsquos transport theoremMoving control volume Lagrangian coordinates
MingsianMingsian R R BaiBai42
But the control volime is only arbitrary
In particular for flow
( ) 0
con
(continuity equati
stant
on)
Divergence the
0 incompress
Note is t
ible
orem h
V
v
vt
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
e mother of the other integral theorems
MingsianMingsian R R BaiBai43
Ex Archimedes principle ( ) The buoyant force ( ) on a body with volume immersed in a fluid is equal to the weight of the fluid
ˆ displaced by the object ie In the followin
B
V
gVρ= minusf k
阿基米德原理
浮力
ˆg figure the ˆ with and being the fluid density gravitational
acceleration depth and outward normal unit vector to the surface
fhydraulic pressure g z
g z
S
ρ
ρ
= minusp n
n
1
the modified divergence theorem
ˆ
Hint
scalar fuctionV S
vdV v dS v Cnabla = isinint int n
MingsianMingsian R R BaiBai44
Levelsurfacefluid
V
S
airyx
z
ˆf g zρ= minusp n
ˆPf ( ) ( )
ˆ ˆ ˆ ˆ ˆ (0 0 ) v v
B fS S V
V V
g z g zdS dS dV
g dV g dV gV
ρ ρ
ρ ρ ρ
= = minus = minus nabla
= minus + + = minus = minus
int int int
int int
f p n
i j k k k
k
MingsianMingsian R R BaiBai45
Note This can also be thought of as simple balance of forces (resultant force that acts on and weight of filled with fluid)S V
2
~ a little history about George Green Recall the diveregence theorem
ˆ
Let
II th
e Green
are arbitrary fun
s identiti
ctions
es
V S
v dV n v dS
v u vu v C
nablasdot = sdot
= nablaisin
int intv v
v
ˆ ( ) ( )
ˆ ( )V S
V S
u v dV n u v dS
u v u v dV un v dS
nablasdot nabla = sdot nabla
nabla sdotnabla + nablasdotnabla = sdotnabla
int int
int int
MingsianMingsian R R BaiBai46
2 -( ) -(1)V S
vu v u v dV u dSnpart
nabla sdotnabla + nabla =partint int firs(Greens identt ity)
2
If
( ) --(2)
(1) (2) V S
u vuv u v u dV v dSn
harrpart
nabla sdotnabla + nabla =part
minus
int int
( )2 2
V S
dv duu v v u dV u v dSdn dn
⎛ ⎞nabla minus nabla = minus⎜ ⎟⎝ ⎠int int
second(Greens identity)
MingsianMingsian R R BaiBai47
( )
( )
( ) ( )
1st id ( )
Greens identities in the 1D case integration by parts
b b bb ba a
a a ab
ba
ab
ba
a
v udx v u v u dx u v uv dx uv
u v u v vu dx vu
uv vu dx uv vu
primeprime prime prime prime prime prime primeprime prime= minus rArr + =
prime prime primeprime primeharr + =
primeprime primeprime prime primerArr minus = minus
int int int
int
2
2nd id ( )
Good for any adjoint operators in addition to Good also for and integration by parts Important for theory amp reciprocity
From th
Gree
e 1s
2-
t
ns functi
identity
D 1-
tak 1
D
e
on
u
nablararr
=
int
2
V S
vv dV dSnpartnabla =part
rArr int int2Ex 0 st 0 on the boundary 0 ( 1st id u u unabla = = rArr equiv 以 証)
MingsianMingsian R R BaiBai48
1
Thm1 Stokes theorem Let be defined in Let be a piecewise-smooth orientable surface bounded by a piec
III Stokes theorem and
ewise-smooth simple clos
Greens
ed curve Then
theore
m
v C R S
C
isinv
( )
ˆwhere is a unit normal to according t right hand rulo the e
ˆS C
n S
n v dS v dRsdot nablatimes = sdotint intvv v
(not necessarily planar)S
Cn vnablatimes v vv
MingsianMingsian R R BaiBai49
Pf
( )( )
circulatioWe have shown in deriving thatthe in -direction
ˆor
ˆ
n
z
z
j j j j j
j j j j jj j
vz v dx dy
v dR n v S
v dR n v S
nablatimes
=nablatimes
sdot = sdot nablatimes Δ
sdot = sdot nablatimes Δ
intsum sumint
v
v
vv v
vv v
C
S
jS
jC
jP
ˆ jn
jSΔ
MingsianMingsian R R BaiBai50
( )
( )ext surface
After the interior boundaries cancel out only the exterior boundaries (Why)
ˆ
ˆ0 QED
Alternative forms scalar
survive
va
j
j j j j jC
jC S
v dR n v S
S v dR n v dS
φ
sdot = sdot nablatimes Δ
Δ rarr rArr sdot = sdot nablatimes
sum sumint
int int
vv v
vv v
( )
riable constant vector vector variable
1 Let
2 Let
ˆ
ˆ S C
S C
a p
v n dR dR
n p dS dR p
a
v a p
φ φφ= rArr timesnabla =
timesnabla times= rArr timestimes =
int int
int int
v v
v v
v
v v
vv vv v
MingsianMingsian R R BaiBai51
( ) ( )
( )
( )
( ) ( )
1
Thm2 Greens theorem ( planar version of Stokes theorem)ˆ ˆ Let
pf By Stokes theoremˆ
ˆ ˆ ˆ ˆ ˆ ˆ
S C
S C
S C
v P x y i Q x y j C
n v dS v dR
k Pi Qj dS Pi Qj dx i dy
Q P dS Pdx Qdyx y
⎛ ⎞part partminus = +⎜ ⎟
= + isin
sdot nablatimes = sdot
⎡ ⎤sdot nablatimes + =
part part⎝ ⎠
+ sdot +⎣ ⎦
int int
int
int
int
int
v
vv v
( )
( )
( )
ˆ
ˆˆ ˆ
ˆˆ ˆ ˆ ˆ( ) ( )
0
QEDS C
j
i j kQ PPi Qj i j k
x y z x yP Q
Q P dS Pdx Qdyx y
⎛ ⎞part part part part partnablatimes + = = + + minus⎜ ⎟part part part part part⎝ ⎠
⎛ ⎞part partrArr minus = +⎜ ⎟part part⎝ ⎠int int
L L
S
C
i
j
x
y
MingsianMingsian R R BaiBai52
Potential TheorypotentiGeorge Green coined the term (l
a )
rarr 位能 勢能
例子 重力場 電磁場 流場 聲場 彈性力學
1
2
Thm1 If is in a simply connected region then the following statements are equivalent(i) There exists a scala
Conservative Field
potential functior such that
(ii
n
) 0
v
C
C
v
v R
φφ=nabla
nablatimes
isin
isin
=r
r
r
r in (Irrotational field curl-free)
cf 0 st solenoidal field divergence-free
0
(iii) in (Conservative field)C
Rv w v w
v R Rd
nablasdot = exist nabla
sdot =
= times
int
r r r
r
r
r
MingsianMingsian R R BaiBai53
(iv) is connecting and
independent o
Ex Think
f the pa
of dynamics is the force
thB
A
v dR A B
v
sdotintrr
r
1 2
1
0
Pf
(i) (ii) ( ) 0 (zero identity cg rule) irrotational
ˆ(ii) (iii) ( ) 0 (Stokes theorem)
(iii) (iv) ( ) 0
C S
C C C
C
v
v dR n v dS
v dR v dR
v dR
φ
=
rarr nablatimes = nablatimes nabla =
rarr sdot = sdot nablatimes =
rarr sdot = minus sdot =
rArr sdot =
int int
int int int
int
r
rr
rr r
r rr r
rr r
2
independent of pathC
v dRsdotintr
A
B
1C
2C
MingsianMingsian R R BaiBai54
1 2
(iv) (i) constant ( ) ( )
--(1)
But (2)
(1) (2) QED
Note on application
Check
B
C C A
v dR v dR B A d
v dR d
d dx dy dz dRx y z
v dR dR v R
φ φ φ
φφ φ φφ φ
φ φ
rarr sdot = sdot = = minus =
rArr sdot =part part part
= + + = nabla sdot minus minus minus minus minuspart part part
sdot = nabla sdot rArr = nabla forall
int int intr rr r
rr
r
r r rr r
use simpler path (iv)if 0
find (i)v
φ⎧
nablatimes = ⎨⎩
rr
A
B
x
y
Cz
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai10
Ex Plane equationˆˆ ˆ( ) Normal vector
Ex 0 is the necessary condition for searching extrema (from multivariate Tylor expansion)
Ex Revisit of the exampl
u x y z ax by cz d u ai bj ck
u
= + + = rArr nabla = + +
nabla =v
多變數函數求極值
2 21 1 2 2
e in Linear Algebra using method of What is the maximum and minimum distance of the origin to the following curve
Lagrange multipli s
6 8
er
5 5x x x x+ + =
MingsianMingsian R R BaiBai11
1 2
2 21 2( )
2 21 1 2 2
Sol
This problem can be posed as a problemmin max ( )
constrained optimization
5 6 5 8 0 ( )which can
objectivecost function
constraint
equatioals
nx x
J x x
st g x x x x
= +
= + + minus =
2 2 2 21 2 1 1 2 2
o be converted into an unconstrained optimization problem by defining the ( ) (5 6 5 8)where is referred to as the thatLa
L
g ora
agr
nge ften h multi
angi
plie as
an
r ph
L x x x x x xλλ
= + + + + minus
ysical meaning in certain problemsThus the extreme values of will occur wh en or L J gJ nabla = nabla nabla0
MingsianMingsian R R BaiBai12
1 1 21
2 1 22
2 21 1 2 2
1 2 1 2
2 (10 6 ) 0 (1)
2 (6 10 ) 0 (2)
5 6 5 8 0 (constraint)
(1) (2) (2 16 )( ) 0 18 or (3)Substituting Eq (3) into Eqs (1) and (2) l
L x x xxL x x xxL x x x x
x x x x
λ
λ
λλ λ
part = + + =partpart = + + =partpart = + + minus =part+ + + = rArr = minus = minus
1 2 1 2eads to
and which corresponds to 14Thus minimummaximum distance 12x x x x J= = minus =
=
MingsianMingsian R R BaiBai13
22 2
1
1 1
2 2
The result above is identical to the previously obtained result in Linear Algebr
ellipse
45 rotatio
a
1 ( )4
1 11 1
n12
xx
x xx x
deg⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠
prime+ =prime
primeminus= rArr
prime2x prime
2x
1x prime
1x
2
MingsianMingsian R R BaiBai14
Curl 旋度
curl curl In Cartesian coordinates
spatial operator indep
ˆˆ ˆ
ˆˆ ˆ( )
ende
(
nt of coordi
) (
D
nates
ef
)
x y z
y x x xz z
i j k
vx y z
v v v
v v v vv vi j ky z x z x y
v v
part part partnablatimes =part part part
part part part partpart part= minus minus minus + minuspart
times
part part part part part
nabla
rarr
r
r r
0vnablatimes =rr 0vnablatimes ne
rr
MingsianMingsian R R BaiBai15
circulaPhysica
tion o spin ol interpretati
f fluidr or vortion
y
citΓ
( )0 0x dx y dy+ +
( )0 0x y ( )0 0x dx y+
Γ
1
2
3
4
( )0 0x y dy+
MingsianMingsian R R BaiBai16
yy
vv dy
ypart
+part
Γdyminus
dyyvv x
x partpart
+
dy
dx
dxminus
y yy
v vv dx dy
x ypart part
+ +part part
x xx
v vv dx dyx y
part part+ +part part
yy
vv dx
xpart
+part
xx
vv dxx
part+part
xv
yv
vv
dλ paddle weel
MingsianMingsian R R BaiBai17
contour
1 2 3 4
circulation
x x y y x x y y
x
v d
v d v d v d v d
v
λ
λ λ λ λrarr
Γ sdot
= + + +
asymp
intint int int int
vv
12
xv dxx
part+part ydx v
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
+ 12
y yv vdx dy
x ypart part
+ +part part
x
dy
v
⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦
+
12
xv dxx
part+part
( )xy
v dy dx vy
⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦
part+ minus +part
12
yvdy
ypart
+part
( )
( )
Similar process can be applied to the circulation in the and directionsThus the total circulation per unit area i curl s
y xz
dy
v v dxdy v dxdyx y
xv v
y
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎝ ⎠
minus
part part= minus = nablatimespart part
=nablatimes
v
v v
MingsianMingsian R R BaiBai18
Identities of differential operators ( p380 Greenberg )
( )
( ) ( )( )( )
( ) ( )( ) is required
Example of deriv
ˆ ˆˆ ˆ ˆ ˆ
ation
x y z
x y z
v v vx y z
uv u v u v
uv i j k uv i uv j uv kx y z
uv u v u v
u v v u u v
u v u vv v u u
+ +part part partpart part part
⎛ ⎞⎜ ⎟⎝ ⎠
nablasdot = nabla sdot + nablasdot
part part partnablasdot = + + sdot + + =part part part
nablatimes =nabla times + nablatimes
nablasdot times = sdotnablatimes minus sdotnablatimes
nablatimes times = nabla minus + minusnablanablasdot sdotsdot678 678
v
v v v
vL
v v v
v v v v v v
v v v v v v v v( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )2 cf
x y zu u ux y z
A B C A
u v
v w v w w v v w w v
v v v C B A B C
+ +part part partpart part part
sdotnabla
nabla sdot = sdotnabla + sdotnabla + times nablatimes + times
times
nablatimes
nablatimes nablatimes =nabla nabla times = sdot minussdot sdotminusnabla
678v
v v v
v v v v v vv v
v v v v v v
v v
v
v v
ˆ ˆˆ ˆ ˆ ˆ( ) ( ) ( )x y z x y zv i v j v k i j k u v v v ux y z x y zpart part part part part part+ + sdot + + = + +part part part part part part
v v
MingsianMingsian R R BaiBai19
2 2 22
2 2 2
0 (dc direct current) solenoidal0 (cg center of gravity) conservative potent
Laplacia
Zero identitie
n
s
ial
x y z
vu
⎧⎪⎨⎪⎩
part part partΔ =nabla nablasdotnabla = + +part part part
nablasdotnablatimes = rarr
nablatimesnabla = rarr
v
v
These differential operators will take more complicated and irregular form for the other coordinate systems
MingsianMingsian R R BaiBai20
2 2 3
2 2
ˆˆ ˆEx 2
ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ 2 0 2
2 0 0 2yx z
u x y v x yi z j k
u i j k u xyi x j k xyi x jx y z
vv vv xy xyx y z
⎛ ⎞⎜ ⎟⎝ ⎠
= = minus +
part part partnabla = + + = + + = +part part part
partpart partnablasdot = + + = + minus =part part part
v
v
MingsianMingsian R R BaiBai21
( ) ( ) ( )
( ) ( )
2 2 2 2
2 2 22
2 2 2
2
ˆˆ ˆ
ˆˆ ˆ
ˆ ˆˆ ˆ ˆ 0 3 0 0 0 3
2 0 0 2
ˆˆ ˆ
y yx xz z
x y z
i j kv vv vv vv i j k
x y z y z x z x yv v v
z i j x k z i x k
u u uu y yx y zv v v
i j kx y z
⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠ ⎝ ⎠
⎛
⎝
part partpart partpart partpart part partnablatimes = = minus minus minus + minuspart part part part part part part part part
= + minus minus + minus = minus
part part partnabla = + + = + + =part part part
nablatimes nablatimes =nabla nablasdot minusnabla
part part part= + +part part part
v
v v v
( ) ( )( ) ( ) ( ) ( )
( )
2 2 22 3
2 2 2ˆˆ ˆ2 2
ˆ ˆˆ ˆ ˆ ˆ 2 2 0 2 0 0 0 0 6 0 0 0
ˆ 2 6
xy x yi z j kx y z
yi xj k y i z j k
x z j
⎛ ⎞⎞⎜ ⎟⎜ ⎟
⎠ ⎝ ⎠⎡ ⎤⎣ ⎦
part part partminus + + minus +part part part
= + + minus + + + + minus + + minus
= +
MingsianMingsian R R BaiBai22
( )v w r w r r wnablatimes =nablatimes times = nablasdot minus nablasdotv v v v v v v ( )r w+ sdotnablav v ( ) constant w
w rminus sdotnablav
v v
Q
Ex Rigid body rotation
MingsianMingsian R R BaiBai23
( )( ) ( )
( ) ( )
ˆˆ ˆ
ˆ ˆˆ ˆ ˆ ˆ 1 1 1 3
ˆˆ ˆ
ˆˆ ˆ
Thus 3 2 is linked directly
x y z
x y z
r xi yj zk
r i j k xi yj zkx y z
w r w w w xi yj zkx y z
w i w j w k w
w r w r w r w w wv
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎝ ⎠
= + +
part part partnablasdot = + + + + = + + =part part part
part part partsdotnabla = + + + +part part part
= + + =
nablatimes times == nablasdot minus sdotnabla = minus =rArrnablatimes
v
v
v v
v
v v v v v v v v v
v spinn to ing
MingsianMingsian R R BaiBai24
Line surface and volume integrals
x
yC
B
A
isΔ
( )ii ηξ
Piecewise smooth curve
Line integral
MingsianMingsian R R BaiBai25
0 1lim ( ) ( ) or ( )
Intepretation of a line integral as an area ( )
i
Bn
i i isA
C
i C
F s F x y ds F x y d
x
s
F y ds
ξ ηΔ rarr =
Δ =sum int
int
int
x
y
z
C
( )z F x y=
MingsianMingsian R R BaiBai26
the area is always on the left of an obs
Positive dierver trave
rection of irsing on the
ntegrationcurve
RC
CCW CW
If is a regular closed curve piecewise smo
( ) ( )
(contou
oth simple ( )
r integral)
x y zC C
x y zC
F dR F F F dx dy dz
F dx F dy
C
F dz
sdot = + + + +
= + +
lowast
int int
int
i j k i j kv v
無交叉
work dW F dR= sdotv v
MingsianMingsian R R BaiBai27
0 1
Surface integral
lim ( ) ( )i
n
i i i iS i S
F S F x y z dSξ η ζΔ rarr =
Δ =sum intint( ) i i iξ η ζ
iSΔ
0 1
mass density
lim ( ) ( )
Ex
Volume integral
( )
i
n
i i i iV i V
V
F V F x y z dV
M x y z dV
ξ η ζ
ρ
Δ rarr =Δ =
=
sum intintint
intintint( ) i i iξ η ζ
iVΔ
MingsianMingsian R R BaiBai28
Ref (1)Hildebrand advanced Calculus for application Prentice-Hall 1976 p306-313(2) Greenberg p383-389 (3) Arfken ch2
1 2 3
1 2 3
1 2 3
1 2 3
(
Orthogonal coordinate systems base vectors are not constant in s
) ( )( )( )( )
ˆˆ ˆ
pacep x y z p u u ux x u u uy y u u uz z u u u
r xi yj zk
primerarr
= +
rarr
===
+v
1uv
2uv3uv
1u
2u
3uP coord curve
xy
z
O
X Curvilinear Coordinates
MingsianMingsian R R BaiBai29
1
11
1 1 1
1
1
1 1 1
11 1
1 1
2 2 2 3 3 3
Tangent vector to the at
where is the arc length
1
where unit vector
Similarly
curv
e
u psr rU
u s us
rs
U h uds rh udu sU h u U h u
partpart part= =part part part
part =part
there4 =part= =part
= =
v vv
vQ
v v
vv
v vv v
1u cood curve
1uv
1du 1dS
drv
P
MingsianMingsian R R BaiBai30
3 31 2 1 21 2 3
1 2 32
curv
In summary scale factor
If is the arc length along a in any directione
k k k
kk k
k k
k k k
U h u
ds rh Udu u
ds h dus
du dudu du du dudr r r r U U Uds u ds u ds u ds ds ds ds
dr drds ds
=
part= = =part
=
part part part= + + = + +part part part
= sdot
v v
v v
v v v v v v v
v v 3 3
1 1
23 3 32 2
21 1 1
2 2 2 2 2 2 2 2 2 21 1 2 2 3 3 1 2 3
1 (for orthogonal coord)
jii j
i j
ji ii j i
i j i
dududr U Uds ds ds
dudu duU U hds ds ds
ds h du h du h du ds ds ds
= =
= = =
⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
= sdot
rarr = sdot =
rarr = + + = + +
sum sum
sumsum sum
v v v
v v
MingsianMingsian R R BaiBai31
1 1 2 2 3 3
1 2 3
are mutually vectors having with arc length
k k k k kds h du U du
U du U du U duds ds ds
= =
rArr perp
v
v v v
1uv
2uv
3uv
1dS
2dS
3dS
dS
MingsianMingsian R R BaiBai32
( ) ( )
( ) ( )
1 1 2 2 3 3 1 2 3 1 2 3 1 2 3
1 2 3 1 2 3
1
1 2 2 3 3 2 2 2 3 3 3 2 3 2 3 1
Differential volume
Differential On const coord surface
area
Gradi t
en
d U du U du U du u u u h h h du du du
d h h h du du du
ud U du U du h u du h u du h h du du u
df
τ
τ
σ
= times sdot = times sdot
=
=
= times = times =
= nabla
v v v v v v
v vv v v v
1 2 31 2 3
1 2 3 1 1 2 2 3 31 2 3
1 1 1 2 2 2 3 3 3
f f ff dr df du du duu u u
r r rdr du du du U du U du U duu u u
h u du h u du h u du
part part partsdot rarr = + +part part part
part part part= + + = + +part part part
= + +
v
v v v v v vv
v v v
=1 (right hand rule)
MingsianMingsian R R BaiBai33
Rectangular Cartesian Coordinates
ij
k
MingsianMingsian R R BaiBai34
Circular Cylindrical Coordinatescossin
xyz z
ρ ϕρ ϕ
===
Ref Arfken
( )P zρ ϕ
MingsianMingsian R R BaiBai35
Polar Spherical Coordinatessin cossin sincos
x ry rz r
θ ϕθ ϕθ
===
Ref Arfken
0
0
0
sin cos sin sin coscos cos cos sin sin
sin cos
θ ϕ θ ϕ θθ ϕ θ ϕ θϕ ϕ
= + +
= + minus= minus +
r i j kθ i j kφ i j
2
2
sinsin
dS r d ddV r dr d d
θ θ ϕ
θ θ ϕ
=
=( )P r θ ϕ
MingsianMingsian R R BaiBai36
Integral TheoremsHeart of this chapter
Interchange of dimensionality Div Curl Grad in integrals High-dimensional integration
rarr
1
Thm1 Divergence theorem Let be a closed region bounded by piecewise smooth orientable surface and let (con
I Divergence theor
tinuous derivatives up to the first order
em (Gauss theorem)
)
VS v Cisin
r
ˆ If denotes the outward unit normal on
the ˆn V S
v dV n v dS
n S
nablasdot = sdotint intr r
MingsianMingsian R R BaiBai37
jS
jP
jV
S
0
ˆBy definition ( ) lim
ˆ( )
ˆ( )
j
j
j
j
Sj V
j
j j S
j j Sj j
n v dSv P
V
v P V n v dS
v P V n v dS
Δ rarr
⎧ ⎫sdot⎪ ⎪nabla sdot ⎨ ⎬Δ⎪ ⎪⎩ ⎭
nabla sdot Δ asymp sdot
nabla sdot Δ = sdot
int
intsum sumint
rr
r r
r r
Pf
MingsianMingsian R R BaiBai38
0
After interior surfaces cancel out only exterior surface (Why Opposite normal vectors)
ˆ( )
volumes exterior surfaces
ˆlim
QE
survive
j
j
j j Sj j
V SV
v p V n v dS
v dV n v dSΔ rarr
nabla sdot Δ = sdot
rArr nablasdot = sdot
sum sumint
int int
r r
r r
D
MingsianMingsian R R BaiBai39
( )
scalar variable constant vectorvector variable
Alternative formsLet be a an arbitrary and
a
ˆ
1 Le
(
ˆt
V
V S
S
V
v ap
v d
v dV vn d
V n v dS
v va
va dV v a v a
S
nablasdot = sdot
= rArr
nablasdot = nabla sdot + nabla
=
sdot
nablaint intint int
int
r
v
r r
r r
r r rQ ( )
( )
ˆ)
2 Let ˆV S
V
V S
dV n va dS
v a p
a p
p dV n p d
d p
S
V a
⎛ ⎞= sdot⎜ ⎟
⎝ ⎠
= times rArr
nablasdot times = sdot nabla
times
times
nabla = times
int int
int int
int
r
r r r
r r v
r
vQ
r
( ) ( ) ( )
( ) ( )
ˆ
ˆV S
V S
a p dV n a p dS
a p dV a n p dS
⎛ ⎞⎡ ⎤minus sdot nablatimes = sdot times⎣ ⎦⎜ ⎟⎜ ⎟⎜ ⎟rArr minus sdot nablatimes = minus sdot times⎜ ⎟⎝ ⎠
int int
int int
r rv v
r rv v
MingsianMingsian R R BaiBai40
( )
( )
From Leibnitzs rule
(1-D) ( )
( ) ( )
B t
A t
d f x t dxdt
f x t dx B ttpart prime= +part
int( )
( )( ( ) ) ( )
B t
A tf B t t A tprimeminusint ( ( ) )
(assume ( ) ( ) )
f A t t
A t B t const= =
ContinuityEx The Equation
rate of increase of mass
mass densityV
dM d dVdt dt
ρ
ρ
= int
flow velocityvv
MingsianMingsian R R BaiBai41
ˆ ˆ ( )
(
Conservation
) ( Divergence theorem)
of mass
( ) 0
V S S
V
V
dV n v dS n v dSt
v dV
v dVt
ρ ρ ρ
ρ
ρ ρ⎡ ⎤⎢ ⎥⎣ ⎦
part = minus sdot = minus sdotpart
= minus nablasdot
partrArr +nablasdot =part
int int int
int
int
v v
vQ
v
ˆOn the other hand rate influof mass through n ox i t
Extended to 3-D for time-independent fixed boundary
(3-D) ( ) ( )
Thus
S
V V
V
S V n v dS
Sd f x y z t dV f x y z t dVdt tdM dVdt t
ρ
ρ
= minus sdot
part=part
part=part
int
int int
intv
Fixed control volume (Eulerian coordinates)Otherwise Raynoldrsquos transport theoremMoving control volume Lagrangian coordinates
MingsianMingsian R R BaiBai42
But the control volime is only arbitrary
In particular for flow
( ) 0
con
(continuity equati
stant
on)
Divergence the
0 incompress
Note is t
ible
orem h
V
v
vt
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
e mother of the other integral theorems
MingsianMingsian R R BaiBai43
Ex Archimedes principle ( ) The buoyant force ( ) on a body with volume immersed in a fluid is equal to the weight of the fluid
ˆ displaced by the object ie In the followin
B
V
gVρ= minusf k
阿基米德原理
浮力
ˆg figure the ˆ with and being the fluid density gravitational
acceleration depth and outward normal unit vector to the surface
fhydraulic pressure g z
g z
S
ρ
ρ
= minusp n
n
1
the modified divergence theorem
ˆ
Hint
scalar fuctionV S
vdV v dS v Cnabla = isinint int n
MingsianMingsian R R BaiBai44
Levelsurfacefluid
V
S
airyx
z
ˆf g zρ= minusp n
ˆPf ( ) ( )
ˆ ˆ ˆ ˆ ˆ (0 0 ) v v
B fS S V
V V
g z g zdS dS dV
g dV g dV gV
ρ ρ
ρ ρ ρ
= = minus = minus nabla
= minus + + = minus = minus
int int int
int int
f p n
i j k k k
k
MingsianMingsian R R BaiBai45
Note This can also be thought of as simple balance of forces (resultant force that acts on and weight of filled with fluid)S V
2
~ a little history about George Green Recall the diveregence theorem
ˆ
Let
II th
e Green
are arbitrary fun
s identiti
ctions
es
V S
v dV n v dS
v u vu v C
nablasdot = sdot
= nablaisin
int intv v
v
ˆ ( ) ( )
ˆ ( )V S
V S
u v dV n u v dS
u v u v dV un v dS
nablasdot nabla = sdot nabla
nabla sdotnabla + nablasdotnabla = sdotnabla
int int
int int
MingsianMingsian R R BaiBai46
2 -( ) -(1)V S
vu v u v dV u dSnpart
nabla sdotnabla + nabla =partint int firs(Greens identt ity)
2
If
( ) --(2)
(1) (2) V S
u vuv u v u dV v dSn
harrpart
nabla sdotnabla + nabla =part
minus
int int
( )2 2
V S
dv duu v v u dV u v dSdn dn
⎛ ⎞nabla minus nabla = minus⎜ ⎟⎝ ⎠int int
second(Greens identity)
MingsianMingsian R R BaiBai47
( )
( )
( ) ( )
1st id ( )
Greens identities in the 1D case integration by parts
b b bb ba a
a a ab
ba
ab
ba
a
v udx v u v u dx u v uv dx uv
u v u v vu dx vu
uv vu dx uv vu
primeprime prime prime prime prime prime primeprime prime= minus rArr + =
prime prime primeprime primeharr + =
primeprime primeprime prime primerArr minus = minus
int int int
int
2
2nd id ( )
Good for any adjoint operators in addition to Good also for and integration by parts Important for theory amp reciprocity
From th
Gree
e 1s
2-
t
ns functi
identity
D 1-
tak 1
D
e
on
u
nablararr
=
int
2
V S
vv dV dSnpartnabla =part
rArr int int2Ex 0 st 0 on the boundary 0 ( 1st id u u unabla = = rArr equiv 以 証)
MingsianMingsian R R BaiBai48
1
Thm1 Stokes theorem Let be defined in Let be a piecewise-smooth orientable surface bounded by a piec
III Stokes theorem and
ewise-smooth simple clos
Greens
ed curve Then
theore
m
v C R S
C
isinv
( )
ˆwhere is a unit normal to according t right hand rulo the e
ˆS C
n S
n v dS v dRsdot nablatimes = sdotint intvv v
(not necessarily planar)S
Cn vnablatimes v vv
MingsianMingsian R R BaiBai49
Pf
( )( )
circulatioWe have shown in deriving thatthe in -direction
ˆor
ˆ
n
z
z
j j j j j
j j j j jj j
vz v dx dy
v dR n v S
v dR n v S
nablatimes
=nablatimes
sdot = sdot nablatimes Δ
sdot = sdot nablatimes Δ
intsum sumint
v
v
vv v
vv v
C
S
jS
jC
jP
ˆ jn
jSΔ
MingsianMingsian R R BaiBai50
( )
( )ext surface
After the interior boundaries cancel out only the exterior boundaries (Why)
ˆ
ˆ0 QED
Alternative forms scalar
survive
va
j
j j j j jC
jC S
v dR n v S
S v dR n v dS
φ
sdot = sdot nablatimes Δ
Δ rarr rArr sdot = sdot nablatimes
sum sumint
int int
vv v
vv v
( )
riable constant vector vector variable
1 Let
2 Let
ˆ
ˆ S C
S C
a p
v n dR dR
n p dS dR p
a
v a p
φ φφ= rArr timesnabla =
timesnabla times= rArr timestimes =
int int
int int
v v
v v
v
v v
vv vv v
MingsianMingsian R R BaiBai51
( ) ( )
( )
( )
( ) ( )
1
Thm2 Greens theorem ( planar version of Stokes theorem)ˆ ˆ Let
pf By Stokes theoremˆ
ˆ ˆ ˆ ˆ ˆ ˆ
S C
S C
S C
v P x y i Q x y j C
n v dS v dR
k Pi Qj dS Pi Qj dx i dy
Q P dS Pdx Qdyx y
⎛ ⎞part partminus = +⎜ ⎟
= + isin
sdot nablatimes = sdot
⎡ ⎤sdot nablatimes + =
part part⎝ ⎠
+ sdot +⎣ ⎦
int int
int
int
int
int
v
vv v
( )
( )
( )
ˆ
ˆˆ ˆ
ˆˆ ˆ ˆ ˆ( ) ( )
0
QEDS C
j
i j kQ PPi Qj i j k
x y z x yP Q
Q P dS Pdx Qdyx y
⎛ ⎞part part part part partnablatimes + = = + + minus⎜ ⎟part part part part part⎝ ⎠
⎛ ⎞part partrArr minus = +⎜ ⎟part part⎝ ⎠int int
L L
S
C
i
j
x
y
MingsianMingsian R R BaiBai52
Potential TheorypotentiGeorge Green coined the term (l
a )
rarr 位能 勢能
例子 重力場 電磁場 流場 聲場 彈性力學
1
2
Thm1 If is in a simply connected region then the following statements are equivalent(i) There exists a scala
Conservative Field
potential functior such that
(ii
n
) 0
v
C
C
v
v R
φφ=nabla
nablatimes
isin
isin
=r
r
r
r in (Irrotational field curl-free)
cf 0 st solenoidal field divergence-free
0
(iii) in (Conservative field)C
Rv w v w
v R Rd
nablasdot = exist nabla
sdot =
= times
int
r r r
r
r
r
MingsianMingsian R R BaiBai53
(iv) is connecting and
independent o
Ex Think
f the pa
of dynamics is the force
thB
A
v dR A B
v
sdotintrr
r
1 2
1
0
Pf
(i) (ii) ( ) 0 (zero identity cg rule) irrotational
ˆ(ii) (iii) ( ) 0 (Stokes theorem)
(iii) (iv) ( ) 0
C S
C C C
C
v
v dR n v dS
v dR v dR
v dR
φ
=
rarr nablatimes = nablatimes nabla =
rarr sdot = sdot nablatimes =
rarr sdot = minus sdot =
rArr sdot =
int int
int int int
int
r
rr
rr r
r rr r
rr r
2
independent of pathC
v dRsdotintr
A
B
1C
2C
MingsianMingsian R R BaiBai54
1 2
(iv) (i) constant ( ) ( )
--(1)
But (2)
(1) (2) QED
Note on application
Check
B
C C A
v dR v dR B A d
v dR d
d dx dy dz dRx y z
v dR dR v R
φ φ φ
φφ φ φφ φ
φ φ
rarr sdot = sdot = = minus =
rArr sdot =part part part
= + + = nabla sdot minus minus minus minus minuspart part part
sdot = nabla sdot rArr = nabla forall
int int intr rr r
rr
r
r r rr r
use simpler path (iv)if 0
find (i)v
φ⎧
nablatimes = ⎨⎩
rr
A
B
x
y
Cz
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai11
1 2
2 21 2( )
2 21 1 2 2
Sol
This problem can be posed as a problemmin max ( )
constrained optimization
5 6 5 8 0 ( )which can
objectivecost function
constraint
equatioals
nx x
J x x
st g x x x x
= +
= + + minus =
2 2 2 21 2 1 1 2 2
o be converted into an unconstrained optimization problem by defining the ( ) (5 6 5 8)where is referred to as the thatLa
L
g ora
agr
nge ften h multi
angi
plie as
an
r ph
L x x x x x xλλ
= + + + + minus
ysical meaning in certain problemsThus the extreme values of will occur wh en or L J gJ nabla = nabla nabla0
MingsianMingsian R R BaiBai12
1 1 21
2 1 22
2 21 1 2 2
1 2 1 2
2 (10 6 ) 0 (1)
2 (6 10 ) 0 (2)
5 6 5 8 0 (constraint)
(1) (2) (2 16 )( ) 0 18 or (3)Substituting Eq (3) into Eqs (1) and (2) l
L x x xxL x x xxL x x x x
x x x x
λ
λ
λλ λ
part = + + =partpart = + + =partpart = + + minus =part+ + + = rArr = minus = minus
1 2 1 2eads to
and which corresponds to 14Thus minimummaximum distance 12x x x x J= = minus =
=
MingsianMingsian R R BaiBai13
22 2
1
1 1
2 2
The result above is identical to the previously obtained result in Linear Algebr
ellipse
45 rotatio
a
1 ( )4
1 11 1
n12
xx
x xx x
deg⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠
prime+ =prime
primeminus= rArr
prime2x prime
2x
1x prime
1x
2
MingsianMingsian R R BaiBai14
Curl 旋度
curl curl In Cartesian coordinates
spatial operator indep
ˆˆ ˆ
ˆˆ ˆ( )
ende
(
nt of coordi
) (
D
nates
ef
)
x y z
y x x xz z
i j k
vx y z
v v v
v v v vv vi j ky z x z x y
v v
part part partnablatimes =part part part
part part part partpart part= minus minus minus + minuspart
times
part part part part part
nabla
rarr
r
r r
0vnablatimes =rr 0vnablatimes ne
rr
MingsianMingsian R R BaiBai15
circulaPhysica
tion o spin ol interpretati
f fluidr or vortion
y
citΓ
( )0 0x dx y dy+ +
( )0 0x y ( )0 0x dx y+
Γ
1
2
3
4
( )0 0x y dy+
MingsianMingsian R R BaiBai16
yy
vv dy
ypart
+part
Γdyminus
dyyvv x
x partpart
+
dy
dx
dxminus
y yy
v vv dx dy
x ypart part
+ +part part
x xx
v vv dx dyx y
part part+ +part part
yy
vv dx
xpart
+part
xx
vv dxx
part+part
xv
yv
vv
dλ paddle weel
MingsianMingsian R R BaiBai17
contour
1 2 3 4
circulation
x x y y x x y y
x
v d
v d v d v d v d
v
λ
λ λ λ λrarr
Γ sdot
= + + +
asymp
intint int int int
vv
12
xv dxx
part+part ydx v
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
+ 12
y yv vdx dy
x ypart part
+ +part part
x
dy
v
⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦
+
12
xv dxx
part+part
( )xy
v dy dx vy
⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦
part+ minus +part
12
yvdy
ypart
+part
( )
( )
Similar process can be applied to the circulation in the and directionsThus the total circulation per unit area i curl s
y xz
dy
v v dxdy v dxdyx y
xv v
y
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎝ ⎠
minus
part part= minus = nablatimespart part
=nablatimes
v
v v
MingsianMingsian R R BaiBai18
Identities of differential operators ( p380 Greenberg )
( )
( ) ( )( )( )
( ) ( )( ) is required
Example of deriv
ˆ ˆˆ ˆ ˆ ˆ
ation
x y z
x y z
v v vx y z
uv u v u v
uv i j k uv i uv j uv kx y z
uv u v u v
u v v u u v
u v u vv v u u
+ +part part partpart part part
⎛ ⎞⎜ ⎟⎝ ⎠
nablasdot = nabla sdot + nablasdot
part part partnablasdot = + + sdot + + =part part part
nablatimes =nabla times + nablatimes
nablasdot times = sdotnablatimes minus sdotnablatimes
nablatimes times = nabla minus + minusnablanablasdot sdotsdot678 678
v
v v v
vL
v v v
v v v v v v
v v v v v v v v( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )2 cf
x y zu u ux y z
A B C A
u v
v w v w w v v w w v
v v v C B A B C
+ +part part partpart part part
sdotnabla
nabla sdot = sdotnabla + sdotnabla + times nablatimes + times
times
nablatimes
nablatimes nablatimes =nabla nabla times = sdot minussdot sdotminusnabla
678v
v v v
v v v v v vv v
v v v v v v
v v
v
v v
ˆ ˆˆ ˆ ˆ ˆ( ) ( ) ( )x y z x y zv i v j v k i j k u v v v ux y z x y zpart part part part part part+ + sdot + + = + +part part part part part part
v v
MingsianMingsian R R BaiBai19
2 2 22
2 2 2
0 (dc direct current) solenoidal0 (cg center of gravity) conservative potent
Laplacia
Zero identitie
n
s
ial
x y z
vu
⎧⎪⎨⎪⎩
part part partΔ =nabla nablasdotnabla = + +part part part
nablasdotnablatimes = rarr
nablatimesnabla = rarr
v
v
These differential operators will take more complicated and irregular form for the other coordinate systems
MingsianMingsian R R BaiBai20
2 2 3
2 2
ˆˆ ˆEx 2
ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ 2 0 2
2 0 0 2yx z
u x y v x yi z j k
u i j k u xyi x j k xyi x jx y z
vv vv xy xyx y z
⎛ ⎞⎜ ⎟⎝ ⎠
= = minus +
part part partnabla = + + = + + = +part part part
partpart partnablasdot = + + = + minus =part part part
v
v
MingsianMingsian R R BaiBai21
( ) ( ) ( )
( ) ( )
2 2 2 2
2 2 22
2 2 2
2
ˆˆ ˆ
ˆˆ ˆ
ˆ ˆˆ ˆ ˆ 0 3 0 0 0 3
2 0 0 2
ˆˆ ˆ
y yx xz z
x y z
i j kv vv vv vv i j k
x y z y z x z x yv v v
z i j x k z i x k
u u uu y yx y zv v v
i j kx y z
⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠ ⎝ ⎠
⎛
⎝
part partpart partpart partpart part partnablatimes = = minus minus minus + minuspart part part part part part part part part
= + minus minus + minus = minus
part part partnabla = + + = + + =part part part
nablatimes nablatimes =nabla nablasdot minusnabla
part part part= + +part part part
v
v v v
( ) ( )( ) ( ) ( ) ( )
( )
2 2 22 3
2 2 2ˆˆ ˆ2 2
ˆ ˆˆ ˆ ˆ ˆ 2 2 0 2 0 0 0 0 6 0 0 0
ˆ 2 6
xy x yi z j kx y z
yi xj k y i z j k
x z j
⎛ ⎞⎞⎜ ⎟⎜ ⎟
⎠ ⎝ ⎠⎡ ⎤⎣ ⎦
part part partminus + + minus +part part part
= + + minus + + + + minus + + minus
= +
MingsianMingsian R R BaiBai22
( )v w r w r r wnablatimes =nablatimes times = nablasdot minus nablasdotv v v v v v v ( )r w+ sdotnablav v ( ) constant w
w rminus sdotnablav
v v
Q
Ex Rigid body rotation
MingsianMingsian R R BaiBai23
( )( ) ( )
( ) ( )
ˆˆ ˆ
ˆ ˆˆ ˆ ˆ ˆ 1 1 1 3
ˆˆ ˆ
ˆˆ ˆ
Thus 3 2 is linked directly
x y z
x y z
r xi yj zk
r i j k xi yj zkx y z
w r w w w xi yj zkx y z
w i w j w k w
w r w r w r w w wv
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎝ ⎠
= + +
part part partnablasdot = + + + + = + + =part part part
part part partsdotnabla = + + + +part part part
= + + =
nablatimes times == nablasdot minus sdotnabla = minus =rArrnablatimes
v
v
v v
v
v v v v v v v v v
v spinn to ing
MingsianMingsian R R BaiBai24
Line surface and volume integrals
x
yC
B
A
isΔ
( )ii ηξ
Piecewise smooth curve
Line integral
MingsianMingsian R R BaiBai25
0 1lim ( ) ( ) or ( )
Intepretation of a line integral as an area ( )
i
Bn
i i isA
C
i C
F s F x y ds F x y d
x
s
F y ds
ξ ηΔ rarr =
Δ =sum int
int
int
x
y
z
C
( )z F x y=
MingsianMingsian R R BaiBai26
the area is always on the left of an obs
Positive dierver trave
rection of irsing on the
ntegrationcurve
RC
CCW CW
If is a regular closed curve piecewise smo
( ) ( )
(contou
oth simple ( )
r integral)
x y zC C
x y zC
F dR F F F dx dy dz
F dx F dy
C
F dz
sdot = + + + +
= + +
lowast
int int
int
i j k i j kv v
無交叉
work dW F dR= sdotv v
MingsianMingsian R R BaiBai27
0 1
Surface integral
lim ( ) ( )i
n
i i i iS i S
F S F x y z dSξ η ζΔ rarr =
Δ =sum intint( ) i i iξ η ζ
iSΔ
0 1
mass density
lim ( ) ( )
Ex
Volume integral
( )
i
n
i i i iV i V
V
F V F x y z dV
M x y z dV
ξ η ζ
ρ
Δ rarr =Δ =
=
sum intintint
intintint( ) i i iξ η ζ
iVΔ
MingsianMingsian R R BaiBai28
Ref (1)Hildebrand advanced Calculus for application Prentice-Hall 1976 p306-313(2) Greenberg p383-389 (3) Arfken ch2
1 2 3
1 2 3
1 2 3
1 2 3
(
Orthogonal coordinate systems base vectors are not constant in s
) ( )( )( )( )
ˆˆ ˆ
pacep x y z p u u ux x u u uy y u u uz z u u u
r xi yj zk
primerarr
= +
rarr
===
+v
1uv
2uv3uv
1u
2u
3uP coord curve
xy
z
O
X Curvilinear Coordinates
MingsianMingsian R R BaiBai29
1
11
1 1 1
1
1
1 1 1
11 1
1 1
2 2 2 3 3 3
Tangent vector to the at
where is the arc length
1
where unit vector
Similarly
curv
e
u psr rU
u s us
rs
U h uds rh udu sU h u U h u
partpart part= =part part part
part =part
there4 =part= =part
= =
v vv
vQ
v v
vv
v vv v
1u cood curve
1uv
1du 1dS
drv
P
MingsianMingsian R R BaiBai30
3 31 2 1 21 2 3
1 2 32
curv
In summary scale factor
If is the arc length along a in any directione
k k k
kk k
k k
k k k
U h u
ds rh Udu u
ds h dus
du dudu du du dudr r r r U U Uds u ds u ds u ds ds ds ds
dr drds ds
=
part= = =part
=
part part part= + + = + +part part part
= sdot
v v
v v
v v v v v v v
v v 3 3
1 1
23 3 32 2
21 1 1
2 2 2 2 2 2 2 2 2 21 1 2 2 3 3 1 2 3
1 (for orthogonal coord)
jii j
i j
ji ii j i
i j i
dududr U Uds ds ds
dudu duU U hds ds ds
ds h du h du h du ds ds ds
= =
= = =
⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
= sdot
rarr = sdot =
rarr = + + = + +
sum sum
sumsum sum
v v v
v v
MingsianMingsian R R BaiBai31
1 1 2 2 3 3
1 2 3
are mutually vectors having with arc length
k k k k kds h du U du
U du U du U duds ds ds
= =
rArr perp
v
v v v
1uv
2uv
3uv
1dS
2dS
3dS
dS
MingsianMingsian R R BaiBai32
( ) ( )
( ) ( )
1 1 2 2 3 3 1 2 3 1 2 3 1 2 3
1 2 3 1 2 3
1
1 2 2 3 3 2 2 2 3 3 3 2 3 2 3 1
Differential volume
Differential On const coord surface
area
Gradi t
en
d U du U du U du u u u h h h du du du
d h h h du du du
ud U du U du h u du h u du h h du du u
df
τ
τ
σ
= times sdot = times sdot
=
=
= times = times =
= nabla
v v v v v v
v vv v v v
1 2 31 2 3
1 2 3 1 1 2 2 3 31 2 3
1 1 1 2 2 2 3 3 3
f f ff dr df du du duu u u
r r rdr du du du U du U du U duu u u
h u du h u du h u du
part part partsdot rarr = + +part part part
part part part= + + = + +part part part
= + +
v
v v v v v vv
v v v
=1 (right hand rule)
MingsianMingsian R R BaiBai33
Rectangular Cartesian Coordinates
ij
k
MingsianMingsian R R BaiBai34
Circular Cylindrical Coordinatescossin
xyz z
ρ ϕρ ϕ
===
Ref Arfken
( )P zρ ϕ
MingsianMingsian R R BaiBai35
Polar Spherical Coordinatessin cossin sincos
x ry rz r
θ ϕθ ϕθ
===
Ref Arfken
0
0
0
sin cos sin sin coscos cos cos sin sin
sin cos
θ ϕ θ ϕ θθ ϕ θ ϕ θϕ ϕ
= + +
= + minus= minus +
r i j kθ i j kφ i j
2
2
sinsin
dS r d ddV r dr d d
θ θ ϕ
θ θ ϕ
=
=( )P r θ ϕ
MingsianMingsian R R BaiBai36
Integral TheoremsHeart of this chapter
Interchange of dimensionality Div Curl Grad in integrals High-dimensional integration
rarr
1
Thm1 Divergence theorem Let be a closed region bounded by piecewise smooth orientable surface and let (con
I Divergence theor
tinuous derivatives up to the first order
em (Gauss theorem)
)
VS v Cisin
r
ˆ If denotes the outward unit normal on
the ˆn V S
v dV n v dS
n S
nablasdot = sdotint intr r
MingsianMingsian R R BaiBai37
jS
jP
jV
S
0
ˆBy definition ( ) lim
ˆ( )
ˆ( )
j
j
j
j
Sj V
j
j j S
j j Sj j
n v dSv P
V
v P V n v dS
v P V n v dS
Δ rarr
⎧ ⎫sdot⎪ ⎪nabla sdot ⎨ ⎬Δ⎪ ⎪⎩ ⎭
nabla sdot Δ asymp sdot
nabla sdot Δ = sdot
int
intsum sumint
rr
r r
r r
Pf
MingsianMingsian R R BaiBai38
0
After interior surfaces cancel out only exterior surface (Why Opposite normal vectors)
ˆ( )
volumes exterior surfaces
ˆlim
QE
survive
j
j
j j Sj j
V SV
v p V n v dS
v dV n v dSΔ rarr
nabla sdot Δ = sdot
rArr nablasdot = sdot
sum sumint
int int
r r
r r
D
MingsianMingsian R R BaiBai39
( )
scalar variable constant vectorvector variable
Alternative formsLet be a an arbitrary and
a
ˆ
1 Le
(
ˆt
V
V S
S
V
v ap
v d
v dV vn d
V n v dS
v va
va dV v a v a
S
nablasdot = sdot
= rArr
nablasdot = nabla sdot + nabla
=
sdot
nablaint intint int
int
r
v
r r
r r
r r rQ ( )
( )
ˆ)
2 Let ˆV S
V
V S
dV n va dS
v a p
a p
p dV n p d
d p
S
V a
⎛ ⎞= sdot⎜ ⎟
⎝ ⎠
= times rArr
nablasdot times = sdot nabla
times
times
nabla = times
int int
int int
int
r
r r r
r r v
r
vQ
r
( ) ( ) ( )
( ) ( )
ˆ
ˆV S
V S
a p dV n a p dS
a p dV a n p dS
⎛ ⎞⎡ ⎤minus sdot nablatimes = sdot times⎣ ⎦⎜ ⎟⎜ ⎟⎜ ⎟rArr minus sdot nablatimes = minus sdot times⎜ ⎟⎝ ⎠
int int
int int
r rv v
r rv v
MingsianMingsian R R BaiBai40
( )
( )
From Leibnitzs rule
(1-D) ( )
( ) ( )
B t
A t
d f x t dxdt
f x t dx B ttpart prime= +part
int( )
( )( ( ) ) ( )
B t
A tf B t t A tprimeminusint ( ( ) )
(assume ( ) ( ) )
f A t t
A t B t const= =
ContinuityEx The Equation
rate of increase of mass
mass densityV
dM d dVdt dt
ρ
ρ
= int
flow velocityvv
MingsianMingsian R R BaiBai41
ˆ ˆ ( )
(
Conservation
) ( Divergence theorem)
of mass
( ) 0
V S S
V
V
dV n v dS n v dSt
v dV
v dVt
ρ ρ ρ
ρ
ρ ρ⎡ ⎤⎢ ⎥⎣ ⎦
part = minus sdot = minus sdotpart
= minus nablasdot
partrArr +nablasdot =part
int int int
int
int
v v
vQ
v
ˆOn the other hand rate influof mass through n ox i t
Extended to 3-D for time-independent fixed boundary
(3-D) ( ) ( )
Thus
S
V V
V
S V n v dS
Sd f x y z t dV f x y z t dVdt tdM dVdt t
ρ
ρ
= minus sdot
part=part
part=part
int
int int
intv
Fixed control volume (Eulerian coordinates)Otherwise Raynoldrsquos transport theoremMoving control volume Lagrangian coordinates
MingsianMingsian R R BaiBai42
But the control volime is only arbitrary
In particular for flow
( ) 0
con
(continuity equati
stant
on)
Divergence the
0 incompress
Note is t
ible
orem h
V
v
vt
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
e mother of the other integral theorems
MingsianMingsian R R BaiBai43
Ex Archimedes principle ( ) The buoyant force ( ) on a body with volume immersed in a fluid is equal to the weight of the fluid
ˆ displaced by the object ie In the followin
B
V
gVρ= minusf k
阿基米德原理
浮力
ˆg figure the ˆ with and being the fluid density gravitational
acceleration depth and outward normal unit vector to the surface
fhydraulic pressure g z
g z
S
ρ
ρ
= minusp n
n
1
the modified divergence theorem
ˆ
Hint
scalar fuctionV S
vdV v dS v Cnabla = isinint int n
MingsianMingsian R R BaiBai44
Levelsurfacefluid
V
S
airyx
z
ˆf g zρ= minusp n
ˆPf ( ) ( )
ˆ ˆ ˆ ˆ ˆ (0 0 ) v v
B fS S V
V V
g z g zdS dS dV
g dV g dV gV
ρ ρ
ρ ρ ρ
= = minus = minus nabla
= minus + + = minus = minus
int int int
int int
f p n
i j k k k
k
MingsianMingsian R R BaiBai45
Note This can also be thought of as simple balance of forces (resultant force that acts on and weight of filled with fluid)S V
2
~ a little history about George Green Recall the diveregence theorem
ˆ
Let
II th
e Green
are arbitrary fun
s identiti
ctions
es
V S
v dV n v dS
v u vu v C
nablasdot = sdot
= nablaisin
int intv v
v
ˆ ( ) ( )
ˆ ( )V S
V S
u v dV n u v dS
u v u v dV un v dS
nablasdot nabla = sdot nabla
nabla sdotnabla + nablasdotnabla = sdotnabla
int int
int int
MingsianMingsian R R BaiBai46
2 -( ) -(1)V S
vu v u v dV u dSnpart
nabla sdotnabla + nabla =partint int firs(Greens identt ity)
2
If
( ) --(2)
(1) (2) V S
u vuv u v u dV v dSn
harrpart
nabla sdotnabla + nabla =part
minus
int int
( )2 2
V S
dv duu v v u dV u v dSdn dn
⎛ ⎞nabla minus nabla = minus⎜ ⎟⎝ ⎠int int
second(Greens identity)
MingsianMingsian R R BaiBai47
( )
( )
( ) ( )
1st id ( )
Greens identities in the 1D case integration by parts
b b bb ba a
a a ab
ba
ab
ba
a
v udx v u v u dx u v uv dx uv
u v u v vu dx vu
uv vu dx uv vu
primeprime prime prime prime prime prime primeprime prime= minus rArr + =
prime prime primeprime primeharr + =
primeprime primeprime prime primerArr minus = minus
int int int
int
2
2nd id ( )
Good for any adjoint operators in addition to Good also for and integration by parts Important for theory amp reciprocity
From th
Gree
e 1s
2-
t
ns functi
identity
D 1-
tak 1
D
e
on
u
nablararr
=
int
2
V S
vv dV dSnpartnabla =part
rArr int int2Ex 0 st 0 on the boundary 0 ( 1st id u u unabla = = rArr equiv 以 証)
MingsianMingsian R R BaiBai48
1
Thm1 Stokes theorem Let be defined in Let be a piecewise-smooth orientable surface bounded by a piec
III Stokes theorem and
ewise-smooth simple clos
Greens
ed curve Then
theore
m
v C R S
C
isinv
( )
ˆwhere is a unit normal to according t right hand rulo the e
ˆS C
n S
n v dS v dRsdot nablatimes = sdotint intvv v
(not necessarily planar)S
Cn vnablatimes v vv
MingsianMingsian R R BaiBai49
Pf
( )( )
circulatioWe have shown in deriving thatthe in -direction
ˆor
ˆ
n
z
z
j j j j j
j j j j jj j
vz v dx dy
v dR n v S
v dR n v S
nablatimes
=nablatimes
sdot = sdot nablatimes Δ
sdot = sdot nablatimes Δ
intsum sumint
v
v
vv v
vv v
C
S
jS
jC
jP
ˆ jn
jSΔ
MingsianMingsian R R BaiBai50
( )
( )ext surface
After the interior boundaries cancel out only the exterior boundaries (Why)
ˆ
ˆ0 QED
Alternative forms scalar
survive
va
j
j j j j jC
jC S
v dR n v S
S v dR n v dS
φ
sdot = sdot nablatimes Δ
Δ rarr rArr sdot = sdot nablatimes
sum sumint
int int
vv v
vv v
( )
riable constant vector vector variable
1 Let
2 Let
ˆ
ˆ S C
S C
a p
v n dR dR
n p dS dR p
a
v a p
φ φφ= rArr timesnabla =
timesnabla times= rArr timestimes =
int int
int int
v v
v v
v
v v
vv vv v
MingsianMingsian R R BaiBai51
( ) ( )
( )
( )
( ) ( )
1
Thm2 Greens theorem ( planar version of Stokes theorem)ˆ ˆ Let
pf By Stokes theoremˆ
ˆ ˆ ˆ ˆ ˆ ˆ
S C
S C
S C
v P x y i Q x y j C
n v dS v dR
k Pi Qj dS Pi Qj dx i dy
Q P dS Pdx Qdyx y
⎛ ⎞part partminus = +⎜ ⎟
= + isin
sdot nablatimes = sdot
⎡ ⎤sdot nablatimes + =
part part⎝ ⎠
+ sdot +⎣ ⎦
int int
int
int
int
int
v
vv v
( )
( )
( )
ˆ
ˆˆ ˆ
ˆˆ ˆ ˆ ˆ( ) ( )
0
QEDS C
j
i j kQ PPi Qj i j k
x y z x yP Q
Q P dS Pdx Qdyx y
⎛ ⎞part part part part partnablatimes + = = + + minus⎜ ⎟part part part part part⎝ ⎠
⎛ ⎞part partrArr minus = +⎜ ⎟part part⎝ ⎠int int
L L
S
C
i
j
x
y
MingsianMingsian R R BaiBai52
Potential TheorypotentiGeorge Green coined the term (l
a )
rarr 位能 勢能
例子 重力場 電磁場 流場 聲場 彈性力學
1
2
Thm1 If is in a simply connected region then the following statements are equivalent(i) There exists a scala
Conservative Field
potential functior such that
(ii
n
) 0
v
C
C
v
v R
φφ=nabla
nablatimes
isin
isin
=r
r
r
r in (Irrotational field curl-free)
cf 0 st solenoidal field divergence-free
0
(iii) in (Conservative field)C
Rv w v w
v R Rd
nablasdot = exist nabla
sdot =
= times
int
r r r
r
r
r
MingsianMingsian R R BaiBai53
(iv) is connecting and
independent o
Ex Think
f the pa
of dynamics is the force
thB
A
v dR A B
v
sdotintrr
r
1 2
1
0
Pf
(i) (ii) ( ) 0 (zero identity cg rule) irrotational
ˆ(ii) (iii) ( ) 0 (Stokes theorem)
(iii) (iv) ( ) 0
C S
C C C
C
v
v dR n v dS
v dR v dR
v dR
φ
=
rarr nablatimes = nablatimes nabla =
rarr sdot = sdot nablatimes =
rarr sdot = minus sdot =
rArr sdot =
int int
int int int
int
r
rr
rr r
r rr r
rr r
2
independent of pathC
v dRsdotintr
A
B
1C
2C
MingsianMingsian R R BaiBai54
1 2
(iv) (i) constant ( ) ( )
--(1)
But (2)
(1) (2) QED
Note on application
Check
B
C C A
v dR v dR B A d
v dR d
d dx dy dz dRx y z
v dR dR v R
φ φ φ
φφ φ φφ φ
φ φ
rarr sdot = sdot = = minus =
rArr sdot =part part part
= + + = nabla sdot minus minus minus minus minuspart part part
sdot = nabla sdot rArr = nabla forall
int int intr rr r
rr
r
r r rr r
use simpler path (iv)if 0
find (i)v
φ⎧
nablatimes = ⎨⎩
rr
A
B
x
y
Cz
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai12
1 1 21
2 1 22
2 21 1 2 2
1 2 1 2
2 (10 6 ) 0 (1)
2 (6 10 ) 0 (2)
5 6 5 8 0 (constraint)
(1) (2) (2 16 )( ) 0 18 or (3)Substituting Eq (3) into Eqs (1) and (2) l
L x x xxL x x xxL x x x x
x x x x
λ
λ
λλ λ
part = + + =partpart = + + =partpart = + + minus =part+ + + = rArr = minus = minus
1 2 1 2eads to
and which corresponds to 14Thus minimummaximum distance 12x x x x J= = minus =
=
MingsianMingsian R R BaiBai13
22 2
1
1 1
2 2
The result above is identical to the previously obtained result in Linear Algebr
ellipse
45 rotatio
a
1 ( )4
1 11 1
n12
xx
x xx x
deg⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠
prime+ =prime
primeminus= rArr
prime2x prime
2x
1x prime
1x
2
MingsianMingsian R R BaiBai14
Curl 旋度
curl curl In Cartesian coordinates
spatial operator indep
ˆˆ ˆ
ˆˆ ˆ( )
ende
(
nt of coordi
) (
D
nates
ef
)
x y z
y x x xz z
i j k
vx y z
v v v
v v v vv vi j ky z x z x y
v v
part part partnablatimes =part part part
part part part partpart part= minus minus minus + minuspart
times
part part part part part
nabla
rarr
r
r r
0vnablatimes =rr 0vnablatimes ne
rr
MingsianMingsian R R BaiBai15
circulaPhysica
tion o spin ol interpretati
f fluidr or vortion
y
citΓ
( )0 0x dx y dy+ +
( )0 0x y ( )0 0x dx y+
Γ
1
2
3
4
( )0 0x y dy+
MingsianMingsian R R BaiBai16
yy
vv dy
ypart
+part
Γdyminus
dyyvv x
x partpart
+
dy
dx
dxminus
y yy
v vv dx dy
x ypart part
+ +part part
x xx
v vv dx dyx y
part part+ +part part
yy
vv dx
xpart
+part
xx
vv dxx
part+part
xv
yv
vv
dλ paddle weel
MingsianMingsian R R BaiBai17
contour
1 2 3 4
circulation
x x y y x x y y
x
v d
v d v d v d v d
v
λ
λ λ λ λrarr
Γ sdot
= + + +
asymp
intint int int int
vv
12
xv dxx
part+part ydx v
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
+ 12
y yv vdx dy
x ypart part
+ +part part
x
dy
v
⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦
+
12
xv dxx
part+part
( )xy
v dy dx vy
⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦
part+ minus +part
12
yvdy
ypart
+part
( )
( )
Similar process can be applied to the circulation in the and directionsThus the total circulation per unit area i curl s
y xz
dy
v v dxdy v dxdyx y
xv v
y
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎝ ⎠
minus
part part= minus = nablatimespart part
=nablatimes
v
v v
MingsianMingsian R R BaiBai18
Identities of differential operators ( p380 Greenberg )
( )
( ) ( )( )( )
( ) ( )( ) is required
Example of deriv
ˆ ˆˆ ˆ ˆ ˆ
ation
x y z
x y z
v v vx y z
uv u v u v
uv i j k uv i uv j uv kx y z
uv u v u v
u v v u u v
u v u vv v u u
+ +part part partpart part part
⎛ ⎞⎜ ⎟⎝ ⎠
nablasdot = nabla sdot + nablasdot
part part partnablasdot = + + sdot + + =part part part
nablatimes =nabla times + nablatimes
nablasdot times = sdotnablatimes minus sdotnablatimes
nablatimes times = nabla minus + minusnablanablasdot sdotsdot678 678
v
v v v
vL
v v v
v v v v v v
v v v v v v v v( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )2 cf
x y zu u ux y z
A B C A
u v
v w v w w v v w w v
v v v C B A B C
+ +part part partpart part part
sdotnabla
nabla sdot = sdotnabla + sdotnabla + times nablatimes + times
times
nablatimes
nablatimes nablatimes =nabla nabla times = sdot minussdot sdotminusnabla
678v
v v v
v v v v v vv v
v v v v v v
v v
v
v v
ˆ ˆˆ ˆ ˆ ˆ( ) ( ) ( )x y z x y zv i v j v k i j k u v v v ux y z x y zpart part part part part part+ + sdot + + = + +part part part part part part
v v
MingsianMingsian R R BaiBai19
2 2 22
2 2 2
0 (dc direct current) solenoidal0 (cg center of gravity) conservative potent
Laplacia
Zero identitie
n
s
ial
x y z
vu
⎧⎪⎨⎪⎩
part part partΔ =nabla nablasdotnabla = + +part part part
nablasdotnablatimes = rarr
nablatimesnabla = rarr
v
v
These differential operators will take more complicated and irregular form for the other coordinate systems
MingsianMingsian R R BaiBai20
2 2 3
2 2
ˆˆ ˆEx 2
ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ 2 0 2
2 0 0 2yx z
u x y v x yi z j k
u i j k u xyi x j k xyi x jx y z
vv vv xy xyx y z
⎛ ⎞⎜ ⎟⎝ ⎠
= = minus +
part part partnabla = + + = + + = +part part part
partpart partnablasdot = + + = + minus =part part part
v
v
MingsianMingsian R R BaiBai21
( ) ( ) ( )
( ) ( )
2 2 2 2
2 2 22
2 2 2
2
ˆˆ ˆ
ˆˆ ˆ
ˆ ˆˆ ˆ ˆ 0 3 0 0 0 3
2 0 0 2
ˆˆ ˆ
y yx xz z
x y z
i j kv vv vv vv i j k
x y z y z x z x yv v v
z i j x k z i x k
u u uu y yx y zv v v
i j kx y z
⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠ ⎝ ⎠
⎛
⎝
part partpart partpart partpart part partnablatimes = = minus minus minus + minuspart part part part part part part part part
= + minus minus + minus = minus
part part partnabla = + + = + + =part part part
nablatimes nablatimes =nabla nablasdot minusnabla
part part part= + +part part part
v
v v v
( ) ( )( ) ( ) ( ) ( )
( )
2 2 22 3
2 2 2ˆˆ ˆ2 2
ˆ ˆˆ ˆ ˆ ˆ 2 2 0 2 0 0 0 0 6 0 0 0
ˆ 2 6
xy x yi z j kx y z
yi xj k y i z j k
x z j
⎛ ⎞⎞⎜ ⎟⎜ ⎟
⎠ ⎝ ⎠⎡ ⎤⎣ ⎦
part part partminus + + minus +part part part
= + + minus + + + + minus + + minus
= +
MingsianMingsian R R BaiBai22
( )v w r w r r wnablatimes =nablatimes times = nablasdot minus nablasdotv v v v v v v ( )r w+ sdotnablav v ( ) constant w
w rminus sdotnablav
v v
Q
Ex Rigid body rotation
MingsianMingsian R R BaiBai23
( )( ) ( )
( ) ( )
ˆˆ ˆ
ˆ ˆˆ ˆ ˆ ˆ 1 1 1 3
ˆˆ ˆ
ˆˆ ˆ
Thus 3 2 is linked directly
x y z
x y z
r xi yj zk
r i j k xi yj zkx y z
w r w w w xi yj zkx y z
w i w j w k w
w r w r w r w w wv
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎝ ⎠
= + +
part part partnablasdot = + + + + = + + =part part part
part part partsdotnabla = + + + +part part part
= + + =
nablatimes times == nablasdot minus sdotnabla = minus =rArrnablatimes
v
v
v v
v
v v v v v v v v v
v spinn to ing
MingsianMingsian R R BaiBai24
Line surface and volume integrals
x
yC
B
A
isΔ
( )ii ηξ
Piecewise smooth curve
Line integral
MingsianMingsian R R BaiBai25
0 1lim ( ) ( ) or ( )
Intepretation of a line integral as an area ( )
i
Bn
i i isA
C
i C
F s F x y ds F x y d
x
s
F y ds
ξ ηΔ rarr =
Δ =sum int
int
int
x
y
z
C
( )z F x y=
MingsianMingsian R R BaiBai26
the area is always on the left of an obs
Positive dierver trave
rection of irsing on the
ntegrationcurve
RC
CCW CW
If is a regular closed curve piecewise smo
( ) ( )
(contou
oth simple ( )
r integral)
x y zC C
x y zC
F dR F F F dx dy dz
F dx F dy
C
F dz
sdot = + + + +
= + +
lowast
int int
int
i j k i j kv v
無交叉
work dW F dR= sdotv v
MingsianMingsian R R BaiBai27
0 1
Surface integral
lim ( ) ( )i
n
i i i iS i S
F S F x y z dSξ η ζΔ rarr =
Δ =sum intint( ) i i iξ η ζ
iSΔ
0 1
mass density
lim ( ) ( )
Ex
Volume integral
( )
i
n
i i i iV i V
V
F V F x y z dV
M x y z dV
ξ η ζ
ρ
Δ rarr =Δ =
=
sum intintint
intintint( ) i i iξ η ζ
iVΔ
MingsianMingsian R R BaiBai28
Ref (1)Hildebrand advanced Calculus for application Prentice-Hall 1976 p306-313(2) Greenberg p383-389 (3) Arfken ch2
1 2 3
1 2 3
1 2 3
1 2 3
(
Orthogonal coordinate systems base vectors are not constant in s
) ( )( )( )( )
ˆˆ ˆ
pacep x y z p u u ux x u u uy y u u uz z u u u
r xi yj zk
primerarr
= +
rarr
===
+v
1uv
2uv3uv
1u
2u
3uP coord curve
xy
z
O
X Curvilinear Coordinates
MingsianMingsian R R BaiBai29
1
11
1 1 1
1
1
1 1 1
11 1
1 1
2 2 2 3 3 3
Tangent vector to the at
where is the arc length
1
where unit vector
Similarly
curv
e
u psr rU
u s us
rs
U h uds rh udu sU h u U h u
partpart part= =part part part
part =part
there4 =part= =part
= =
v vv
vQ
v v
vv
v vv v
1u cood curve
1uv
1du 1dS
drv
P
MingsianMingsian R R BaiBai30
3 31 2 1 21 2 3
1 2 32
curv
In summary scale factor
If is the arc length along a in any directione
k k k
kk k
k k
k k k
U h u
ds rh Udu u
ds h dus
du dudu du du dudr r r r U U Uds u ds u ds u ds ds ds ds
dr drds ds
=
part= = =part
=
part part part= + + = + +part part part
= sdot
v v
v v
v v v v v v v
v v 3 3
1 1
23 3 32 2
21 1 1
2 2 2 2 2 2 2 2 2 21 1 2 2 3 3 1 2 3
1 (for orthogonal coord)
jii j
i j
ji ii j i
i j i
dududr U Uds ds ds
dudu duU U hds ds ds
ds h du h du h du ds ds ds
= =
= = =
⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
= sdot
rarr = sdot =
rarr = + + = + +
sum sum
sumsum sum
v v v
v v
MingsianMingsian R R BaiBai31
1 1 2 2 3 3
1 2 3
are mutually vectors having with arc length
k k k k kds h du U du
U du U du U duds ds ds
= =
rArr perp
v
v v v
1uv
2uv
3uv
1dS
2dS
3dS
dS
MingsianMingsian R R BaiBai32
( ) ( )
( ) ( )
1 1 2 2 3 3 1 2 3 1 2 3 1 2 3
1 2 3 1 2 3
1
1 2 2 3 3 2 2 2 3 3 3 2 3 2 3 1
Differential volume
Differential On const coord surface
area
Gradi t
en
d U du U du U du u u u h h h du du du
d h h h du du du
ud U du U du h u du h u du h h du du u
df
τ
τ
σ
= times sdot = times sdot
=
=
= times = times =
= nabla
v v v v v v
v vv v v v
1 2 31 2 3
1 2 3 1 1 2 2 3 31 2 3
1 1 1 2 2 2 3 3 3
f f ff dr df du du duu u u
r r rdr du du du U du U du U duu u u
h u du h u du h u du
part part partsdot rarr = + +part part part
part part part= + + = + +part part part
= + +
v
v v v v v vv
v v v
=1 (right hand rule)
MingsianMingsian R R BaiBai33
Rectangular Cartesian Coordinates
ij
k
MingsianMingsian R R BaiBai34
Circular Cylindrical Coordinatescossin
xyz z
ρ ϕρ ϕ
===
Ref Arfken
( )P zρ ϕ
MingsianMingsian R R BaiBai35
Polar Spherical Coordinatessin cossin sincos
x ry rz r
θ ϕθ ϕθ
===
Ref Arfken
0
0
0
sin cos sin sin coscos cos cos sin sin
sin cos
θ ϕ θ ϕ θθ ϕ θ ϕ θϕ ϕ
= + +
= + minus= minus +
r i j kθ i j kφ i j
2
2
sinsin
dS r d ddV r dr d d
θ θ ϕ
θ θ ϕ
=
=( )P r θ ϕ
MingsianMingsian R R BaiBai36
Integral TheoremsHeart of this chapter
Interchange of dimensionality Div Curl Grad in integrals High-dimensional integration
rarr
1
Thm1 Divergence theorem Let be a closed region bounded by piecewise smooth orientable surface and let (con
I Divergence theor
tinuous derivatives up to the first order
em (Gauss theorem)
)
VS v Cisin
r
ˆ If denotes the outward unit normal on
the ˆn V S
v dV n v dS
n S
nablasdot = sdotint intr r
MingsianMingsian R R BaiBai37
jS
jP
jV
S
0
ˆBy definition ( ) lim
ˆ( )
ˆ( )
j
j
j
j
Sj V
j
j j S
j j Sj j
n v dSv P
V
v P V n v dS
v P V n v dS
Δ rarr
⎧ ⎫sdot⎪ ⎪nabla sdot ⎨ ⎬Δ⎪ ⎪⎩ ⎭
nabla sdot Δ asymp sdot
nabla sdot Δ = sdot
int
intsum sumint
rr
r r
r r
Pf
MingsianMingsian R R BaiBai38
0
After interior surfaces cancel out only exterior surface (Why Opposite normal vectors)
ˆ( )
volumes exterior surfaces
ˆlim
QE
survive
j
j
j j Sj j
V SV
v p V n v dS
v dV n v dSΔ rarr
nabla sdot Δ = sdot
rArr nablasdot = sdot
sum sumint
int int
r r
r r
D
MingsianMingsian R R BaiBai39
( )
scalar variable constant vectorvector variable
Alternative formsLet be a an arbitrary and
a
ˆ
1 Le
(
ˆt
V
V S
S
V
v ap
v d
v dV vn d
V n v dS
v va
va dV v a v a
S
nablasdot = sdot
= rArr
nablasdot = nabla sdot + nabla
=
sdot
nablaint intint int
int
r
v
r r
r r
r r rQ ( )
( )
ˆ)
2 Let ˆV S
V
V S
dV n va dS
v a p
a p
p dV n p d
d p
S
V a
⎛ ⎞= sdot⎜ ⎟
⎝ ⎠
= times rArr
nablasdot times = sdot nabla
times
times
nabla = times
int int
int int
int
r
r r r
r r v
r
vQ
r
( ) ( ) ( )
( ) ( )
ˆ
ˆV S
V S
a p dV n a p dS
a p dV a n p dS
⎛ ⎞⎡ ⎤minus sdot nablatimes = sdot times⎣ ⎦⎜ ⎟⎜ ⎟⎜ ⎟rArr minus sdot nablatimes = minus sdot times⎜ ⎟⎝ ⎠
int int
int int
r rv v
r rv v
MingsianMingsian R R BaiBai40
( )
( )
From Leibnitzs rule
(1-D) ( )
( ) ( )
B t
A t
d f x t dxdt
f x t dx B ttpart prime= +part
int( )
( )( ( ) ) ( )
B t
A tf B t t A tprimeminusint ( ( ) )
(assume ( ) ( ) )
f A t t
A t B t const= =
ContinuityEx The Equation
rate of increase of mass
mass densityV
dM d dVdt dt
ρ
ρ
= int
flow velocityvv
MingsianMingsian R R BaiBai41
ˆ ˆ ( )
(
Conservation
) ( Divergence theorem)
of mass
( ) 0
V S S
V
V
dV n v dS n v dSt
v dV
v dVt
ρ ρ ρ
ρ
ρ ρ⎡ ⎤⎢ ⎥⎣ ⎦
part = minus sdot = minus sdotpart
= minus nablasdot
partrArr +nablasdot =part
int int int
int
int
v v
vQ
v
ˆOn the other hand rate influof mass through n ox i t
Extended to 3-D for time-independent fixed boundary
(3-D) ( ) ( )
Thus
S
V V
V
S V n v dS
Sd f x y z t dV f x y z t dVdt tdM dVdt t
ρ
ρ
= minus sdot
part=part
part=part
int
int int
intv
Fixed control volume (Eulerian coordinates)Otherwise Raynoldrsquos transport theoremMoving control volume Lagrangian coordinates
MingsianMingsian R R BaiBai42
But the control volime is only arbitrary
In particular for flow
( ) 0
con
(continuity equati
stant
on)
Divergence the
0 incompress
Note is t
ible
orem h
V
v
vt
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
e mother of the other integral theorems
MingsianMingsian R R BaiBai43
Ex Archimedes principle ( ) The buoyant force ( ) on a body with volume immersed in a fluid is equal to the weight of the fluid
ˆ displaced by the object ie In the followin
B
V
gVρ= minusf k
阿基米德原理
浮力
ˆg figure the ˆ with and being the fluid density gravitational
acceleration depth and outward normal unit vector to the surface
fhydraulic pressure g z
g z
S
ρ
ρ
= minusp n
n
1
the modified divergence theorem
ˆ
Hint
scalar fuctionV S
vdV v dS v Cnabla = isinint int n
MingsianMingsian R R BaiBai44
Levelsurfacefluid
V
S
airyx
z
ˆf g zρ= minusp n
ˆPf ( ) ( )
ˆ ˆ ˆ ˆ ˆ (0 0 ) v v
B fS S V
V V
g z g zdS dS dV
g dV g dV gV
ρ ρ
ρ ρ ρ
= = minus = minus nabla
= minus + + = minus = minus
int int int
int int
f p n
i j k k k
k
MingsianMingsian R R BaiBai45
Note This can also be thought of as simple balance of forces (resultant force that acts on and weight of filled with fluid)S V
2
~ a little history about George Green Recall the diveregence theorem
ˆ
Let
II th
e Green
are arbitrary fun
s identiti
ctions
es
V S
v dV n v dS
v u vu v C
nablasdot = sdot
= nablaisin
int intv v
v
ˆ ( ) ( )
ˆ ( )V S
V S
u v dV n u v dS
u v u v dV un v dS
nablasdot nabla = sdot nabla
nabla sdotnabla + nablasdotnabla = sdotnabla
int int
int int
MingsianMingsian R R BaiBai46
2 -( ) -(1)V S
vu v u v dV u dSnpart
nabla sdotnabla + nabla =partint int firs(Greens identt ity)
2
If
( ) --(2)
(1) (2) V S
u vuv u v u dV v dSn
harrpart
nabla sdotnabla + nabla =part
minus
int int
( )2 2
V S
dv duu v v u dV u v dSdn dn
⎛ ⎞nabla minus nabla = minus⎜ ⎟⎝ ⎠int int
second(Greens identity)
MingsianMingsian R R BaiBai47
( )
( )
( ) ( )
1st id ( )
Greens identities in the 1D case integration by parts
b b bb ba a
a a ab
ba
ab
ba
a
v udx v u v u dx u v uv dx uv
u v u v vu dx vu
uv vu dx uv vu
primeprime prime prime prime prime prime primeprime prime= minus rArr + =
prime prime primeprime primeharr + =
primeprime primeprime prime primerArr minus = minus
int int int
int
2
2nd id ( )
Good for any adjoint operators in addition to Good also for and integration by parts Important for theory amp reciprocity
From th
Gree
e 1s
2-
t
ns functi
identity
D 1-
tak 1
D
e
on
u
nablararr
=
int
2
V S
vv dV dSnpartnabla =part
rArr int int2Ex 0 st 0 on the boundary 0 ( 1st id u u unabla = = rArr equiv 以 証)
MingsianMingsian R R BaiBai48
1
Thm1 Stokes theorem Let be defined in Let be a piecewise-smooth orientable surface bounded by a piec
III Stokes theorem and
ewise-smooth simple clos
Greens
ed curve Then
theore
m
v C R S
C
isinv
( )
ˆwhere is a unit normal to according t right hand rulo the e
ˆS C
n S
n v dS v dRsdot nablatimes = sdotint intvv v
(not necessarily planar)S
Cn vnablatimes v vv
MingsianMingsian R R BaiBai49
Pf
( )( )
circulatioWe have shown in deriving thatthe in -direction
ˆor
ˆ
n
z
z
j j j j j
j j j j jj j
vz v dx dy
v dR n v S
v dR n v S
nablatimes
=nablatimes
sdot = sdot nablatimes Δ
sdot = sdot nablatimes Δ
intsum sumint
v
v
vv v
vv v
C
S
jS
jC
jP
ˆ jn
jSΔ
MingsianMingsian R R BaiBai50
( )
( )ext surface
After the interior boundaries cancel out only the exterior boundaries (Why)
ˆ
ˆ0 QED
Alternative forms scalar
survive
va
j
j j j j jC
jC S
v dR n v S
S v dR n v dS
φ
sdot = sdot nablatimes Δ
Δ rarr rArr sdot = sdot nablatimes
sum sumint
int int
vv v
vv v
( )
riable constant vector vector variable
1 Let
2 Let
ˆ
ˆ S C
S C
a p
v n dR dR
n p dS dR p
a
v a p
φ φφ= rArr timesnabla =
timesnabla times= rArr timestimes =
int int
int int
v v
v v
v
v v
vv vv v
MingsianMingsian R R BaiBai51
( ) ( )
( )
( )
( ) ( )
1
Thm2 Greens theorem ( planar version of Stokes theorem)ˆ ˆ Let
pf By Stokes theoremˆ
ˆ ˆ ˆ ˆ ˆ ˆ
S C
S C
S C
v P x y i Q x y j C
n v dS v dR
k Pi Qj dS Pi Qj dx i dy
Q P dS Pdx Qdyx y
⎛ ⎞part partminus = +⎜ ⎟
= + isin
sdot nablatimes = sdot
⎡ ⎤sdot nablatimes + =
part part⎝ ⎠
+ sdot +⎣ ⎦
int int
int
int
int
int
v
vv v
( )
( )
( )
ˆ
ˆˆ ˆ
ˆˆ ˆ ˆ ˆ( ) ( )
0
QEDS C
j
i j kQ PPi Qj i j k
x y z x yP Q
Q P dS Pdx Qdyx y
⎛ ⎞part part part part partnablatimes + = = + + minus⎜ ⎟part part part part part⎝ ⎠
⎛ ⎞part partrArr minus = +⎜ ⎟part part⎝ ⎠int int
L L
S
C
i
j
x
y
MingsianMingsian R R BaiBai52
Potential TheorypotentiGeorge Green coined the term (l
a )
rarr 位能 勢能
例子 重力場 電磁場 流場 聲場 彈性力學
1
2
Thm1 If is in a simply connected region then the following statements are equivalent(i) There exists a scala
Conservative Field
potential functior such that
(ii
n
) 0
v
C
C
v
v R
φφ=nabla
nablatimes
isin
isin
=r
r
r
r in (Irrotational field curl-free)
cf 0 st solenoidal field divergence-free
0
(iii) in (Conservative field)C
Rv w v w
v R Rd
nablasdot = exist nabla
sdot =
= times
int
r r r
r
r
r
MingsianMingsian R R BaiBai53
(iv) is connecting and
independent o
Ex Think
f the pa
of dynamics is the force
thB
A
v dR A B
v
sdotintrr
r
1 2
1
0
Pf
(i) (ii) ( ) 0 (zero identity cg rule) irrotational
ˆ(ii) (iii) ( ) 0 (Stokes theorem)
(iii) (iv) ( ) 0
C S
C C C
C
v
v dR n v dS
v dR v dR
v dR
φ
=
rarr nablatimes = nablatimes nabla =
rarr sdot = sdot nablatimes =
rarr sdot = minus sdot =
rArr sdot =
int int
int int int
int
r
rr
rr r
r rr r
rr r
2
independent of pathC
v dRsdotintr
A
B
1C
2C
MingsianMingsian R R BaiBai54
1 2
(iv) (i) constant ( ) ( )
--(1)
But (2)
(1) (2) QED
Note on application
Check
B
C C A
v dR v dR B A d
v dR d
d dx dy dz dRx y z
v dR dR v R
φ φ φ
φφ φ φφ φ
φ φ
rarr sdot = sdot = = minus =
rArr sdot =part part part
= + + = nabla sdot minus minus minus minus minuspart part part
sdot = nabla sdot rArr = nabla forall
int int intr rr r
rr
r
r r rr r
use simpler path (iv)if 0
find (i)v
φ⎧
nablatimes = ⎨⎩
rr
A
B
x
y
Cz
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai13
22 2
1
1 1
2 2
The result above is identical to the previously obtained result in Linear Algebr
ellipse
45 rotatio
a
1 ( )4
1 11 1
n12
xx
x xx x
deg⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠
prime+ =prime
primeminus= rArr
prime2x prime
2x
1x prime
1x
2
MingsianMingsian R R BaiBai14
Curl 旋度
curl curl In Cartesian coordinates
spatial operator indep
ˆˆ ˆ
ˆˆ ˆ( )
ende
(
nt of coordi
) (
D
nates
ef
)
x y z
y x x xz z
i j k
vx y z
v v v
v v v vv vi j ky z x z x y
v v
part part partnablatimes =part part part
part part part partpart part= minus minus minus + minuspart
times
part part part part part
nabla
rarr
r
r r
0vnablatimes =rr 0vnablatimes ne
rr
MingsianMingsian R R BaiBai15
circulaPhysica
tion o spin ol interpretati
f fluidr or vortion
y
citΓ
( )0 0x dx y dy+ +
( )0 0x y ( )0 0x dx y+
Γ
1
2
3
4
( )0 0x y dy+
MingsianMingsian R R BaiBai16
yy
vv dy
ypart
+part
Γdyminus
dyyvv x
x partpart
+
dy
dx
dxminus
y yy
v vv dx dy
x ypart part
+ +part part
x xx
v vv dx dyx y
part part+ +part part
yy
vv dx
xpart
+part
xx
vv dxx
part+part
xv
yv
vv
dλ paddle weel
MingsianMingsian R R BaiBai17
contour
1 2 3 4
circulation
x x y y x x y y
x
v d
v d v d v d v d
v
λ
λ λ λ λrarr
Γ sdot
= + + +
asymp
intint int int int
vv
12
xv dxx
part+part ydx v
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
+ 12
y yv vdx dy
x ypart part
+ +part part
x
dy
v
⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦
+
12
xv dxx
part+part
( )xy
v dy dx vy
⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦
part+ minus +part
12
yvdy
ypart
+part
( )
( )
Similar process can be applied to the circulation in the and directionsThus the total circulation per unit area i curl s
y xz
dy
v v dxdy v dxdyx y
xv v
y
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎝ ⎠
minus
part part= minus = nablatimespart part
=nablatimes
v
v v
MingsianMingsian R R BaiBai18
Identities of differential operators ( p380 Greenberg )
( )
( ) ( )( )( )
( ) ( )( ) is required
Example of deriv
ˆ ˆˆ ˆ ˆ ˆ
ation
x y z
x y z
v v vx y z
uv u v u v
uv i j k uv i uv j uv kx y z
uv u v u v
u v v u u v
u v u vv v u u
+ +part part partpart part part
⎛ ⎞⎜ ⎟⎝ ⎠
nablasdot = nabla sdot + nablasdot
part part partnablasdot = + + sdot + + =part part part
nablatimes =nabla times + nablatimes
nablasdot times = sdotnablatimes minus sdotnablatimes
nablatimes times = nabla minus + minusnablanablasdot sdotsdot678 678
v
v v v
vL
v v v
v v v v v v
v v v v v v v v( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )2 cf
x y zu u ux y z
A B C A
u v
v w v w w v v w w v
v v v C B A B C
+ +part part partpart part part
sdotnabla
nabla sdot = sdotnabla + sdotnabla + times nablatimes + times
times
nablatimes
nablatimes nablatimes =nabla nabla times = sdot minussdot sdotminusnabla
678v
v v v
v v v v v vv v
v v v v v v
v v
v
v v
ˆ ˆˆ ˆ ˆ ˆ( ) ( ) ( )x y z x y zv i v j v k i j k u v v v ux y z x y zpart part part part part part+ + sdot + + = + +part part part part part part
v v
MingsianMingsian R R BaiBai19
2 2 22
2 2 2
0 (dc direct current) solenoidal0 (cg center of gravity) conservative potent
Laplacia
Zero identitie
n
s
ial
x y z
vu
⎧⎪⎨⎪⎩
part part partΔ =nabla nablasdotnabla = + +part part part
nablasdotnablatimes = rarr
nablatimesnabla = rarr
v
v
These differential operators will take more complicated and irregular form for the other coordinate systems
MingsianMingsian R R BaiBai20
2 2 3
2 2
ˆˆ ˆEx 2
ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ 2 0 2
2 0 0 2yx z
u x y v x yi z j k
u i j k u xyi x j k xyi x jx y z
vv vv xy xyx y z
⎛ ⎞⎜ ⎟⎝ ⎠
= = minus +
part part partnabla = + + = + + = +part part part
partpart partnablasdot = + + = + minus =part part part
v
v
MingsianMingsian R R BaiBai21
( ) ( ) ( )
( ) ( )
2 2 2 2
2 2 22
2 2 2
2
ˆˆ ˆ
ˆˆ ˆ
ˆ ˆˆ ˆ ˆ 0 3 0 0 0 3
2 0 0 2
ˆˆ ˆ
y yx xz z
x y z
i j kv vv vv vv i j k
x y z y z x z x yv v v
z i j x k z i x k
u u uu y yx y zv v v
i j kx y z
⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠ ⎝ ⎠
⎛
⎝
part partpart partpart partpart part partnablatimes = = minus minus minus + minuspart part part part part part part part part
= + minus minus + minus = minus
part part partnabla = + + = + + =part part part
nablatimes nablatimes =nabla nablasdot minusnabla
part part part= + +part part part
v
v v v
( ) ( )( ) ( ) ( ) ( )
( )
2 2 22 3
2 2 2ˆˆ ˆ2 2
ˆ ˆˆ ˆ ˆ ˆ 2 2 0 2 0 0 0 0 6 0 0 0
ˆ 2 6
xy x yi z j kx y z
yi xj k y i z j k
x z j
⎛ ⎞⎞⎜ ⎟⎜ ⎟
⎠ ⎝ ⎠⎡ ⎤⎣ ⎦
part part partminus + + minus +part part part
= + + minus + + + + minus + + minus
= +
MingsianMingsian R R BaiBai22
( )v w r w r r wnablatimes =nablatimes times = nablasdot minus nablasdotv v v v v v v ( )r w+ sdotnablav v ( ) constant w
w rminus sdotnablav
v v
Q
Ex Rigid body rotation
MingsianMingsian R R BaiBai23
( )( ) ( )
( ) ( )
ˆˆ ˆ
ˆ ˆˆ ˆ ˆ ˆ 1 1 1 3
ˆˆ ˆ
ˆˆ ˆ
Thus 3 2 is linked directly
x y z
x y z
r xi yj zk
r i j k xi yj zkx y z
w r w w w xi yj zkx y z
w i w j w k w
w r w r w r w w wv
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎝ ⎠
= + +
part part partnablasdot = + + + + = + + =part part part
part part partsdotnabla = + + + +part part part
= + + =
nablatimes times == nablasdot minus sdotnabla = minus =rArrnablatimes
v
v
v v
v
v v v v v v v v v
v spinn to ing
MingsianMingsian R R BaiBai24
Line surface and volume integrals
x
yC
B
A
isΔ
( )ii ηξ
Piecewise smooth curve
Line integral
MingsianMingsian R R BaiBai25
0 1lim ( ) ( ) or ( )
Intepretation of a line integral as an area ( )
i
Bn
i i isA
C
i C
F s F x y ds F x y d
x
s
F y ds
ξ ηΔ rarr =
Δ =sum int
int
int
x
y
z
C
( )z F x y=
MingsianMingsian R R BaiBai26
the area is always on the left of an obs
Positive dierver trave
rection of irsing on the
ntegrationcurve
RC
CCW CW
If is a regular closed curve piecewise smo
( ) ( )
(contou
oth simple ( )
r integral)
x y zC C
x y zC
F dR F F F dx dy dz
F dx F dy
C
F dz
sdot = + + + +
= + +
lowast
int int
int
i j k i j kv v
無交叉
work dW F dR= sdotv v
MingsianMingsian R R BaiBai27
0 1
Surface integral
lim ( ) ( )i
n
i i i iS i S
F S F x y z dSξ η ζΔ rarr =
Δ =sum intint( ) i i iξ η ζ
iSΔ
0 1
mass density
lim ( ) ( )
Ex
Volume integral
( )
i
n
i i i iV i V
V
F V F x y z dV
M x y z dV
ξ η ζ
ρ
Δ rarr =Δ =
=
sum intintint
intintint( ) i i iξ η ζ
iVΔ
MingsianMingsian R R BaiBai28
Ref (1)Hildebrand advanced Calculus for application Prentice-Hall 1976 p306-313(2) Greenberg p383-389 (3) Arfken ch2
1 2 3
1 2 3
1 2 3
1 2 3
(
Orthogonal coordinate systems base vectors are not constant in s
) ( )( )( )( )
ˆˆ ˆ
pacep x y z p u u ux x u u uy y u u uz z u u u
r xi yj zk
primerarr
= +
rarr
===
+v
1uv
2uv3uv
1u
2u
3uP coord curve
xy
z
O
X Curvilinear Coordinates
MingsianMingsian R R BaiBai29
1
11
1 1 1
1
1
1 1 1
11 1
1 1
2 2 2 3 3 3
Tangent vector to the at
where is the arc length
1
where unit vector
Similarly
curv
e
u psr rU
u s us
rs
U h uds rh udu sU h u U h u
partpart part= =part part part
part =part
there4 =part= =part
= =
v vv
vQ
v v
vv
v vv v
1u cood curve
1uv
1du 1dS
drv
P
MingsianMingsian R R BaiBai30
3 31 2 1 21 2 3
1 2 32
curv
In summary scale factor
If is the arc length along a in any directione
k k k
kk k
k k
k k k
U h u
ds rh Udu u
ds h dus
du dudu du du dudr r r r U U Uds u ds u ds u ds ds ds ds
dr drds ds
=
part= = =part
=
part part part= + + = + +part part part
= sdot
v v
v v
v v v v v v v
v v 3 3
1 1
23 3 32 2
21 1 1
2 2 2 2 2 2 2 2 2 21 1 2 2 3 3 1 2 3
1 (for orthogonal coord)
jii j
i j
ji ii j i
i j i
dududr U Uds ds ds
dudu duU U hds ds ds
ds h du h du h du ds ds ds
= =
= = =
⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
= sdot
rarr = sdot =
rarr = + + = + +
sum sum
sumsum sum
v v v
v v
MingsianMingsian R R BaiBai31
1 1 2 2 3 3
1 2 3
are mutually vectors having with arc length
k k k k kds h du U du
U du U du U duds ds ds
= =
rArr perp
v
v v v
1uv
2uv
3uv
1dS
2dS
3dS
dS
MingsianMingsian R R BaiBai32
( ) ( )
( ) ( )
1 1 2 2 3 3 1 2 3 1 2 3 1 2 3
1 2 3 1 2 3
1
1 2 2 3 3 2 2 2 3 3 3 2 3 2 3 1
Differential volume
Differential On const coord surface
area
Gradi t
en
d U du U du U du u u u h h h du du du
d h h h du du du
ud U du U du h u du h u du h h du du u
df
τ
τ
σ
= times sdot = times sdot
=
=
= times = times =
= nabla
v v v v v v
v vv v v v
1 2 31 2 3
1 2 3 1 1 2 2 3 31 2 3
1 1 1 2 2 2 3 3 3
f f ff dr df du du duu u u
r r rdr du du du U du U du U duu u u
h u du h u du h u du
part part partsdot rarr = + +part part part
part part part= + + = + +part part part
= + +
v
v v v v v vv
v v v
=1 (right hand rule)
MingsianMingsian R R BaiBai33
Rectangular Cartesian Coordinates
ij
k
MingsianMingsian R R BaiBai34
Circular Cylindrical Coordinatescossin
xyz z
ρ ϕρ ϕ
===
Ref Arfken
( )P zρ ϕ
MingsianMingsian R R BaiBai35
Polar Spherical Coordinatessin cossin sincos
x ry rz r
θ ϕθ ϕθ
===
Ref Arfken
0
0
0
sin cos sin sin coscos cos cos sin sin
sin cos
θ ϕ θ ϕ θθ ϕ θ ϕ θϕ ϕ
= + +
= + minus= minus +
r i j kθ i j kφ i j
2
2
sinsin
dS r d ddV r dr d d
θ θ ϕ
θ θ ϕ
=
=( )P r θ ϕ
MingsianMingsian R R BaiBai36
Integral TheoremsHeart of this chapter
Interchange of dimensionality Div Curl Grad in integrals High-dimensional integration
rarr
1
Thm1 Divergence theorem Let be a closed region bounded by piecewise smooth orientable surface and let (con
I Divergence theor
tinuous derivatives up to the first order
em (Gauss theorem)
)
VS v Cisin
r
ˆ If denotes the outward unit normal on
the ˆn V S
v dV n v dS
n S
nablasdot = sdotint intr r
MingsianMingsian R R BaiBai37
jS
jP
jV
S
0
ˆBy definition ( ) lim
ˆ( )
ˆ( )
j
j
j
j
Sj V
j
j j S
j j Sj j
n v dSv P
V
v P V n v dS
v P V n v dS
Δ rarr
⎧ ⎫sdot⎪ ⎪nabla sdot ⎨ ⎬Δ⎪ ⎪⎩ ⎭
nabla sdot Δ asymp sdot
nabla sdot Δ = sdot
int
intsum sumint
rr
r r
r r
Pf
MingsianMingsian R R BaiBai38
0
After interior surfaces cancel out only exterior surface (Why Opposite normal vectors)
ˆ( )
volumes exterior surfaces
ˆlim
QE
survive
j
j
j j Sj j
V SV
v p V n v dS
v dV n v dSΔ rarr
nabla sdot Δ = sdot
rArr nablasdot = sdot
sum sumint
int int
r r
r r
D
MingsianMingsian R R BaiBai39
( )
scalar variable constant vectorvector variable
Alternative formsLet be a an arbitrary and
a
ˆ
1 Le
(
ˆt
V
V S
S
V
v ap
v d
v dV vn d
V n v dS
v va
va dV v a v a
S
nablasdot = sdot
= rArr
nablasdot = nabla sdot + nabla
=
sdot
nablaint intint int
int
r
v
r r
r r
r r rQ ( )
( )
ˆ)
2 Let ˆV S
V
V S
dV n va dS
v a p
a p
p dV n p d
d p
S
V a
⎛ ⎞= sdot⎜ ⎟
⎝ ⎠
= times rArr
nablasdot times = sdot nabla
times
times
nabla = times
int int
int int
int
r
r r r
r r v
r
vQ
r
( ) ( ) ( )
( ) ( )
ˆ
ˆV S
V S
a p dV n a p dS
a p dV a n p dS
⎛ ⎞⎡ ⎤minus sdot nablatimes = sdot times⎣ ⎦⎜ ⎟⎜ ⎟⎜ ⎟rArr minus sdot nablatimes = minus sdot times⎜ ⎟⎝ ⎠
int int
int int
r rv v
r rv v
MingsianMingsian R R BaiBai40
( )
( )
From Leibnitzs rule
(1-D) ( )
( ) ( )
B t
A t
d f x t dxdt
f x t dx B ttpart prime= +part
int( )
( )( ( ) ) ( )
B t
A tf B t t A tprimeminusint ( ( ) )
(assume ( ) ( ) )
f A t t
A t B t const= =
ContinuityEx The Equation
rate of increase of mass
mass densityV
dM d dVdt dt
ρ
ρ
= int
flow velocityvv
MingsianMingsian R R BaiBai41
ˆ ˆ ( )
(
Conservation
) ( Divergence theorem)
of mass
( ) 0
V S S
V
V
dV n v dS n v dSt
v dV
v dVt
ρ ρ ρ
ρ
ρ ρ⎡ ⎤⎢ ⎥⎣ ⎦
part = minus sdot = minus sdotpart
= minus nablasdot
partrArr +nablasdot =part
int int int
int
int
v v
vQ
v
ˆOn the other hand rate influof mass through n ox i t
Extended to 3-D for time-independent fixed boundary
(3-D) ( ) ( )
Thus
S
V V
V
S V n v dS
Sd f x y z t dV f x y z t dVdt tdM dVdt t
ρ
ρ
= minus sdot
part=part
part=part
int
int int
intv
Fixed control volume (Eulerian coordinates)Otherwise Raynoldrsquos transport theoremMoving control volume Lagrangian coordinates
MingsianMingsian R R BaiBai42
But the control volime is only arbitrary
In particular for flow
( ) 0
con
(continuity equati
stant
on)
Divergence the
0 incompress
Note is t
ible
orem h
V
v
vt
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
e mother of the other integral theorems
MingsianMingsian R R BaiBai43
Ex Archimedes principle ( ) The buoyant force ( ) on a body with volume immersed in a fluid is equal to the weight of the fluid
ˆ displaced by the object ie In the followin
B
V
gVρ= minusf k
阿基米德原理
浮力
ˆg figure the ˆ with and being the fluid density gravitational
acceleration depth and outward normal unit vector to the surface
fhydraulic pressure g z
g z
S
ρ
ρ
= minusp n
n
1
the modified divergence theorem
ˆ
Hint
scalar fuctionV S
vdV v dS v Cnabla = isinint int n
MingsianMingsian R R BaiBai44
Levelsurfacefluid
V
S
airyx
z
ˆf g zρ= minusp n
ˆPf ( ) ( )
ˆ ˆ ˆ ˆ ˆ (0 0 ) v v
B fS S V
V V
g z g zdS dS dV
g dV g dV gV
ρ ρ
ρ ρ ρ
= = minus = minus nabla
= minus + + = minus = minus
int int int
int int
f p n
i j k k k
k
MingsianMingsian R R BaiBai45
Note This can also be thought of as simple balance of forces (resultant force that acts on and weight of filled with fluid)S V
2
~ a little history about George Green Recall the diveregence theorem
ˆ
Let
II th
e Green
are arbitrary fun
s identiti
ctions
es
V S
v dV n v dS
v u vu v C
nablasdot = sdot
= nablaisin
int intv v
v
ˆ ( ) ( )
ˆ ( )V S
V S
u v dV n u v dS
u v u v dV un v dS
nablasdot nabla = sdot nabla
nabla sdotnabla + nablasdotnabla = sdotnabla
int int
int int
MingsianMingsian R R BaiBai46
2 -( ) -(1)V S
vu v u v dV u dSnpart
nabla sdotnabla + nabla =partint int firs(Greens identt ity)
2
If
( ) --(2)
(1) (2) V S
u vuv u v u dV v dSn
harrpart
nabla sdotnabla + nabla =part
minus
int int
( )2 2
V S
dv duu v v u dV u v dSdn dn
⎛ ⎞nabla minus nabla = minus⎜ ⎟⎝ ⎠int int
second(Greens identity)
MingsianMingsian R R BaiBai47
( )
( )
( ) ( )
1st id ( )
Greens identities in the 1D case integration by parts
b b bb ba a
a a ab
ba
ab
ba
a
v udx v u v u dx u v uv dx uv
u v u v vu dx vu
uv vu dx uv vu
primeprime prime prime prime prime prime primeprime prime= minus rArr + =
prime prime primeprime primeharr + =
primeprime primeprime prime primerArr minus = minus
int int int
int
2
2nd id ( )
Good for any adjoint operators in addition to Good also for and integration by parts Important for theory amp reciprocity
From th
Gree
e 1s
2-
t
ns functi
identity
D 1-
tak 1
D
e
on
u
nablararr
=
int
2
V S
vv dV dSnpartnabla =part
rArr int int2Ex 0 st 0 on the boundary 0 ( 1st id u u unabla = = rArr equiv 以 証)
MingsianMingsian R R BaiBai48
1
Thm1 Stokes theorem Let be defined in Let be a piecewise-smooth orientable surface bounded by a piec
III Stokes theorem and
ewise-smooth simple clos
Greens
ed curve Then
theore
m
v C R S
C
isinv
( )
ˆwhere is a unit normal to according t right hand rulo the e
ˆS C
n S
n v dS v dRsdot nablatimes = sdotint intvv v
(not necessarily planar)S
Cn vnablatimes v vv
MingsianMingsian R R BaiBai49
Pf
( )( )
circulatioWe have shown in deriving thatthe in -direction
ˆor
ˆ
n
z
z
j j j j j
j j j j jj j
vz v dx dy
v dR n v S
v dR n v S
nablatimes
=nablatimes
sdot = sdot nablatimes Δ
sdot = sdot nablatimes Δ
intsum sumint
v
v
vv v
vv v
C
S
jS
jC
jP
ˆ jn
jSΔ
MingsianMingsian R R BaiBai50
( )
( )ext surface
After the interior boundaries cancel out only the exterior boundaries (Why)
ˆ
ˆ0 QED
Alternative forms scalar
survive
va
j
j j j j jC
jC S
v dR n v S
S v dR n v dS
φ
sdot = sdot nablatimes Δ
Δ rarr rArr sdot = sdot nablatimes
sum sumint
int int
vv v
vv v
( )
riable constant vector vector variable
1 Let
2 Let
ˆ
ˆ S C
S C
a p
v n dR dR
n p dS dR p
a
v a p
φ φφ= rArr timesnabla =
timesnabla times= rArr timestimes =
int int
int int
v v
v v
v
v v
vv vv v
MingsianMingsian R R BaiBai51
( ) ( )
( )
( )
( ) ( )
1
Thm2 Greens theorem ( planar version of Stokes theorem)ˆ ˆ Let
pf By Stokes theoremˆ
ˆ ˆ ˆ ˆ ˆ ˆ
S C
S C
S C
v P x y i Q x y j C
n v dS v dR
k Pi Qj dS Pi Qj dx i dy
Q P dS Pdx Qdyx y
⎛ ⎞part partminus = +⎜ ⎟
= + isin
sdot nablatimes = sdot
⎡ ⎤sdot nablatimes + =
part part⎝ ⎠
+ sdot +⎣ ⎦
int int
int
int
int
int
v
vv v
( )
( )
( )
ˆ
ˆˆ ˆ
ˆˆ ˆ ˆ ˆ( ) ( )
0
QEDS C
j
i j kQ PPi Qj i j k
x y z x yP Q
Q P dS Pdx Qdyx y
⎛ ⎞part part part part partnablatimes + = = + + minus⎜ ⎟part part part part part⎝ ⎠
⎛ ⎞part partrArr minus = +⎜ ⎟part part⎝ ⎠int int
L L
S
C
i
j
x
y
MingsianMingsian R R BaiBai52
Potential TheorypotentiGeorge Green coined the term (l
a )
rarr 位能 勢能
例子 重力場 電磁場 流場 聲場 彈性力學
1
2
Thm1 If is in a simply connected region then the following statements are equivalent(i) There exists a scala
Conservative Field
potential functior such that
(ii
n
) 0
v
C
C
v
v R
φφ=nabla
nablatimes
isin
isin
=r
r
r
r in (Irrotational field curl-free)
cf 0 st solenoidal field divergence-free
0
(iii) in (Conservative field)C
Rv w v w
v R Rd
nablasdot = exist nabla
sdot =
= times
int
r r r
r
r
r
MingsianMingsian R R BaiBai53
(iv) is connecting and
independent o
Ex Think
f the pa
of dynamics is the force
thB
A
v dR A B
v
sdotintrr
r
1 2
1
0
Pf
(i) (ii) ( ) 0 (zero identity cg rule) irrotational
ˆ(ii) (iii) ( ) 0 (Stokes theorem)
(iii) (iv) ( ) 0
C S
C C C
C
v
v dR n v dS
v dR v dR
v dR
φ
=
rarr nablatimes = nablatimes nabla =
rarr sdot = sdot nablatimes =
rarr sdot = minus sdot =
rArr sdot =
int int
int int int
int
r
rr
rr r
r rr r
rr r
2
independent of pathC
v dRsdotintr
A
B
1C
2C
MingsianMingsian R R BaiBai54
1 2
(iv) (i) constant ( ) ( )
--(1)
But (2)
(1) (2) QED
Note on application
Check
B
C C A
v dR v dR B A d
v dR d
d dx dy dz dRx y z
v dR dR v R
φ φ φ
φφ φ φφ φ
φ φ
rarr sdot = sdot = = minus =
rArr sdot =part part part
= + + = nabla sdot minus minus minus minus minuspart part part
sdot = nabla sdot rArr = nabla forall
int int intr rr r
rr
r
r r rr r
use simpler path (iv)if 0
find (i)v
φ⎧
nablatimes = ⎨⎩
rr
A
B
x
y
Cz
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai14
Curl 旋度
curl curl In Cartesian coordinates
spatial operator indep
ˆˆ ˆ
ˆˆ ˆ( )
ende
(
nt of coordi
) (
D
nates
ef
)
x y z
y x x xz z
i j k
vx y z
v v v
v v v vv vi j ky z x z x y
v v
part part partnablatimes =part part part
part part part partpart part= minus minus minus + minuspart
times
part part part part part
nabla
rarr
r
r r
0vnablatimes =rr 0vnablatimes ne
rr
MingsianMingsian R R BaiBai15
circulaPhysica
tion o spin ol interpretati
f fluidr or vortion
y
citΓ
( )0 0x dx y dy+ +
( )0 0x y ( )0 0x dx y+
Γ
1
2
3
4
( )0 0x y dy+
MingsianMingsian R R BaiBai16
yy
vv dy
ypart
+part
Γdyminus
dyyvv x
x partpart
+
dy
dx
dxminus
y yy
v vv dx dy
x ypart part
+ +part part
x xx
v vv dx dyx y
part part+ +part part
yy
vv dx
xpart
+part
xx
vv dxx
part+part
xv
yv
vv
dλ paddle weel
MingsianMingsian R R BaiBai17
contour
1 2 3 4
circulation
x x y y x x y y
x
v d
v d v d v d v d
v
λ
λ λ λ λrarr
Γ sdot
= + + +
asymp
intint int int int
vv
12
xv dxx
part+part ydx v
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
+ 12
y yv vdx dy
x ypart part
+ +part part
x
dy
v
⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦
+
12
xv dxx
part+part
( )xy
v dy dx vy
⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦
part+ minus +part
12
yvdy
ypart
+part
( )
( )
Similar process can be applied to the circulation in the and directionsThus the total circulation per unit area i curl s
y xz
dy
v v dxdy v dxdyx y
xv v
y
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎝ ⎠
minus
part part= minus = nablatimespart part
=nablatimes
v
v v
MingsianMingsian R R BaiBai18
Identities of differential operators ( p380 Greenberg )
( )
( ) ( )( )( )
( ) ( )( ) is required
Example of deriv
ˆ ˆˆ ˆ ˆ ˆ
ation
x y z
x y z
v v vx y z
uv u v u v
uv i j k uv i uv j uv kx y z
uv u v u v
u v v u u v
u v u vv v u u
+ +part part partpart part part
⎛ ⎞⎜ ⎟⎝ ⎠
nablasdot = nabla sdot + nablasdot
part part partnablasdot = + + sdot + + =part part part
nablatimes =nabla times + nablatimes
nablasdot times = sdotnablatimes minus sdotnablatimes
nablatimes times = nabla minus + minusnablanablasdot sdotsdot678 678
v
v v v
vL
v v v
v v v v v v
v v v v v v v v( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )2 cf
x y zu u ux y z
A B C A
u v
v w v w w v v w w v
v v v C B A B C
+ +part part partpart part part
sdotnabla
nabla sdot = sdotnabla + sdotnabla + times nablatimes + times
times
nablatimes
nablatimes nablatimes =nabla nabla times = sdot minussdot sdotminusnabla
678v
v v v
v v v v v vv v
v v v v v v
v v
v
v v
ˆ ˆˆ ˆ ˆ ˆ( ) ( ) ( )x y z x y zv i v j v k i j k u v v v ux y z x y zpart part part part part part+ + sdot + + = + +part part part part part part
v v
MingsianMingsian R R BaiBai19
2 2 22
2 2 2
0 (dc direct current) solenoidal0 (cg center of gravity) conservative potent
Laplacia
Zero identitie
n
s
ial
x y z
vu
⎧⎪⎨⎪⎩
part part partΔ =nabla nablasdotnabla = + +part part part
nablasdotnablatimes = rarr
nablatimesnabla = rarr
v
v
These differential operators will take more complicated and irregular form for the other coordinate systems
MingsianMingsian R R BaiBai20
2 2 3
2 2
ˆˆ ˆEx 2
ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ 2 0 2
2 0 0 2yx z
u x y v x yi z j k
u i j k u xyi x j k xyi x jx y z
vv vv xy xyx y z
⎛ ⎞⎜ ⎟⎝ ⎠
= = minus +
part part partnabla = + + = + + = +part part part
partpart partnablasdot = + + = + minus =part part part
v
v
MingsianMingsian R R BaiBai21
( ) ( ) ( )
( ) ( )
2 2 2 2
2 2 22
2 2 2
2
ˆˆ ˆ
ˆˆ ˆ
ˆ ˆˆ ˆ ˆ 0 3 0 0 0 3
2 0 0 2
ˆˆ ˆ
y yx xz z
x y z
i j kv vv vv vv i j k
x y z y z x z x yv v v
z i j x k z i x k
u u uu y yx y zv v v
i j kx y z
⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠ ⎝ ⎠
⎛
⎝
part partpart partpart partpart part partnablatimes = = minus minus minus + minuspart part part part part part part part part
= + minus minus + minus = minus
part part partnabla = + + = + + =part part part
nablatimes nablatimes =nabla nablasdot minusnabla
part part part= + +part part part
v
v v v
( ) ( )( ) ( ) ( ) ( )
( )
2 2 22 3
2 2 2ˆˆ ˆ2 2
ˆ ˆˆ ˆ ˆ ˆ 2 2 0 2 0 0 0 0 6 0 0 0
ˆ 2 6
xy x yi z j kx y z
yi xj k y i z j k
x z j
⎛ ⎞⎞⎜ ⎟⎜ ⎟
⎠ ⎝ ⎠⎡ ⎤⎣ ⎦
part part partminus + + minus +part part part
= + + minus + + + + minus + + minus
= +
MingsianMingsian R R BaiBai22
( )v w r w r r wnablatimes =nablatimes times = nablasdot minus nablasdotv v v v v v v ( )r w+ sdotnablav v ( ) constant w
w rminus sdotnablav
v v
Q
Ex Rigid body rotation
MingsianMingsian R R BaiBai23
( )( ) ( )
( ) ( )
ˆˆ ˆ
ˆ ˆˆ ˆ ˆ ˆ 1 1 1 3
ˆˆ ˆ
ˆˆ ˆ
Thus 3 2 is linked directly
x y z
x y z
r xi yj zk
r i j k xi yj zkx y z
w r w w w xi yj zkx y z
w i w j w k w
w r w r w r w w wv
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎝ ⎠
= + +
part part partnablasdot = + + + + = + + =part part part
part part partsdotnabla = + + + +part part part
= + + =
nablatimes times == nablasdot minus sdotnabla = minus =rArrnablatimes
v
v
v v
v
v v v v v v v v v
v spinn to ing
MingsianMingsian R R BaiBai24
Line surface and volume integrals
x
yC
B
A
isΔ
( )ii ηξ
Piecewise smooth curve
Line integral
MingsianMingsian R R BaiBai25
0 1lim ( ) ( ) or ( )
Intepretation of a line integral as an area ( )
i
Bn
i i isA
C
i C
F s F x y ds F x y d
x
s
F y ds
ξ ηΔ rarr =
Δ =sum int
int
int
x
y
z
C
( )z F x y=
MingsianMingsian R R BaiBai26
the area is always on the left of an obs
Positive dierver trave
rection of irsing on the
ntegrationcurve
RC
CCW CW
If is a regular closed curve piecewise smo
( ) ( )
(contou
oth simple ( )
r integral)
x y zC C
x y zC
F dR F F F dx dy dz
F dx F dy
C
F dz
sdot = + + + +
= + +
lowast
int int
int
i j k i j kv v
無交叉
work dW F dR= sdotv v
MingsianMingsian R R BaiBai27
0 1
Surface integral
lim ( ) ( )i
n
i i i iS i S
F S F x y z dSξ η ζΔ rarr =
Δ =sum intint( ) i i iξ η ζ
iSΔ
0 1
mass density
lim ( ) ( )
Ex
Volume integral
( )
i
n
i i i iV i V
V
F V F x y z dV
M x y z dV
ξ η ζ
ρ
Δ rarr =Δ =
=
sum intintint
intintint( ) i i iξ η ζ
iVΔ
MingsianMingsian R R BaiBai28
Ref (1)Hildebrand advanced Calculus for application Prentice-Hall 1976 p306-313(2) Greenberg p383-389 (3) Arfken ch2
1 2 3
1 2 3
1 2 3
1 2 3
(
Orthogonal coordinate systems base vectors are not constant in s
) ( )( )( )( )
ˆˆ ˆ
pacep x y z p u u ux x u u uy y u u uz z u u u
r xi yj zk
primerarr
= +
rarr
===
+v
1uv
2uv3uv
1u
2u
3uP coord curve
xy
z
O
X Curvilinear Coordinates
MingsianMingsian R R BaiBai29
1
11
1 1 1
1
1
1 1 1
11 1
1 1
2 2 2 3 3 3
Tangent vector to the at
where is the arc length
1
where unit vector
Similarly
curv
e
u psr rU
u s us
rs
U h uds rh udu sU h u U h u
partpart part= =part part part
part =part
there4 =part= =part
= =
v vv
vQ
v v
vv
v vv v
1u cood curve
1uv
1du 1dS
drv
P
MingsianMingsian R R BaiBai30
3 31 2 1 21 2 3
1 2 32
curv
In summary scale factor
If is the arc length along a in any directione
k k k
kk k
k k
k k k
U h u
ds rh Udu u
ds h dus
du dudu du du dudr r r r U U Uds u ds u ds u ds ds ds ds
dr drds ds
=
part= = =part
=
part part part= + + = + +part part part
= sdot
v v
v v
v v v v v v v
v v 3 3
1 1
23 3 32 2
21 1 1
2 2 2 2 2 2 2 2 2 21 1 2 2 3 3 1 2 3
1 (for orthogonal coord)
jii j
i j
ji ii j i
i j i
dududr U Uds ds ds
dudu duU U hds ds ds
ds h du h du h du ds ds ds
= =
= = =
⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
= sdot
rarr = sdot =
rarr = + + = + +
sum sum
sumsum sum
v v v
v v
MingsianMingsian R R BaiBai31
1 1 2 2 3 3
1 2 3
are mutually vectors having with arc length
k k k k kds h du U du
U du U du U duds ds ds
= =
rArr perp
v
v v v
1uv
2uv
3uv
1dS
2dS
3dS
dS
MingsianMingsian R R BaiBai32
( ) ( )
( ) ( )
1 1 2 2 3 3 1 2 3 1 2 3 1 2 3
1 2 3 1 2 3
1
1 2 2 3 3 2 2 2 3 3 3 2 3 2 3 1
Differential volume
Differential On const coord surface
area
Gradi t
en
d U du U du U du u u u h h h du du du
d h h h du du du
ud U du U du h u du h u du h h du du u
df
τ
τ
σ
= times sdot = times sdot
=
=
= times = times =
= nabla
v v v v v v
v vv v v v
1 2 31 2 3
1 2 3 1 1 2 2 3 31 2 3
1 1 1 2 2 2 3 3 3
f f ff dr df du du duu u u
r r rdr du du du U du U du U duu u u
h u du h u du h u du
part part partsdot rarr = + +part part part
part part part= + + = + +part part part
= + +
v
v v v v v vv
v v v
=1 (right hand rule)
MingsianMingsian R R BaiBai33
Rectangular Cartesian Coordinates
ij
k
MingsianMingsian R R BaiBai34
Circular Cylindrical Coordinatescossin
xyz z
ρ ϕρ ϕ
===
Ref Arfken
( )P zρ ϕ
MingsianMingsian R R BaiBai35
Polar Spherical Coordinatessin cossin sincos
x ry rz r
θ ϕθ ϕθ
===
Ref Arfken
0
0
0
sin cos sin sin coscos cos cos sin sin
sin cos
θ ϕ θ ϕ θθ ϕ θ ϕ θϕ ϕ
= + +
= + minus= minus +
r i j kθ i j kφ i j
2
2
sinsin
dS r d ddV r dr d d
θ θ ϕ
θ θ ϕ
=
=( )P r θ ϕ
MingsianMingsian R R BaiBai36
Integral TheoremsHeart of this chapter
Interchange of dimensionality Div Curl Grad in integrals High-dimensional integration
rarr
1
Thm1 Divergence theorem Let be a closed region bounded by piecewise smooth orientable surface and let (con
I Divergence theor
tinuous derivatives up to the first order
em (Gauss theorem)
)
VS v Cisin
r
ˆ If denotes the outward unit normal on
the ˆn V S
v dV n v dS
n S
nablasdot = sdotint intr r
MingsianMingsian R R BaiBai37
jS
jP
jV
S
0
ˆBy definition ( ) lim
ˆ( )
ˆ( )
j
j
j
j
Sj V
j
j j S
j j Sj j
n v dSv P
V
v P V n v dS
v P V n v dS
Δ rarr
⎧ ⎫sdot⎪ ⎪nabla sdot ⎨ ⎬Δ⎪ ⎪⎩ ⎭
nabla sdot Δ asymp sdot
nabla sdot Δ = sdot
int
intsum sumint
rr
r r
r r
Pf
MingsianMingsian R R BaiBai38
0
After interior surfaces cancel out only exterior surface (Why Opposite normal vectors)
ˆ( )
volumes exterior surfaces
ˆlim
QE
survive
j
j
j j Sj j
V SV
v p V n v dS
v dV n v dSΔ rarr
nabla sdot Δ = sdot
rArr nablasdot = sdot
sum sumint
int int
r r
r r
D
MingsianMingsian R R BaiBai39
( )
scalar variable constant vectorvector variable
Alternative formsLet be a an arbitrary and
a
ˆ
1 Le
(
ˆt
V
V S
S
V
v ap
v d
v dV vn d
V n v dS
v va
va dV v a v a
S
nablasdot = sdot
= rArr
nablasdot = nabla sdot + nabla
=
sdot
nablaint intint int
int
r
v
r r
r r
r r rQ ( )
( )
ˆ)
2 Let ˆV S
V
V S
dV n va dS
v a p
a p
p dV n p d
d p
S
V a
⎛ ⎞= sdot⎜ ⎟
⎝ ⎠
= times rArr
nablasdot times = sdot nabla
times
times
nabla = times
int int
int int
int
r
r r r
r r v
r
vQ
r
( ) ( ) ( )
( ) ( )
ˆ
ˆV S
V S
a p dV n a p dS
a p dV a n p dS
⎛ ⎞⎡ ⎤minus sdot nablatimes = sdot times⎣ ⎦⎜ ⎟⎜ ⎟⎜ ⎟rArr minus sdot nablatimes = minus sdot times⎜ ⎟⎝ ⎠
int int
int int
r rv v
r rv v
MingsianMingsian R R BaiBai40
( )
( )
From Leibnitzs rule
(1-D) ( )
( ) ( )
B t
A t
d f x t dxdt
f x t dx B ttpart prime= +part
int( )
( )( ( ) ) ( )
B t
A tf B t t A tprimeminusint ( ( ) )
(assume ( ) ( ) )
f A t t
A t B t const= =
ContinuityEx The Equation
rate of increase of mass
mass densityV
dM d dVdt dt
ρ
ρ
= int
flow velocityvv
MingsianMingsian R R BaiBai41
ˆ ˆ ( )
(
Conservation
) ( Divergence theorem)
of mass
( ) 0
V S S
V
V
dV n v dS n v dSt
v dV
v dVt
ρ ρ ρ
ρ
ρ ρ⎡ ⎤⎢ ⎥⎣ ⎦
part = minus sdot = minus sdotpart
= minus nablasdot
partrArr +nablasdot =part
int int int
int
int
v v
vQ
v
ˆOn the other hand rate influof mass through n ox i t
Extended to 3-D for time-independent fixed boundary
(3-D) ( ) ( )
Thus
S
V V
V
S V n v dS
Sd f x y z t dV f x y z t dVdt tdM dVdt t
ρ
ρ
= minus sdot
part=part
part=part
int
int int
intv
Fixed control volume (Eulerian coordinates)Otherwise Raynoldrsquos transport theoremMoving control volume Lagrangian coordinates
MingsianMingsian R R BaiBai42
But the control volime is only arbitrary
In particular for flow
( ) 0
con
(continuity equati
stant
on)
Divergence the
0 incompress
Note is t
ible
orem h
V
v
vt
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
e mother of the other integral theorems
MingsianMingsian R R BaiBai43
Ex Archimedes principle ( ) The buoyant force ( ) on a body with volume immersed in a fluid is equal to the weight of the fluid
ˆ displaced by the object ie In the followin
B
V
gVρ= minusf k
阿基米德原理
浮力
ˆg figure the ˆ with and being the fluid density gravitational
acceleration depth and outward normal unit vector to the surface
fhydraulic pressure g z
g z
S
ρ
ρ
= minusp n
n
1
the modified divergence theorem
ˆ
Hint
scalar fuctionV S
vdV v dS v Cnabla = isinint int n
MingsianMingsian R R BaiBai44
Levelsurfacefluid
V
S
airyx
z
ˆf g zρ= minusp n
ˆPf ( ) ( )
ˆ ˆ ˆ ˆ ˆ (0 0 ) v v
B fS S V
V V
g z g zdS dS dV
g dV g dV gV
ρ ρ
ρ ρ ρ
= = minus = minus nabla
= minus + + = minus = minus
int int int
int int
f p n
i j k k k
k
MingsianMingsian R R BaiBai45
Note This can also be thought of as simple balance of forces (resultant force that acts on and weight of filled with fluid)S V
2
~ a little history about George Green Recall the diveregence theorem
ˆ
Let
II th
e Green
are arbitrary fun
s identiti
ctions
es
V S
v dV n v dS
v u vu v C
nablasdot = sdot
= nablaisin
int intv v
v
ˆ ( ) ( )
ˆ ( )V S
V S
u v dV n u v dS
u v u v dV un v dS
nablasdot nabla = sdot nabla
nabla sdotnabla + nablasdotnabla = sdotnabla
int int
int int
MingsianMingsian R R BaiBai46
2 -( ) -(1)V S
vu v u v dV u dSnpart
nabla sdotnabla + nabla =partint int firs(Greens identt ity)
2
If
( ) --(2)
(1) (2) V S
u vuv u v u dV v dSn
harrpart
nabla sdotnabla + nabla =part
minus
int int
( )2 2
V S
dv duu v v u dV u v dSdn dn
⎛ ⎞nabla minus nabla = minus⎜ ⎟⎝ ⎠int int
second(Greens identity)
MingsianMingsian R R BaiBai47
( )
( )
( ) ( )
1st id ( )
Greens identities in the 1D case integration by parts
b b bb ba a
a a ab
ba
ab
ba
a
v udx v u v u dx u v uv dx uv
u v u v vu dx vu
uv vu dx uv vu
primeprime prime prime prime prime prime primeprime prime= minus rArr + =
prime prime primeprime primeharr + =
primeprime primeprime prime primerArr minus = minus
int int int
int
2
2nd id ( )
Good for any adjoint operators in addition to Good also for and integration by parts Important for theory amp reciprocity
From th
Gree
e 1s
2-
t
ns functi
identity
D 1-
tak 1
D
e
on
u
nablararr
=
int
2
V S
vv dV dSnpartnabla =part
rArr int int2Ex 0 st 0 on the boundary 0 ( 1st id u u unabla = = rArr equiv 以 証)
MingsianMingsian R R BaiBai48
1
Thm1 Stokes theorem Let be defined in Let be a piecewise-smooth orientable surface bounded by a piec
III Stokes theorem and
ewise-smooth simple clos
Greens
ed curve Then
theore
m
v C R S
C
isinv
( )
ˆwhere is a unit normal to according t right hand rulo the e
ˆS C
n S
n v dS v dRsdot nablatimes = sdotint intvv v
(not necessarily planar)S
Cn vnablatimes v vv
MingsianMingsian R R BaiBai49
Pf
( )( )
circulatioWe have shown in deriving thatthe in -direction
ˆor
ˆ
n
z
z
j j j j j
j j j j jj j
vz v dx dy
v dR n v S
v dR n v S
nablatimes
=nablatimes
sdot = sdot nablatimes Δ
sdot = sdot nablatimes Δ
intsum sumint
v
v
vv v
vv v
C
S
jS
jC
jP
ˆ jn
jSΔ
MingsianMingsian R R BaiBai50
( )
( )ext surface
After the interior boundaries cancel out only the exterior boundaries (Why)
ˆ
ˆ0 QED
Alternative forms scalar
survive
va
j
j j j j jC
jC S
v dR n v S
S v dR n v dS
φ
sdot = sdot nablatimes Δ
Δ rarr rArr sdot = sdot nablatimes
sum sumint
int int
vv v
vv v
( )
riable constant vector vector variable
1 Let
2 Let
ˆ
ˆ S C
S C
a p
v n dR dR
n p dS dR p
a
v a p
φ φφ= rArr timesnabla =
timesnabla times= rArr timestimes =
int int
int int
v v
v v
v
v v
vv vv v
MingsianMingsian R R BaiBai51
( ) ( )
( )
( )
( ) ( )
1
Thm2 Greens theorem ( planar version of Stokes theorem)ˆ ˆ Let
pf By Stokes theoremˆ
ˆ ˆ ˆ ˆ ˆ ˆ
S C
S C
S C
v P x y i Q x y j C
n v dS v dR
k Pi Qj dS Pi Qj dx i dy
Q P dS Pdx Qdyx y
⎛ ⎞part partminus = +⎜ ⎟
= + isin
sdot nablatimes = sdot
⎡ ⎤sdot nablatimes + =
part part⎝ ⎠
+ sdot +⎣ ⎦
int int
int
int
int
int
v
vv v
( )
( )
( )
ˆ
ˆˆ ˆ
ˆˆ ˆ ˆ ˆ( ) ( )
0
QEDS C
j
i j kQ PPi Qj i j k
x y z x yP Q
Q P dS Pdx Qdyx y
⎛ ⎞part part part part partnablatimes + = = + + minus⎜ ⎟part part part part part⎝ ⎠
⎛ ⎞part partrArr minus = +⎜ ⎟part part⎝ ⎠int int
L L
S
C
i
j
x
y
MingsianMingsian R R BaiBai52
Potential TheorypotentiGeorge Green coined the term (l
a )
rarr 位能 勢能
例子 重力場 電磁場 流場 聲場 彈性力學
1
2
Thm1 If is in a simply connected region then the following statements are equivalent(i) There exists a scala
Conservative Field
potential functior such that
(ii
n
) 0
v
C
C
v
v R
φφ=nabla
nablatimes
isin
isin
=r
r
r
r in (Irrotational field curl-free)
cf 0 st solenoidal field divergence-free
0
(iii) in (Conservative field)C
Rv w v w
v R Rd
nablasdot = exist nabla
sdot =
= times
int
r r r
r
r
r
MingsianMingsian R R BaiBai53
(iv) is connecting and
independent o
Ex Think
f the pa
of dynamics is the force
thB
A
v dR A B
v
sdotintrr
r
1 2
1
0
Pf
(i) (ii) ( ) 0 (zero identity cg rule) irrotational
ˆ(ii) (iii) ( ) 0 (Stokes theorem)
(iii) (iv) ( ) 0
C S
C C C
C
v
v dR n v dS
v dR v dR
v dR
φ
=
rarr nablatimes = nablatimes nabla =
rarr sdot = sdot nablatimes =
rarr sdot = minus sdot =
rArr sdot =
int int
int int int
int
r
rr
rr r
r rr r
rr r
2
independent of pathC
v dRsdotintr
A
B
1C
2C
MingsianMingsian R R BaiBai54
1 2
(iv) (i) constant ( ) ( )
--(1)
But (2)
(1) (2) QED
Note on application
Check
B
C C A
v dR v dR B A d
v dR d
d dx dy dz dRx y z
v dR dR v R
φ φ φ
φφ φ φφ φ
φ φ
rarr sdot = sdot = = minus =
rArr sdot =part part part
= + + = nabla sdot minus minus minus minus minuspart part part
sdot = nabla sdot rArr = nabla forall
int int intr rr r
rr
r
r r rr r
use simpler path (iv)if 0
find (i)v
φ⎧
nablatimes = ⎨⎩
rr
A
B
x
y
Cz
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai15
circulaPhysica
tion o spin ol interpretati
f fluidr or vortion
y
citΓ
( )0 0x dx y dy+ +
( )0 0x y ( )0 0x dx y+
Γ
1
2
3
4
( )0 0x y dy+
MingsianMingsian R R BaiBai16
yy
vv dy
ypart
+part
Γdyminus
dyyvv x
x partpart
+
dy
dx
dxminus
y yy
v vv dx dy
x ypart part
+ +part part
x xx
v vv dx dyx y
part part+ +part part
yy
vv dx
xpart
+part
xx
vv dxx
part+part
xv
yv
vv
dλ paddle weel
MingsianMingsian R R BaiBai17
contour
1 2 3 4
circulation
x x y y x x y y
x
v d
v d v d v d v d
v
λ
λ λ λ λrarr
Γ sdot
= + + +
asymp
intint int int int
vv
12
xv dxx
part+part ydx v
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
+ 12
y yv vdx dy
x ypart part
+ +part part
x
dy
v
⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦
+
12
xv dxx
part+part
( )xy
v dy dx vy
⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦
part+ minus +part
12
yvdy
ypart
+part
( )
( )
Similar process can be applied to the circulation in the and directionsThus the total circulation per unit area i curl s
y xz
dy
v v dxdy v dxdyx y
xv v
y
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎝ ⎠
minus
part part= minus = nablatimespart part
=nablatimes
v
v v
MingsianMingsian R R BaiBai18
Identities of differential operators ( p380 Greenberg )
( )
( ) ( )( )( )
( ) ( )( ) is required
Example of deriv
ˆ ˆˆ ˆ ˆ ˆ
ation
x y z
x y z
v v vx y z
uv u v u v
uv i j k uv i uv j uv kx y z
uv u v u v
u v v u u v
u v u vv v u u
+ +part part partpart part part
⎛ ⎞⎜ ⎟⎝ ⎠
nablasdot = nabla sdot + nablasdot
part part partnablasdot = + + sdot + + =part part part
nablatimes =nabla times + nablatimes
nablasdot times = sdotnablatimes minus sdotnablatimes
nablatimes times = nabla minus + minusnablanablasdot sdotsdot678 678
v
v v v
vL
v v v
v v v v v v
v v v v v v v v( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )2 cf
x y zu u ux y z
A B C A
u v
v w v w w v v w w v
v v v C B A B C
+ +part part partpart part part
sdotnabla
nabla sdot = sdotnabla + sdotnabla + times nablatimes + times
times
nablatimes
nablatimes nablatimes =nabla nabla times = sdot minussdot sdotminusnabla
678v
v v v
v v v v v vv v
v v v v v v
v v
v
v v
ˆ ˆˆ ˆ ˆ ˆ( ) ( ) ( )x y z x y zv i v j v k i j k u v v v ux y z x y zpart part part part part part+ + sdot + + = + +part part part part part part
v v
MingsianMingsian R R BaiBai19
2 2 22
2 2 2
0 (dc direct current) solenoidal0 (cg center of gravity) conservative potent
Laplacia
Zero identitie
n
s
ial
x y z
vu
⎧⎪⎨⎪⎩
part part partΔ =nabla nablasdotnabla = + +part part part
nablasdotnablatimes = rarr
nablatimesnabla = rarr
v
v
These differential operators will take more complicated and irregular form for the other coordinate systems
MingsianMingsian R R BaiBai20
2 2 3
2 2
ˆˆ ˆEx 2
ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ 2 0 2
2 0 0 2yx z
u x y v x yi z j k
u i j k u xyi x j k xyi x jx y z
vv vv xy xyx y z
⎛ ⎞⎜ ⎟⎝ ⎠
= = minus +
part part partnabla = + + = + + = +part part part
partpart partnablasdot = + + = + minus =part part part
v
v
MingsianMingsian R R BaiBai21
( ) ( ) ( )
( ) ( )
2 2 2 2
2 2 22
2 2 2
2
ˆˆ ˆ
ˆˆ ˆ
ˆ ˆˆ ˆ ˆ 0 3 0 0 0 3
2 0 0 2
ˆˆ ˆ
y yx xz z
x y z
i j kv vv vv vv i j k
x y z y z x z x yv v v
z i j x k z i x k
u u uu y yx y zv v v
i j kx y z
⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠ ⎝ ⎠
⎛
⎝
part partpart partpart partpart part partnablatimes = = minus minus minus + minuspart part part part part part part part part
= + minus minus + minus = minus
part part partnabla = + + = + + =part part part
nablatimes nablatimes =nabla nablasdot minusnabla
part part part= + +part part part
v
v v v
( ) ( )( ) ( ) ( ) ( )
( )
2 2 22 3
2 2 2ˆˆ ˆ2 2
ˆ ˆˆ ˆ ˆ ˆ 2 2 0 2 0 0 0 0 6 0 0 0
ˆ 2 6
xy x yi z j kx y z
yi xj k y i z j k
x z j
⎛ ⎞⎞⎜ ⎟⎜ ⎟
⎠ ⎝ ⎠⎡ ⎤⎣ ⎦
part part partminus + + minus +part part part
= + + minus + + + + minus + + minus
= +
MingsianMingsian R R BaiBai22
( )v w r w r r wnablatimes =nablatimes times = nablasdot minus nablasdotv v v v v v v ( )r w+ sdotnablav v ( ) constant w
w rminus sdotnablav
v v
Q
Ex Rigid body rotation
MingsianMingsian R R BaiBai23
( )( ) ( )
( ) ( )
ˆˆ ˆ
ˆ ˆˆ ˆ ˆ ˆ 1 1 1 3
ˆˆ ˆ
ˆˆ ˆ
Thus 3 2 is linked directly
x y z
x y z
r xi yj zk
r i j k xi yj zkx y z
w r w w w xi yj zkx y z
w i w j w k w
w r w r w r w w wv
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎝ ⎠
= + +
part part partnablasdot = + + + + = + + =part part part
part part partsdotnabla = + + + +part part part
= + + =
nablatimes times == nablasdot minus sdotnabla = minus =rArrnablatimes
v
v
v v
v
v v v v v v v v v
v spinn to ing
MingsianMingsian R R BaiBai24
Line surface and volume integrals
x
yC
B
A
isΔ
( )ii ηξ
Piecewise smooth curve
Line integral
MingsianMingsian R R BaiBai25
0 1lim ( ) ( ) or ( )
Intepretation of a line integral as an area ( )
i
Bn
i i isA
C
i C
F s F x y ds F x y d
x
s
F y ds
ξ ηΔ rarr =
Δ =sum int
int
int
x
y
z
C
( )z F x y=
MingsianMingsian R R BaiBai26
the area is always on the left of an obs
Positive dierver trave
rection of irsing on the
ntegrationcurve
RC
CCW CW
If is a regular closed curve piecewise smo
( ) ( )
(contou
oth simple ( )
r integral)
x y zC C
x y zC
F dR F F F dx dy dz
F dx F dy
C
F dz
sdot = + + + +
= + +
lowast
int int
int
i j k i j kv v
無交叉
work dW F dR= sdotv v
MingsianMingsian R R BaiBai27
0 1
Surface integral
lim ( ) ( )i
n
i i i iS i S
F S F x y z dSξ η ζΔ rarr =
Δ =sum intint( ) i i iξ η ζ
iSΔ
0 1
mass density
lim ( ) ( )
Ex
Volume integral
( )
i
n
i i i iV i V
V
F V F x y z dV
M x y z dV
ξ η ζ
ρ
Δ rarr =Δ =
=
sum intintint
intintint( ) i i iξ η ζ
iVΔ
MingsianMingsian R R BaiBai28
Ref (1)Hildebrand advanced Calculus for application Prentice-Hall 1976 p306-313(2) Greenberg p383-389 (3) Arfken ch2
1 2 3
1 2 3
1 2 3
1 2 3
(
Orthogonal coordinate systems base vectors are not constant in s
) ( )( )( )( )
ˆˆ ˆ
pacep x y z p u u ux x u u uy y u u uz z u u u
r xi yj zk
primerarr
= +
rarr
===
+v
1uv
2uv3uv
1u
2u
3uP coord curve
xy
z
O
X Curvilinear Coordinates
MingsianMingsian R R BaiBai29
1
11
1 1 1
1
1
1 1 1
11 1
1 1
2 2 2 3 3 3
Tangent vector to the at
where is the arc length
1
where unit vector
Similarly
curv
e
u psr rU
u s us
rs
U h uds rh udu sU h u U h u
partpart part= =part part part
part =part
there4 =part= =part
= =
v vv
vQ
v v
vv
v vv v
1u cood curve
1uv
1du 1dS
drv
P
MingsianMingsian R R BaiBai30
3 31 2 1 21 2 3
1 2 32
curv
In summary scale factor
If is the arc length along a in any directione
k k k
kk k
k k
k k k
U h u
ds rh Udu u
ds h dus
du dudu du du dudr r r r U U Uds u ds u ds u ds ds ds ds
dr drds ds
=
part= = =part
=
part part part= + + = + +part part part
= sdot
v v
v v
v v v v v v v
v v 3 3
1 1
23 3 32 2
21 1 1
2 2 2 2 2 2 2 2 2 21 1 2 2 3 3 1 2 3
1 (for orthogonal coord)
jii j
i j
ji ii j i
i j i
dududr U Uds ds ds
dudu duU U hds ds ds
ds h du h du h du ds ds ds
= =
= = =
⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
= sdot
rarr = sdot =
rarr = + + = + +
sum sum
sumsum sum
v v v
v v
MingsianMingsian R R BaiBai31
1 1 2 2 3 3
1 2 3
are mutually vectors having with arc length
k k k k kds h du U du
U du U du U duds ds ds
= =
rArr perp
v
v v v
1uv
2uv
3uv
1dS
2dS
3dS
dS
MingsianMingsian R R BaiBai32
( ) ( )
( ) ( )
1 1 2 2 3 3 1 2 3 1 2 3 1 2 3
1 2 3 1 2 3
1
1 2 2 3 3 2 2 2 3 3 3 2 3 2 3 1
Differential volume
Differential On const coord surface
area
Gradi t
en
d U du U du U du u u u h h h du du du
d h h h du du du
ud U du U du h u du h u du h h du du u
df
τ
τ
σ
= times sdot = times sdot
=
=
= times = times =
= nabla
v v v v v v
v vv v v v
1 2 31 2 3
1 2 3 1 1 2 2 3 31 2 3
1 1 1 2 2 2 3 3 3
f f ff dr df du du duu u u
r r rdr du du du U du U du U duu u u
h u du h u du h u du
part part partsdot rarr = + +part part part
part part part= + + = + +part part part
= + +
v
v v v v v vv
v v v
=1 (right hand rule)
MingsianMingsian R R BaiBai33
Rectangular Cartesian Coordinates
ij
k
MingsianMingsian R R BaiBai34
Circular Cylindrical Coordinatescossin
xyz z
ρ ϕρ ϕ
===
Ref Arfken
( )P zρ ϕ
MingsianMingsian R R BaiBai35
Polar Spherical Coordinatessin cossin sincos
x ry rz r
θ ϕθ ϕθ
===
Ref Arfken
0
0
0
sin cos sin sin coscos cos cos sin sin
sin cos
θ ϕ θ ϕ θθ ϕ θ ϕ θϕ ϕ
= + +
= + minus= minus +
r i j kθ i j kφ i j
2
2
sinsin
dS r d ddV r dr d d
θ θ ϕ
θ θ ϕ
=
=( )P r θ ϕ
MingsianMingsian R R BaiBai36
Integral TheoremsHeart of this chapter
Interchange of dimensionality Div Curl Grad in integrals High-dimensional integration
rarr
1
Thm1 Divergence theorem Let be a closed region bounded by piecewise smooth orientable surface and let (con
I Divergence theor
tinuous derivatives up to the first order
em (Gauss theorem)
)
VS v Cisin
r
ˆ If denotes the outward unit normal on
the ˆn V S
v dV n v dS
n S
nablasdot = sdotint intr r
MingsianMingsian R R BaiBai37
jS
jP
jV
S
0
ˆBy definition ( ) lim
ˆ( )
ˆ( )
j
j
j
j
Sj V
j
j j S
j j Sj j
n v dSv P
V
v P V n v dS
v P V n v dS
Δ rarr
⎧ ⎫sdot⎪ ⎪nabla sdot ⎨ ⎬Δ⎪ ⎪⎩ ⎭
nabla sdot Δ asymp sdot
nabla sdot Δ = sdot
int
intsum sumint
rr
r r
r r
Pf
MingsianMingsian R R BaiBai38
0
After interior surfaces cancel out only exterior surface (Why Opposite normal vectors)
ˆ( )
volumes exterior surfaces
ˆlim
QE
survive
j
j
j j Sj j
V SV
v p V n v dS
v dV n v dSΔ rarr
nabla sdot Δ = sdot
rArr nablasdot = sdot
sum sumint
int int
r r
r r
D
MingsianMingsian R R BaiBai39
( )
scalar variable constant vectorvector variable
Alternative formsLet be a an arbitrary and
a
ˆ
1 Le
(
ˆt
V
V S
S
V
v ap
v d
v dV vn d
V n v dS
v va
va dV v a v a
S
nablasdot = sdot
= rArr
nablasdot = nabla sdot + nabla
=
sdot
nablaint intint int
int
r
v
r r
r r
r r rQ ( )
( )
ˆ)
2 Let ˆV S
V
V S
dV n va dS
v a p
a p
p dV n p d
d p
S
V a
⎛ ⎞= sdot⎜ ⎟
⎝ ⎠
= times rArr
nablasdot times = sdot nabla
times
times
nabla = times
int int
int int
int
r
r r r
r r v
r
vQ
r
( ) ( ) ( )
( ) ( )
ˆ
ˆV S
V S
a p dV n a p dS
a p dV a n p dS
⎛ ⎞⎡ ⎤minus sdot nablatimes = sdot times⎣ ⎦⎜ ⎟⎜ ⎟⎜ ⎟rArr minus sdot nablatimes = minus sdot times⎜ ⎟⎝ ⎠
int int
int int
r rv v
r rv v
MingsianMingsian R R BaiBai40
( )
( )
From Leibnitzs rule
(1-D) ( )
( ) ( )
B t
A t
d f x t dxdt
f x t dx B ttpart prime= +part
int( )
( )( ( ) ) ( )
B t
A tf B t t A tprimeminusint ( ( ) )
(assume ( ) ( ) )
f A t t
A t B t const= =
ContinuityEx The Equation
rate of increase of mass
mass densityV
dM d dVdt dt
ρ
ρ
= int
flow velocityvv
MingsianMingsian R R BaiBai41
ˆ ˆ ( )
(
Conservation
) ( Divergence theorem)
of mass
( ) 0
V S S
V
V
dV n v dS n v dSt
v dV
v dVt
ρ ρ ρ
ρ
ρ ρ⎡ ⎤⎢ ⎥⎣ ⎦
part = minus sdot = minus sdotpart
= minus nablasdot
partrArr +nablasdot =part
int int int
int
int
v v
vQ
v
ˆOn the other hand rate influof mass through n ox i t
Extended to 3-D for time-independent fixed boundary
(3-D) ( ) ( )
Thus
S
V V
V
S V n v dS
Sd f x y z t dV f x y z t dVdt tdM dVdt t
ρ
ρ
= minus sdot
part=part
part=part
int
int int
intv
Fixed control volume (Eulerian coordinates)Otherwise Raynoldrsquos transport theoremMoving control volume Lagrangian coordinates
MingsianMingsian R R BaiBai42
But the control volime is only arbitrary
In particular for flow
( ) 0
con
(continuity equati
stant
on)
Divergence the
0 incompress
Note is t
ible
orem h
V
v
vt
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
e mother of the other integral theorems
MingsianMingsian R R BaiBai43
Ex Archimedes principle ( ) The buoyant force ( ) on a body with volume immersed in a fluid is equal to the weight of the fluid
ˆ displaced by the object ie In the followin
B
V
gVρ= minusf k
阿基米德原理
浮力
ˆg figure the ˆ with and being the fluid density gravitational
acceleration depth and outward normal unit vector to the surface
fhydraulic pressure g z
g z
S
ρ
ρ
= minusp n
n
1
the modified divergence theorem
ˆ
Hint
scalar fuctionV S
vdV v dS v Cnabla = isinint int n
MingsianMingsian R R BaiBai44
Levelsurfacefluid
V
S
airyx
z
ˆf g zρ= minusp n
ˆPf ( ) ( )
ˆ ˆ ˆ ˆ ˆ (0 0 ) v v
B fS S V
V V
g z g zdS dS dV
g dV g dV gV
ρ ρ
ρ ρ ρ
= = minus = minus nabla
= minus + + = minus = minus
int int int
int int
f p n
i j k k k
k
MingsianMingsian R R BaiBai45
Note This can also be thought of as simple balance of forces (resultant force that acts on and weight of filled with fluid)S V
2
~ a little history about George Green Recall the diveregence theorem
ˆ
Let
II th
e Green
are arbitrary fun
s identiti
ctions
es
V S
v dV n v dS
v u vu v C
nablasdot = sdot
= nablaisin
int intv v
v
ˆ ( ) ( )
ˆ ( )V S
V S
u v dV n u v dS
u v u v dV un v dS
nablasdot nabla = sdot nabla
nabla sdotnabla + nablasdotnabla = sdotnabla
int int
int int
MingsianMingsian R R BaiBai46
2 -( ) -(1)V S
vu v u v dV u dSnpart
nabla sdotnabla + nabla =partint int firs(Greens identt ity)
2
If
( ) --(2)
(1) (2) V S
u vuv u v u dV v dSn
harrpart
nabla sdotnabla + nabla =part
minus
int int
( )2 2
V S
dv duu v v u dV u v dSdn dn
⎛ ⎞nabla minus nabla = minus⎜ ⎟⎝ ⎠int int
second(Greens identity)
MingsianMingsian R R BaiBai47
( )
( )
( ) ( )
1st id ( )
Greens identities in the 1D case integration by parts
b b bb ba a
a a ab
ba
ab
ba
a
v udx v u v u dx u v uv dx uv
u v u v vu dx vu
uv vu dx uv vu
primeprime prime prime prime prime prime primeprime prime= minus rArr + =
prime prime primeprime primeharr + =
primeprime primeprime prime primerArr minus = minus
int int int
int
2
2nd id ( )
Good for any adjoint operators in addition to Good also for and integration by parts Important for theory amp reciprocity
From th
Gree
e 1s
2-
t
ns functi
identity
D 1-
tak 1
D
e
on
u
nablararr
=
int
2
V S
vv dV dSnpartnabla =part
rArr int int2Ex 0 st 0 on the boundary 0 ( 1st id u u unabla = = rArr equiv 以 証)
MingsianMingsian R R BaiBai48
1
Thm1 Stokes theorem Let be defined in Let be a piecewise-smooth orientable surface bounded by a piec
III Stokes theorem and
ewise-smooth simple clos
Greens
ed curve Then
theore
m
v C R S
C
isinv
( )
ˆwhere is a unit normal to according t right hand rulo the e
ˆS C
n S
n v dS v dRsdot nablatimes = sdotint intvv v
(not necessarily planar)S
Cn vnablatimes v vv
MingsianMingsian R R BaiBai49
Pf
( )( )
circulatioWe have shown in deriving thatthe in -direction
ˆor
ˆ
n
z
z
j j j j j
j j j j jj j
vz v dx dy
v dR n v S
v dR n v S
nablatimes
=nablatimes
sdot = sdot nablatimes Δ
sdot = sdot nablatimes Δ
intsum sumint
v
v
vv v
vv v
C
S
jS
jC
jP
ˆ jn
jSΔ
MingsianMingsian R R BaiBai50
( )
( )ext surface
After the interior boundaries cancel out only the exterior boundaries (Why)
ˆ
ˆ0 QED
Alternative forms scalar
survive
va
j
j j j j jC
jC S
v dR n v S
S v dR n v dS
φ
sdot = sdot nablatimes Δ
Δ rarr rArr sdot = sdot nablatimes
sum sumint
int int
vv v
vv v
( )
riable constant vector vector variable
1 Let
2 Let
ˆ
ˆ S C
S C
a p
v n dR dR
n p dS dR p
a
v a p
φ φφ= rArr timesnabla =
timesnabla times= rArr timestimes =
int int
int int
v v
v v
v
v v
vv vv v
MingsianMingsian R R BaiBai51
( ) ( )
( )
( )
( ) ( )
1
Thm2 Greens theorem ( planar version of Stokes theorem)ˆ ˆ Let
pf By Stokes theoremˆ
ˆ ˆ ˆ ˆ ˆ ˆ
S C
S C
S C
v P x y i Q x y j C
n v dS v dR
k Pi Qj dS Pi Qj dx i dy
Q P dS Pdx Qdyx y
⎛ ⎞part partminus = +⎜ ⎟
= + isin
sdot nablatimes = sdot
⎡ ⎤sdot nablatimes + =
part part⎝ ⎠
+ sdot +⎣ ⎦
int int
int
int
int
int
v
vv v
( )
( )
( )
ˆ
ˆˆ ˆ
ˆˆ ˆ ˆ ˆ( ) ( )
0
QEDS C
j
i j kQ PPi Qj i j k
x y z x yP Q
Q P dS Pdx Qdyx y
⎛ ⎞part part part part partnablatimes + = = + + minus⎜ ⎟part part part part part⎝ ⎠
⎛ ⎞part partrArr minus = +⎜ ⎟part part⎝ ⎠int int
L L
S
C
i
j
x
y
MingsianMingsian R R BaiBai52
Potential TheorypotentiGeorge Green coined the term (l
a )
rarr 位能 勢能
例子 重力場 電磁場 流場 聲場 彈性力學
1
2
Thm1 If is in a simply connected region then the following statements are equivalent(i) There exists a scala
Conservative Field
potential functior such that
(ii
n
) 0
v
C
C
v
v R
φφ=nabla
nablatimes
isin
isin
=r
r
r
r in (Irrotational field curl-free)
cf 0 st solenoidal field divergence-free
0
(iii) in (Conservative field)C
Rv w v w
v R Rd
nablasdot = exist nabla
sdot =
= times
int
r r r
r
r
r
MingsianMingsian R R BaiBai53
(iv) is connecting and
independent o
Ex Think
f the pa
of dynamics is the force
thB
A
v dR A B
v
sdotintrr
r
1 2
1
0
Pf
(i) (ii) ( ) 0 (zero identity cg rule) irrotational
ˆ(ii) (iii) ( ) 0 (Stokes theorem)
(iii) (iv) ( ) 0
C S
C C C
C
v
v dR n v dS
v dR v dR
v dR
φ
=
rarr nablatimes = nablatimes nabla =
rarr sdot = sdot nablatimes =
rarr sdot = minus sdot =
rArr sdot =
int int
int int int
int
r
rr
rr r
r rr r
rr r
2
independent of pathC
v dRsdotintr
A
B
1C
2C
MingsianMingsian R R BaiBai54
1 2
(iv) (i) constant ( ) ( )
--(1)
But (2)
(1) (2) QED
Note on application
Check
B
C C A
v dR v dR B A d
v dR d
d dx dy dz dRx y z
v dR dR v R
φ φ φ
φφ φ φφ φ
φ φ
rarr sdot = sdot = = minus =
rArr sdot =part part part
= + + = nabla sdot minus minus minus minus minuspart part part
sdot = nabla sdot rArr = nabla forall
int int intr rr r
rr
r
r r rr r
use simpler path (iv)if 0
find (i)v
φ⎧
nablatimes = ⎨⎩
rr
A
B
x
y
Cz
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai16
yy
vv dy
ypart
+part
Γdyminus
dyyvv x
x partpart
+
dy
dx
dxminus
y yy
v vv dx dy
x ypart part
+ +part part
x xx
v vv dx dyx y
part part+ +part part
yy
vv dx
xpart
+part
xx
vv dxx
part+part
xv
yv
vv
dλ paddle weel
MingsianMingsian R R BaiBai17
contour
1 2 3 4
circulation
x x y y x x y y
x
v d
v d v d v d v d
v
λ
λ λ λ λrarr
Γ sdot
= + + +
asymp
intint int int int
vv
12
xv dxx
part+part ydx v
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
+ 12
y yv vdx dy
x ypart part
+ +part part
x
dy
v
⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦
+
12
xv dxx
part+part
( )xy
v dy dx vy
⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦
part+ minus +part
12
yvdy
ypart
+part
( )
( )
Similar process can be applied to the circulation in the and directionsThus the total circulation per unit area i curl s
y xz
dy
v v dxdy v dxdyx y
xv v
y
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎝ ⎠
minus
part part= minus = nablatimespart part
=nablatimes
v
v v
MingsianMingsian R R BaiBai18
Identities of differential operators ( p380 Greenberg )
( )
( ) ( )( )( )
( ) ( )( ) is required
Example of deriv
ˆ ˆˆ ˆ ˆ ˆ
ation
x y z
x y z
v v vx y z
uv u v u v
uv i j k uv i uv j uv kx y z
uv u v u v
u v v u u v
u v u vv v u u
+ +part part partpart part part
⎛ ⎞⎜ ⎟⎝ ⎠
nablasdot = nabla sdot + nablasdot
part part partnablasdot = + + sdot + + =part part part
nablatimes =nabla times + nablatimes
nablasdot times = sdotnablatimes minus sdotnablatimes
nablatimes times = nabla minus + minusnablanablasdot sdotsdot678 678
v
v v v
vL
v v v
v v v v v v
v v v v v v v v( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )2 cf
x y zu u ux y z
A B C A
u v
v w v w w v v w w v
v v v C B A B C
+ +part part partpart part part
sdotnabla
nabla sdot = sdotnabla + sdotnabla + times nablatimes + times
times
nablatimes
nablatimes nablatimes =nabla nabla times = sdot minussdot sdotminusnabla
678v
v v v
v v v v v vv v
v v v v v v
v v
v
v v
ˆ ˆˆ ˆ ˆ ˆ( ) ( ) ( )x y z x y zv i v j v k i j k u v v v ux y z x y zpart part part part part part+ + sdot + + = + +part part part part part part
v v
MingsianMingsian R R BaiBai19
2 2 22
2 2 2
0 (dc direct current) solenoidal0 (cg center of gravity) conservative potent
Laplacia
Zero identitie
n
s
ial
x y z
vu
⎧⎪⎨⎪⎩
part part partΔ =nabla nablasdotnabla = + +part part part
nablasdotnablatimes = rarr
nablatimesnabla = rarr
v
v
These differential operators will take more complicated and irregular form for the other coordinate systems
MingsianMingsian R R BaiBai20
2 2 3
2 2
ˆˆ ˆEx 2
ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ 2 0 2
2 0 0 2yx z
u x y v x yi z j k
u i j k u xyi x j k xyi x jx y z
vv vv xy xyx y z
⎛ ⎞⎜ ⎟⎝ ⎠
= = minus +
part part partnabla = + + = + + = +part part part
partpart partnablasdot = + + = + minus =part part part
v
v
MingsianMingsian R R BaiBai21
( ) ( ) ( )
( ) ( )
2 2 2 2
2 2 22
2 2 2
2
ˆˆ ˆ
ˆˆ ˆ
ˆ ˆˆ ˆ ˆ 0 3 0 0 0 3
2 0 0 2
ˆˆ ˆ
y yx xz z
x y z
i j kv vv vv vv i j k
x y z y z x z x yv v v
z i j x k z i x k
u u uu y yx y zv v v
i j kx y z
⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠ ⎝ ⎠
⎛
⎝
part partpart partpart partpart part partnablatimes = = minus minus minus + minuspart part part part part part part part part
= + minus minus + minus = minus
part part partnabla = + + = + + =part part part
nablatimes nablatimes =nabla nablasdot minusnabla
part part part= + +part part part
v
v v v
( ) ( )( ) ( ) ( ) ( )
( )
2 2 22 3
2 2 2ˆˆ ˆ2 2
ˆ ˆˆ ˆ ˆ ˆ 2 2 0 2 0 0 0 0 6 0 0 0
ˆ 2 6
xy x yi z j kx y z
yi xj k y i z j k
x z j
⎛ ⎞⎞⎜ ⎟⎜ ⎟
⎠ ⎝ ⎠⎡ ⎤⎣ ⎦
part part partminus + + minus +part part part
= + + minus + + + + minus + + minus
= +
MingsianMingsian R R BaiBai22
( )v w r w r r wnablatimes =nablatimes times = nablasdot minus nablasdotv v v v v v v ( )r w+ sdotnablav v ( ) constant w
w rminus sdotnablav
v v
Q
Ex Rigid body rotation
MingsianMingsian R R BaiBai23
( )( ) ( )
( ) ( )
ˆˆ ˆ
ˆ ˆˆ ˆ ˆ ˆ 1 1 1 3
ˆˆ ˆ
ˆˆ ˆ
Thus 3 2 is linked directly
x y z
x y z
r xi yj zk
r i j k xi yj zkx y z
w r w w w xi yj zkx y z
w i w j w k w
w r w r w r w w wv
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎝ ⎠
= + +
part part partnablasdot = + + + + = + + =part part part
part part partsdotnabla = + + + +part part part
= + + =
nablatimes times == nablasdot minus sdotnabla = minus =rArrnablatimes
v
v
v v
v
v v v v v v v v v
v spinn to ing
MingsianMingsian R R BaiBai24
Line surface and volume integrals
x
yC
B
A
isΔ
( )ii ηξ
Piecewise smooth curve
Line integral
MingsianMingsian R R BaiBai25
0 1lim ( ) ( ) or ( )
Intepretation of a line integral as an area ( )
i
Bn
i i isA
C
i C
F s F x y ds F x y d
x
s
F y ds
ξ ηΔ rarr =
Δ =sum int
int
int
x
y
z
C
( )z F x y=
MingsianMingsian R R BaiBai26
the area is always on the left of an obs
Positive dierver trave
rection of irsing on the
ntegrationcurve
RC
CCW CW
If is a regular closed curve piecewise smo
( ) ( )
(contou
oth simple ( )
r integral)
x y zC C
x y zC
F dR F F F dx dy dz
F dx F dy
C
F dz
sdot = + + + +
= + +
lowast
int int
int
i j k i j kv v
無交叉
work dW F dR= sdotv v
MingsianMingsian R R BaiBai27
0 1
Surface integral
lim ( ) ( )i
n
i i i iS i S
F S F x y z dSξ η ζΔ rarr =
Δ =sum intint( ) i i iξ η ζ
iSΔ
0 1
mass density
lim ( ) ( )
Ex
Volume integral
( )
i
n
i i i iV i V
V
F V F x y z dV
M x y z dV
ξ η ζ
ρ
Δ rarr =Δ =
=
sum intintint
intintint( ) i i iξ η ζ
iVΔ
MingsianMingsian R R BaiBai28
Ref (1)Hildebrand advanced Calculus for application Prentice-Hall 1976 p306-313(2) Greenberg p383-389 (3) Arfken ch2
1 2 3
1 2 3
1 2 3
1 2 3
(
Orthogonal coordinate systems base vectors are not constant in s
) ( )( )( )( )
ˆˆ ˆ
pacep x y z p u u ux x u u uy y u u uz z u u u
r xi yj zk
primerarr
= +
rarr
===
+v
1uv
2uv3uv
1u
2u
3uP coord curve
xy
z
O
X Curvilinear Coordinates
MingsianMingsian R R BaiBai29
1
11
1 1 1
1
1
1 1 1
11 1
1 1
2 2 2 3 3 3
Tangent vector to the at
where is the arc length
1
where unit vector
Similarly
curv
e
u psr rU
u s us
rs
U h uds rh udu sU h u U h u
partpart part= =part part part
part =part
there4 =part= =part
= =
v vv
vQ
v v
vv
v vv v
1u cood curve
1uv
1du 1dS
drv
P
MingsianMingsian R R BaiBai30
3 31 2 1 21 2 3
1 2 32
curv
In summary scale factor
If is the arc length along a in any directione
k k k
kk k
k k
k k k
U h u
ds rh Udu u
ds h dus
du dudu du du dudr r r r U U Uds u ds u ds u ds ds ds ds
dr drds ds
=
part= = =part
=
part part part= + + = + +part part part
= sdot
v v
v v
v v v v v v v
v v 3 3
1 1
23 3 32 2
21 1 1
2 2 2 2 2 2 2 2 2 21 1 2 2 3 3 1 2 3
1 (for orthogonal coord)
jii j
i j
ji ii j i
i j i
dududr U Uds ds ds
dudu duU U hds ds ds
ds h du h du h du ds ds ds
= =
= = =
⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
= sdot
rarr = sdot =
rarr = + + = + +
sum sum
sumsum sum
v v v
v v
MingsianMingsian R R BaiBai31
1 1 2 2 3 3
1 2 3
are mutually vectors having with arc length
k k k k kds h du U du
U du U du U duds ds ds
= =
rArr perp
v
v v v
1uv
2uv
3uv
1dS
2dS
3dS
dS
MingsianMingsian R R BaiBai32
( ) ( )
( ) ( )
1 1 2 2 3 3 1 2 3 1 2 3 1 2 3
1 2 3 1 2 3
1
1 2 2 3 3 2 2 2 3 3 3 2 3 2 3 1
Differential volume
Differential On const coord surface
area
Gradi t
en
d U du U du U du u u u h h h du du du
d h h h du du du
ud U du U du h u du h u du h h du du u
df
τ
τ
σ
= times sdot = times sdot
=
=
= times = times =
= nabla
v v v v v v
v vv v v v
1 2 31 2 3
1 2 3 1 1 2 2 3 31 2 3
1 1 1 2 2 2 3 3 3
f f ff dr df du du duu u u
r r rdr du du du U du U du U duu u u
h u du h u du h u du
part part partsdot rarr = + +part part part
part part part= + + = + +part part part
= + +
v
v v v v v vv
v v v
=1 (right hand rule)
MingsianMingsian R R BaiBai33
Rectangular Cartesian Coordinates
ij
k
MingsianMingsian R R BaiBai34
Circular Cylindrical Coordinatescossin
xyz z
ρ ϕρ ϕ
===
Ref Arfken
( )P zρ ϕ
MingsianMingsian R R BaiBai35
Polar Spherical Coordinatessin cossin sincos
x ry rz r
θ ϕθ ϕθ
===
Ref Arfken
0
0
0
sin cos sin sin coscos cos cos sin sin
sin cos
θ ϕ θ ϕ θθ ϕ θ ϕ θϕ ϕ
= + +
= + minus= minus +
r i j kθ i j kφ i j
2
2
sinsin
dS r d ddV r dr d d
θ θ ϕ
θ θ ϕ
=
=( )P r θ ϕ
MingsianMingsian R R BaiBai36
Integral TheoremsHeart of this chapter
Interchange of dimensionality Div Curl Grad in integrals High-dimensional integration
rarr
1
Thm1 Divergence theorem Let be a closed region bounded by piecewise smooth orientable surface and let (con
I Divergence theor
tinuous derivatives up to the first order
em (Gauss theorem)
)
VS v Cisin
r
ˆ If denotes the outward unit normal on
the ˆn V S
v dV n v dS
n S
nablasdot = sdotint intr r
MingsianMingsian R R BaiBai37
jS
jP
jV
S
0
ˆBy definition ( ) lim
ˆ( )
ˆ( )
j
j
j
j
Sj V
j
j j S
j j Sj j
n v dSv P
V
v P V n v dS
v P V n v dS
Δ rarr
⎧ ⎫sdot⎪ ⎪nabla sdot ⎨ ⎬Δ⎪ ⎪⎩ ⎭
nabla sdot Δ asymp sdot
nabla sdot Δ = sdot
int
intsum sumint
rr
r r
r r
Pf
MingsianMingsian R R BaiBai38
0
After interior surfaces cancel out only exterior surface (Why Opposite normal vectors)
ˆ( )
volumes exterior surfaces
ˆlim
QE
survive
j
j
j j Sj j
V SV
v p V n v dS
v dV n v dSΔ rarr
nabla sdot Δ = sdot
rArr nablasdot = sdot
sum sumint
int int
r r
r r
D
MingsianMingsian R R BaiBai39
( )
scalar variable constant vectorvector variable
Alternative formsLet be a an arbitrary and
a
ˆ
1 Le
(
ˆt
V
V S
S
V
v ap
v d
v dV vn d
V n v dS
v va
va dV v a v a
S
nablasdot = sdot
= rArr
nablasdot = nabla sdot + nabla
=
sdot
nablaint intint int
int
r
v
r r
r r
r r rQ ( )
( )
ˆ)
2 Let ˆV S
V
V S
dV n va dS
v a p
a p
p dV n p d
d p
S
V a
⎛ ⎞= sdot⎜ ⎟
⎝ ⎠
= times rArr
nablasdot times = sdot nabla
times
times
nabla = times
int int
int int
int
r
r r r
r r v
r
vQ
r
( ) ( ) ( )
( ) ( )
ˆ
ˆV S
V S
a p dV n a p dS
a p dV a n p dS
⎛ ⎞⎡ ⎤minus sdot nablatimes = sdot times⎣ ⎦⎜ ⎟⎜ ⎟⎜ ⎟rArr minus sdot nablatimes = minus sdot times⎜ ⎟⎝ ⎠
int int
int int
r rv v
r rv v
MingsianMingsian R R BaiBai40
( )
( )
From Leibnitzs rule
(1-D) ( )
( ) ( )
B t
A t
d f x t dxdt
f x t dx B ttpart prime= +part
int( )
( )( ( ) ) ( )
B t
A tf B t t A tprimeminusint ( ( ) )
(assume ( ) ( ) )
f A t t
A t B t const= =
ContinuityEx The Equation
rate of increase of mass
mass densityV
dM d dVdt dt
ρ
ρ
= int
flow velocityvv
MingsianMingsian R R BaiBai41
ˆ ˆ ( )
(
Conservation
) ( Divergence theorem)
of mass
( ) 0
V S S
V
V
dV n v dS n v dSt
v dV
v dVt
ρ ρ ρ
ρ
ρ ρ⎡ ⎤⎢ ⎥⎣ ⎦
part = minus sdot = minus sdotpart
= minus nablasdot
partrArr +nablasdot =part
int int int
int
int
v v
vQ
v
ˆOn the other hand rate influof mass through n ox i t
Extended to 3-D for time-independent fixed boundary
(3-D) ( ) ( )
Thus
S
V V
V
S V n v dS
Sd f x y z t dV f x y z t dVdt tdM dVdt t
ρ
ρ
= minus sdot
part=part
part=part
int
int int
intv
Fixed control volume (Eulerian coordinates)Otherwise Raynoldrsquos transport theoremMoving control volume Lagrangian coordinates
MingsianMingsian R R BaiBai42
But the control volime is only arbitrary
In particular for flow
( ) 0
con
(continuity equati
stant
on)
Divergence the
0 incompress
Note is t
ible
orem h
V
v
vt
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
e mother of the other integral theorems
MingsianMingsian R R BaiBai43
Ex Archimedes principle ( ) The buoyant force ( ) on a body with volume immersed in a fluid is equal to the weight of the fluid
ˆ displaced by the object ie In the followin
B
V
gVρ= minusf k
阿基米德原理
浮力
ˆg figure the ˆ with and being the fluid density gravitational
acceleration depth and outward normal unit vector to the surface
fhydraulic pressure g z
g z
S
ρ
ρ
= minusp n
n
1
the modified divergence theorem
ˆ
Hint
scalar fuctionV S
vdV v dS v Cnabla = isinint int n
MingsianMingsian R R BaiBai44
Levelsurfacefluid
V
S
airyx
z
ˆf g zρ= minusp n
ˆPf ( ) ( )
ˆ ˆ ˆ ˆ ˆ (0 0 ) v v
B fS S V
V V
g z g zdS dS dV
g dV g dV gV
ρ ρ
ρ ρ ρ
= = minus = minus nabla
= minus + + = minus = minus
int int int
int int
f p n
i j k k k
k
MingsianMingsian R R BaiBai45
Note This can also be thought of as simple balance of forces (resultant force that acts on and weight of filled with fluid)S V
2
~ a little history about George Green Recall the diveregence theorem
ˆ
Let
II th
e Green
are arbitrary fun
s identiti
ctions
es
V S
v dV n v dS
v u vu v C
nablasdot = sdot
= nablaisin
int intv v
v
ˆ ( ) ( )
ˆ ( )V S
V S
u v dV n u v dS
u v u v dV un v dS
nablasdot nabla = sdot nabla
nabla sdotnabla + nablasdotnabla = sdotnabla
int int
int int
MingsianMingsian R R BaiBai46
2 -( ) -(1)V S
vu v u v dV u dSnpart
nabla sdotnabla + nabla =partint int firs(Greens identt ity)
2
If
( ) --(2)
(1) (2) V S
u vuv u v u dV v dSn
harrpart
nabla sdotnabla + nabla =part
minus
int int
( )2 2
V S
dv duu v v u dV u v dSdn dn
⎛ ⎞nabla minus nabla = minus⎜ ⎟⎝ ⎠int int
second(Greens identity)
MingsianMingsian R R BaiBai47
( )
( )
( ) ( )
1st id ( )
Greens identities in the 1D case integration by parts
b b bb ba a
a a ab
ba
ab
ba
a
v udx v u v u dx u v uv dx uv
u v u v vu dx vu
uv vu dx uv vu
primeprime prime prime prime prime prime primeprime prime= minus rArr + =
prime prime primeprime primeharr + =
primeprime primeprime prime primerArr minus = minus
int int int
int
2
2nd id ( )
Good for any adjoint operators in addition to Good also for and integration by parts Important for theory amp reciprocity
From th
Gree
e 1s
2-
t
ns functi
identity
D 1-
tak 1
D
e
on
u
nablararr
=
int
2
V S
vv dV dSnpartnabla =part
rArr int int2Ex 0 st 0 on the boundary 0 ( 1st id u u unabla = = rArr equiv 以 証)
MingsianMingsian R R BaiBai48
1
Thm1 Stokes theorem Let be defined in Let be a piecewise-smooth orientable surface bounded by a piec
III Stokes theorem and
ewise-smooth simple clos
Greens
ed curve Then
theore
m
v C R S
C
isinv
( )
ˆwhere is a unit normal to according t right hand rulo the e
ˆS C
n S
n v dS v dRsdot nablatimes = sdotint intvv v
(not necessarily planar)S
Cn vnablatimes v vv
MingsianMingsian R R BaiBai49
Pf
( )( )
circulatioWe have shown in deriving thatthe in -direction
ˆor
ˆ
n
z
z
j j j j j
j j j j jj j
vz v dx dy
v dR n v S
v dR n v S
nablatimes
=nablatimes
sdot = sdot nablatimes Δ
sdot = sdot nablatimes Δ
intsum sumint
v
v
vv v
vv v
C
S
jS
jC
jP
ˆ jn
jSΔ
MingsianMingsian R R BaiBai50
( )
( )ext surface
After the interior boundaries cancel out only the exterior boundaries (Why)
ˆ
ˆ0 QED
Alternative forms scalar
survive
va
j
j j j j jC
jC S
v dR n v S
S v dR n v dS
φ
sdot = sdot nablatimes Δ
Δ rarr rArr sdot = sdot nablatimes
sum sumint
int int
vv v
vv v
( )
riable constant vector vector variable
1 Let
2 Let
ˆ
ˆ S C
S C
a p
v n dR dR
n p dS dR p
a
v a p
φ φφ= rArr timesnabla =
timesnabla times= rArr timestimes =
int int
int int
v v
v v
v
v v
vv vv v
MingsianMingsian R R BaiBai51
( ) ( )
( )
( )
( ) ( )
1
Thm2 Greens theorem ( planar version of Stokes theorem)ˆ ˆ Let
pf By Stokes theoremˆ
ˆ ˆ ˆ ˆ ˆ ˆ
S C
S C
S C
v P x y i Q x y j C
n v dS v dR
k Pi Qj dS Pi Qj dx i dy
Q P dS Pdx Qdyx y
⎛ ⎞part partminus = +⎜ ⎟
= + isin
sdot nablatimes = sdot
⎡ ⎤sdot nablatimes + =
part part⎝ ⎠
+ sdot +⎣ ⎦
int int
int
int
int
int
v
vv v
( )
( )
( )
ˆ
ˆˆ ˆ
ˆˆ ˆ ˆ ˆ( ) ( )
0
QEDS C
j
i j kQ PPi Qj i j k
x y z x yP Q
Q P dS Pdx Qdyx y
⎛ ⎞part part part part partnablatimes + = = + + minus⎜ ⎟part part part part part⎝ ⎠
⎛ ⎞part partrArr minus = +⎜ ⎟part part⎝ ⎠int int
L L
S
C
i
j
x
y
MingsianMingsian R R BaiBai52
Potential TheorypotentiGeorge Green coined the term (l
a )
rarr 位能 勢能
例子 重力場 電磁場 流場 聲場 彈性力學
1
2
Thm1 If is in a simply connected region then the following statements are equivalent(i) There exists a scala
Conservative Field
potential functior such that
(ii
n
) 0
v
C
C
v
v R
φφ=nabla
nablatimes
isin
isin
=r
r
r
r in (Irrotational field curl-free)
cf 0 st solenoidal field divergence-free
0
(iii) in (Conservative field)C
Rv w v w
v R Rd
nablasdot = exist nabla
sdot =
= times
int
r r r
r
r
r
MingsianMingsian R R BaiBai53
(iv) is connecting and
independent o
Ex Think
f the pa
of dynamics is the force
thB
A
v dR A B
v
sdotintrr
r
1 2
1
0
Pf
(i) (ii) ( ) 0 (zero identity cg rule) irrotational
ˆ(ii) (iii) ( ) 0 (Stokes theorem)
(iii) (iv) ( ) 0
C S
C C C
C
v
v dR n v dS
v dR v dR
v dR
φ
=
rarr nablatimes = nablatimes nabla =
rarr sdot = sdot nablatimes =
rarr sdot = minus sdot =
rArr sdot =
int int
int int int
int
r
rr
rr r
r rr r
rr r
2
independent of pathC
v dRsdotintr
A
B
1C
2C
MingsianMingsian R R BaiBai54
1 2
(iv) (i) constant ( ) ( )
--(1)
But (2)
(1) (2) QED
Note on application
Check
B
C C A
v dR v dR B A d
v dR d
d dx dy dz dRx y z
v dR dR v R
φ φ φ
φφ φ φφ φ
φ φ
rarr sdot = sdot = = minus =
rArr sdot =part part part
= + + = nabla sdot minus minus minus minus minuspart part part
sdot = nabla sdot rArr = nabla forall
int int intr rr r
rr
r
r r rr r
use simpler path (iv)if 0
find (i)v
φ⎧
nablatimes = ⎨⎩
rr
A
B
x
y
Cz
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai17
contour
1 2 3 4
circulation
x x y y x x y y
x
v d
v d v d v d v d
v
λ
λ λ λ λrarr
Γ sdot
= + + +
asymp
intint int int int
vv
12
xv dxx
part+part ydx v
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
+ 12
y yv vdx dy
x ypart part
+ +part part
x
dy
v
⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦
+
12
xv dxx
part+part
( )xy
v dy dx vy
⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦
part+ minus +part
12
yvdy
ypart
+part
( )
( )
Similar process can be applied to the circulation in the and directionsThus the total circulation per unit area i curl s
y xz
dy
v v dxdy v dxdyx y
xv v
y
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎝ ⎠
minus
part part= minus = nablatimespart part
=nablatimes
v
v v
MingsianMingsian R R BaiBai18
Identities of differential operators ( p380 Greenberg )
( )
( ) ( )( )( )
( ) ( )( ) is required
Example of deriv
ˆ ˆˆ ˆ ˆ ˆ
ation
x y z
x y z
v v vx y z
uv u v u v
uv i j k uv i uv j uv kx y z
uv u v u v
u v v u u v
u v u vv v u u
+ +part part partpart part part
⎛ ⎞⎜ ⎟⎝ ⎠
nablasdot = nabla sdot + nablasdot
part part partnablasdot = + + sdot + + =part part part
nablatimes =nabla times + nablatimes
nablasdot times = sdotnablatimes minus sdotnablatimes
nablatimes times = nabla minus + minusnablanablasdot sdotsdot678 678
v
v v v
vL
v v v
v v v v v v
v v v v v v v v( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )2 cf
x y zu u ux y z
A B C A
u v
v w v w w v v w w v
v v v C B A B C
+ +part part partpart part part
sdotnabla
nabla sdot = sdotnabla + sdotnabla + times nablatimes + times
times
nablatimes
nablatimes nablatimes =nabla nabla times = sdot minussdot sdotminusnabla
678v
v v v
v v v v v vv v
v v v v v v
v v
v
v v
ˆ ˆˆ ˆ ˆ ˆ( ) ( ) ( )x y z x y zv i v j v k i j k u v v v ux y z x y zpart part part part part part+ + sdot + + = + +part part part part part part
v v
MingsianMingsian R R BaiBai19
2 2 22
2 2 2
0 (dc direct current) solenoidal0 (cg center of gravity) conservative potent
Laplacia
Zero identitie
n
s
ial
x y z
vu
⎧⎪⎨⎪⎩
part part partΔ =nabla nablasdotnabla = + +part part part
nablasdotnablatimes = rarr
nablatimesnabla = rarr
v
v
These differential operators will take more complicated and irregular form for the other coordinate systems
MingsianMingsian R R BaiBai20
2 2 3
2 2
ˆˆ ˆEx 2
ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ 2 0 2
2 0 0 2yx z
u x y v x yi z j k
u i j k u xyi x j k xyi x jx y z
vv vv xy xyx y z
⎛ ⎞⎜ ⎟⎝ ⎠
= = minus +
part part partnabla = + + = + + = +part part part
partpart partnablasdot = + + = + minus =part part part
v
v
MingsianMingsian R R BaiBai21
( ) ( ) ( )
( ) ( )
2 2 2 2
2 2 22
2 2 2
2
ˆˆ ˆ
ˆˆ ˆ
ˆ ˆˆ ˆ ˆ 0 3 0 0 0 3
2 0 0 2
ˆˆ ˆ
y yx xz z
x y z
i j kv vv vv vv i j k
x y z y z x z x yv v v
z i j x k z i x k
u u uu y yx y zv v v
i j kx y z
⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠ ⎝ ⎠
⎛
⎝
part partpart partpart partpart part partnablatimes = = minus minus minus + minuspart part part part part part part part part
= + minus minus + minus = minus
part part partnabla = + + = + + =part part part
nablatimes nablatimes =nabla nablasdot minusnabla
part part part= + +part part part
v
v v v
( ) ( )( ) ( ) ( ) ( )
( )
2 2 22 3
2 2 2ˆˆ ˆ2 2
ˆ ˆˆ ˆ ˆ ˆ 2 2 0 2 0 0 0 0 6 0 0 0
ˆ 2 6
xy x yi z j kx y z
yi xj k y i z j k
x z j
⎛ ⎞⎞⎜ ⎟⎜ ⎟
⎠ ⎝ ⎠⎡ ⎤⎣ ⎦
part part partminus + + minus +part part part
= + + minus + + + + minus + + minus
= +
MingsianMingsian R R BaiBai22
( )v w r w r r wnablatimes =nablatimes times = nablasdot minus nablasdotv v v v v v v ( )r w+ sdotnablav v ( ) constant w
w rminus sdotnablav
v v
Q
Ex Rigid body rotation
MingsianMingsian R R BaiBai23
( )( ) ( )
( ) ( )
ˆˆ ˆ
ˆ ˆˆ ˆ ˆ ˆ 1 1 1 3
ˆˆ ˆ
ˆˆ ˆ
Thus 3 2 is linked directly
x y z
x y z
r xi yj zk
r i j k xi yj zkx y z
w r w w w xi yj zkx y z
w i w j w k w
w r w r w r w w wv
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎝ ⎠
= + +
part part partnablasdot = + + + + = + + =part part part
part part partsdotnabla = + + + +part part part
= + + =
nablatimes times == nablasdot minus sdotnabla = minus =rArrnablatimes
v
v
v v
v
v v v v v v v v v
v spinn to ing
MingsianMingsian R R BaiBai24
Line surface and volume integrals
x
yC
B
A
isΔ
( )ii ηξ
Piecewise smooth curve
Line integral
MingsianMingsian R R BaiBai25
0 1lim ( ) ( ) or ( )
Intepretation of a line integral as an area ( )
i
Bn
i i isA
C
i C
F s F x y ds F x y d
x
s
F y ds
ξ ηΔ rarr =
Δ =sum int
int
int
x
y
z
C
( )z F x y=
MingsianMingsian R R BaiBai26
the area is always on the left of an obs
Positive dierver trave
rection of irsing on the
ntegrationcurve
RC
CCW CW
If is a regular closed curve piecewise smo
( ) ( )
(contou
oth simple ( )
r integral)
x y zC C
x y zC
F dR F F F dx dy dz
F dx F dy
C
F dz
sdot = + + + +
= + +
lowast
int int
int
i j k i j kv v
無交叉
work dW F dR= sdotv v
MingsianMingsian R R BaiBai27
0 1
Surface integral
lim ( ) ( )i
n
i i i iS i S
F S F x y z dSξ η ζΔ rarr =
Δ =sum intint( ) i i iξ η ζ
iSΔ
0 1
mass density
lim ( ) ( )
Ex
Volume integral
( )
i
n
i i i iV i V
V
F V F x y z dV
M x y z dV
ξ η ζ
ρ
Δ rarr =Δ =
=
sum intintint
intintint( ) i i iξ η ζ
iVΔ
MingsianMingsian R R BaiBai28
Ref (1)Hildebrand advanced Calculus for application Prentice-Hall 1976 p306-313(2) Greenberg p383-389 (3) Arfken ch2
1 2 3
1 2 3
1 2 3
1 2 3
(
Orthogonal coordinate systems base vectors are not constant in s
) ( )( )( )( )
ˆˆ ˆ
pacep x y z p u u ux x u u uy y u u uz z u u u
r xi yj zk
primerarr
= +
rarr
===
+v
1uv
2uv3uv
1u
2u
3uP coord curve
xy
z
O
X Curvilinear Coordinates
MingsianMingsian R R BaiBai29
1
11
1 1 1
1
1
1 1 1
11 1
1 1
2 2 2 3 3 3
Tangent vector to the at
where is the arc length
1
where unit vector
Similarly
curv
e
u psr rU
u s us
rs
U h uds rh udu sU h u U h u
partpart part= =part part part
part =part
there4 =part= =part
= =
v vv
vQ
v v
vv
v vv v
1u cood curve
1uv
1du 1dS
drv
P
MingsianMingsian R R BaiBai30
3 31 2 1 21 2 3
1 2 32
curv
In summary scale factor
If is the arc length along a in any directione
k k k
kk k
k k
k k k
U h u
ds rh Udu u
ds h dus
du dudu du du dudr r r r U U Uds u ds u ds u ds ds ds ds
dr drds ds
=
part= = =part
=
part part part= + + = + +part part part
= sdot
v v
v v
v v v v v v v
v v 3 3
1 1
23 3 32 2
21 1 1
2 2 2 2 2 2 2 2 2 21 1 2 2 3 3 1 2 3
1 (for orthogonal coord)
jii j
i j
ji ii j i
i j i
dududr U Uds ds ds
dudu duU U hds ds ds
ds h du h du h du ds ds ds
= =
= = =
⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
= sdot
rarr = sdot =
rarr = + + = + +
sum sum
sumsum sum
v v v
v v
MingsianMingsian R R BaiBai31
1 1 2 2 3 3
1 2 3
are mutually vectors having with arc length
k k k k kds h du U du
U du U du U duds ds ds
= =
rArr perp
v
v v v
1uv
2uv
3uv
1dS
2dS
3dS
dS
MingsianMingsian R R BaiBai32
( ) ( )
( ) ( )
1 1 2 2 3 3 1 2 3 1 2 3 1 2 3
1 2 3 1 2 3
1
1 2 2 3 3 2 2 2 3 3 3 2 3 2 3 1
Differential volume
Differential On const coord surface
area
Gradi t
en
d U du U du U du u u u h h h du du du
d h h h du du du
ud U du U du h u du h u du h h du du u
df
τ
τ
σ
= times sdot = times sdot
=
=
= times = times =
= nabla
v v v v v v
v vv v v v
1 2 31 2 3
1 2 3 1 1 2 2 3 31 2 3
1 1 1 2 2 2 3 3 3
f f ff dr df du du duu u u
r r rdr du du du U du U du U duu u u
h u du h u du h u du
part part partsdot rarr = + +part part part
part part part= + + = + +part part part
= + +
v
v v v v v vv
v v v
=1 (right hand rule)
MingsianMingsian R R BaiBai33
Rectangular Cartesian Coordinates
ij
k
MingsianMingsian R R BaiBai34
Circular Cylindrical Coordinatescossin
xyz z
ρ ϕρ ϕ
===
Ref Arfken
( )P zρ ϕ
MingsianMingsian R R BaiBai35
Polar Spherical Coordinatessin cossin sincos
x ry rz r
θ ϕθ ϕθ
===
Ref Arfken
0
0
0
sin cos sin sin coscos cos cos sin sin
sin cos
θ ϕ θ ϕ θθ ϕ θ ϕ θϕ ϕ
= + +
= + minus= minus +
r i j kθ i j kφ i j
2
2
sinsin
dS r d ddV r dr d d
θ θ ϕ
θ θ ϕ
=
=( )P r θ ϕ
MingsianMingsian R R BaiBai36
Integral TheoremsHeart of this chapter
Interchange of dimensionality Div Curl Grad in integrals High-dimensional integration
rarr
1
Thm1 Divergence theorem Let be a closed region bounded by piecewise smooth orientable surface and let (con
I Divergence theor
tinuous derivatives up to the first order
em (Gauss theorem)
)
VS v Cisin
r
ˆ If denotes the outward unit normal on
the ˆn V S
v dV n v dS
n S
nablasdot = sdotint intr r
MingsianMingsian R R BaiBai37
jS
jP
jV
S
0
ˆBy definition ( ) lim
ˆ( )
ˆ( )
j
j
j
j
Sj V
j
j j S
j j Sj j
n v dSv P
V
v P V n v dS
v P V n v dS
Δ rarr
⎧ ⎫sdot⎪ ⎪nabla sdot ⎨ ⎬Δ⎪ ⎪⎩ ⎭
nabla sdot Δ asymp sdot
nabla sdot Δ = sdot
int
intsum sumint
rr
r r
r r
Pf
MingsianMingsian R R BaiBai38
0
After interior surfaces cancel out only exterior surface (Why Opposite normal vectors)
ˆ( )
volumes exterior surfaces
ˆlim
QE
survive
j
j
j j Sj j
V SV
v p V n v dS
v dV n v dSΔ rarr
nabla sdot Δ = sdot
rArr nablasdot = sdot
sum sumint
int int
r r
r r
D
MingsianMingsian R R BaiBai39
( )
scalar variable constant vectorvector variable
Alternative formsLet be a an arbitrary and
a
ˆ
1 Le
(
ˆt
V
V S
S
V
v ap
v d
v dV vn d
V n v dS
v va
va dV v a v a
S
nablasdot = sdot
= rArr
nablasdot = nabla sdot + nabla
=
sdot
nablaint intint int
int
r
v
r r
r r
r r rQ ( )
( )
ˆ)
2 Let ˆV S
V
V S
dV n va dS
v a p
a p
p dV n p d
d p
S
V a
⎛ ⎞= sdot⎜ ⎟
⎝ ⎠
= times rArr
nablasdot times = sdot nabla
times
times
nabla = times
int int
int int
int
r
r r r
r r v
r
vQ
r
( ) ( ) ( )
( ) ( )
ˆ
ˆV S
V S
a p dV n a p dS
a p dV a n p dS
⎛ ⎞⎡ ⎤minus sdot nablatimes = sdot times⎣ ⎦⎜ ⎟⎜ ⎟⎜ ⎟rArr minus sdot nablatimes = minus sdot times⎜ ⎟⎝ ⎠
int int
int int
r rv v
r rv v
MingsianMingsian R R BaiBai40
( )
( )
From Leibnitzs rule
(1-D) ( )
( ) ( )
B t
A t
d f x t dxdt
f x t dx B ttpart prime= +part
int( )
( )( ( ) ) ( )
B t
A tf B t t A tprimeminusint ( ( ) )
(assume ( ) ( ) )
f A t t
A t B t const= =
ContinuityEx The Equation
rate of increase of mass
mass densityV
dM d dVdt dt
ρ
ρ
= int
flow velocityvv
MingsianMingsian R R BaiBai41
ˆ ˆ ( )
(
Conservation
) ( Divergence theorem)
of mass
( ) 0
V S S
V
V
dV n v dS n v dSt
v dV
v dVt
ρ ρ ρ
ρ
ρ ρ⎡ ⎤⎢ ⎥⎣ ⎦
part = minus sdot = minus sdotpart
= minus nablasdot
partrArr +nablasdot =part
int int int
int
int
v v
vQ
v
ˆOn the other hand rate influof mass through n ox i t
Extended to 3-D for time-independent fixed boundary
(3-D) ( ) ( )
Thus
S
V V
V
S V n v dS
Sd f x y z t dV f x y z t dVdt tdM dVdt t
ρ
ρ
= minus sdot
part=part
part=part
int
int int
intv
Fixed control volume (Eulerian coordinates)Otherwise Raynoldrsquos transport theoremMoving control volume Lagrangian coordinates
MingsianMingsian R R BaiBai42
But the control volime is only arbitrary
In particular for flow
( ) 0
con
(continuity equati
stant
on)
Divergence the
0 incompress
Note is t
ible
orem h
V
v
vt
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
e mother of the other integral theorems
MingsianMingsian R R BaiBai43
Ex Archimedes principle ( ) The buoyant force ( ) on a body with volume immersed in a fluid is equal to the weight of the fluid
ˆ displaced by the object ie In the followin
B
V
gVρ= minusf k
阿基米德原理
浮力
ˆg figure the ˆ with and being the fluid density gravitational
acceleration depth and outward normal unit vector to the surface
fhydraulic pressure g z
g z
S
ρ
ρ
= minusp n
n
1
the modified divergence theorem
ˆ
Hint
scalar fuctionV S
vdV v dS v Cnabla = isinint int n
MingsianMingsian R R BaiBai44
Levelsurfacefluid
V
S
airyx
z
ˆf g zρ= minusp n
ˆPf ( ) ( )
ˆ ˆ ˆ ˆ ˆ (0 0 ) v v
B fS S V
V V
g z g zdS dS dV
g dV g dV gV
ρ ρ
ρ ρ ρ
= = minus = minus nabla
= minus + + = minus = minus
int int int
int int
f p n
i j k k k
k
MingsianMingsian R R BaiBai45
Note This can also be thought of as simple balance of forces (resultant force that acts on and weight of filled with fluid)S V
2
~ a little history about George Green Recall the diveregence theorem
ˆ
Let
II th
e Green
are arbitrary fun
s identiti
ctions
es
V S
v dV n v dS
v u vu v C
nablasdot = sdot
= nablaisin
int intv v
v
ˆ ( ) ( )
ˆ ( )V S
V S
u v dV n u v dS
u v u v dV un v dS
nablasdot nabla = sdot nabla
nabla sdotnabla + nablasdotnabla = sdotnabla
int int
int int
MingsianMingsian R R BaiBai46
2 -( ) -(1)V S
vu v u v dV u dSnpart
nabla sdotnabla + nabla =partint int firs(Greens identt ity)
2
If
( ) --(2)
(1) (2) V S
u vuv u v u dV v dSn
harrpart
nabla sdotnabla + nabla =part
minus
int int
( )2 2
V S
dv duu v v u dV u v dSdn dn
⎛ ⎞nabla minus nabla = minus⎜ ⎟⎝ ⎠int int
second(Greens identity)
MingsianMingsian R R BaiBai47
( )
( )
( ) ( )
1st id ( )
Greens identities in the 1D case integration by parts
b b bb ba a
a a ab
ba
ab
ba
a
v udx v u v u dx u v uv dx uv
u v u v vu dx vu
uv vu dx uv vu
primeprime prime prime prime prime prime primeprime prime= minus rArr + =
prime prime primeprime primeharr + =
primeprime primeprime prime primerArr minus = minus
int int int
int
2
2nd id ( )
Good for any adjoint operators in addition to Good also for and integration by parts Important for theory amp reciprocity
From th
Gree
e 1s
2-
t
ns functi
identity
D 1-
tak 1
D
e
on
u
nablararr
=
int
2
V S
vv dV dSnpartnabla =part
rArr int int2Ex 0 st 0 on the boundary 0 ( 1st id u u unabla = = rArr equiv 以 証)
MingsianMingsian R R BaiBai48
1
Thm1 Stokes theorem Let be defined in Let be a piecewise-smooth orientable surface bounded by a piec
III Stokes theorem and
ewise-smooth simple clos
Greens
ed curve Then
theore
m
v C R S
C
isinv
( )
ˆwhere is a unit normal to according t right hand rulo the e
ˆS C
n S
n v dS v dRsdot nablatimes = sdotint intvv v
(not necessarily planar)S
Cn vnablatimes v vv
MingsianMingsian R R BaiBai49
Pf
( )( )
circulatioWe have shown in deriving thatthe in -direction
ˆor
ˆ
n
z
z
j j j j j
j j j j jj j
vz v dx dy
v dR n v S
v dR n v S
nablatimes
=nablatimes
sdot = sdot nablatimes Δ
sdot = sdot nablatimes Δ
intsum sumint
v
v
vv v
vv v
C
S
jS
jC
jP
ˆ jn
jSΔ
MingsianMingsian R R BaiBai50
( )
( )ext surface
After the interior boundaries cancel out only the exterior boundaries (Why)
ˆ
ˆ0 QED
Alternative forms scalar
survive
va
j
j j j j jC
jC S
v dR n v S
S v dR n v dS
φ
sdot = sdot nablatimes Δ
Δ rarr rArr sdot = sdot nablatimes
sum sumint
int int
vv v
vv v
( )
riable constant vector vector variable
1 Let
2 Let
ˆ
ˆ S C
S C
a p
v n dR dR
n p dS dR p
a
v a p
φ φφ= rArr timesnabla =
timesnabla times= rArr timestimes =
int int
int int
v v
v v
v
v v
vv vv v
MingsianMingsian R R BaiBai51
( ) ( )
( )
( )
( ) ( )
1
Thm2 Greens theorem ( planar version of Stokes theorem)ˆ ˆ Let
pf By Stokes theoremˆ
ˆ ˆ ˆ ˆ ˆ ˆ
S C
S C
S C
v P x y i Q x y j C
n v dS v dR
k Pi Qj dS Pi Qj dx i dy
Q P dS Pdx Qdyx y
⎛ ⎞part partminus = +⎜ ⎟
= + isin
sdot nablatimes = sdot
⎡ ⎤sdot nablatimes + =
part part⎝ ⎠
+ sdot +⎣ ⎦
int int
int
int
int
int
v
vv v
( )
( )
( )
ˆ
ˆˆ ˆ
ˆˆ ˆ ˆ ˆ( ) ( )
0
QEDS C
j
i j kQ PPi Qj i j k
x y z x yP Q
Q P dS Pdx Qdyx y
⎛ ⎞part part part part partnablatimes + = = + + minus⎜ ⎟part part part part part⎝ ⎠
⎛ ⎞part partrArr minus = +⎜ ⎟part part⎝ ⎠int int
L L
S
C
i
j
x
y
MingsianMingsian R R BaiBai52
Potential TheorypotentiGeorge Green coined the term (l
a )
rarr 位能 勢能
例子 重力場 電磁場 流場 聲場 彈性力學
1
2
Thm1 If is in a simply connected region then the following statements are equivalent(i) There exists a scala
Conservative Field
potential functior such that
(ii
n
) 0
v
C
C
v
v R
φφ=nabla
nablatimes
isin
isin
=r
r
r
r in (Irrotational field curl-free)
cf 0 st solenoidal field divergence-free
0
(iii) in (Conservative field)C
Rv w v w
v R Rd
nablasdot = exist nabla
sdot =
= times
int
r r r
r
r
r
MingsianMingsian R R BaiBai53
(iv) is connecting and
independent o
Ex Think
f the pa
of dynamics is the force
thB
A
v dR A B
v
sdotintrr
r
1 2
1
0
Pf
(i) (ii) ( ) 0 (zero identity cg rule) irrotational
ˆ(ii) (iii) ( ) 0 (Stokes theorem)
(iii) (iv) ( ) 0
C S
C C C
C
v
v dR n v dS
v dR v dR
v dR
φ
=
rarr nablatimes = nablatimes nabla =
rarr sdot = sdot nablatimes =
rarr sdot = minus sdot =
rArr sdot =
int int
int int int
int
r
rr
rr r
r rr r
rr r
2
independent of pathC
v dRsdotintr
A
B
1C
2C
MingsianMingsian R R BaiBai54
1 2
(iv) (i) constant ( ) ( )
--(1)
But (2)
(1) (2) QED
Note on application
Check
B
C C A
v dR v dR B A d
v dR d
d dx dy dz dRx y z
v dR dR v R
φ φ φ
φφ φ φφ φ
φ φ
rarr sdot = sdot = = minus =
rArr sdot =part part part
= + + = nabla sdot minus minus minus minus minuspart part part
sdot = nabla sdot rArr = nabla forall
int int intr rr r
rr
r
r r rr r
use simpler path (iv)if 0
find (i)v
φ⎧
nablatimes = ⎨⎩
rr
A
B
x
y
Cz
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai18
Identities of differential operators ( p380 Greenberg )
( )
( ) ( )( )( )
( ) ( )( ) is required
Example of deriv
ˆ ˆˆ ˆ ˆ ˆ
ation
x y z
x y z
v v vx y z
uv u v u v
uv i j k uv i uv j uv kx y z
uv u v u v
u v v u u v
u v u vv v u u
+ +part part partpart part part
⎛ ⎞⎜ ⎟⎝ ⎠
nablasdot = nabla sdot + nablasdot
part part partnablasdot = + + sdot + + =part part part
nablatimes =nabla times + nablatimes
nablasdot times = sdotnablatimes minus sdotnablatimes
nablatimes times = nabla minus + minusnablanablasdot sdotsdot678 678
v
v v v
vL
v v v
v v v v v v
v v v v v v v v( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )2 cf
x y zu u ux y z
A B C A
u v
v w v w w v v w w v
v v v C B A B C
+ +part part partpart part part
sdotnabla
nabla sdot = sdotnabla + sdotnabla + times nablatimes + times
times
nablatimes
nablatimes nablatimes =nabla nabla times = sdot minussdot sdotminusnabla
678v
v v v
v v v v v vv v
v v v v v v
v v
v
v v
ˆ ˆˆ ˆ ˆ ˆ( ) ( ) ( )x y z x y zv i v j v k i j k u v v v ux y z x y zpart part part part part part+ + sdot + + = + +part part part part part part
v v
MingsianMingsian R R BaiBai19
2 2 22
2 2 2
0 (dc direct current) solenoidal0 (cg center of gravity) conservative potent
Laplacia
Zero identitie
n
s
ial
x y z
vu
⎧⎪⎨⎪⎩
part part partΔ =nabla nablasdotnabla = + +part part part
nablasdotnablatimes = rarr
nablatimesnabla = rarr
v
v
These differential operators will take more complicated and irregular form for the other coordinate systems
MingsianMingsian R R BaiBai20
2 2 3
2 2
ˆˆ ˆEx 2
ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ 2 0 2
2 0 0 2yx z
u x y v x yi z j k
u i j k u xyi x j k xyi x jx y z
vv vv xy xyx y z
⎛ ⎞⎜ ⎟⎝ ⎠
= = minus +
part part partnabla = + + = + + = +part part part
partpart partnablasdot = + + = + minus =part part part
v
v
MingsianMingsian R R BaiBai21
( ) ( ) ( )
( ) ( )
2 2 2 2
2 2 22
2 2 2
2
ˆˆ ˆ
ˆˆ ˆ
ˆ ˆˆ ˆ ˆ 0 3 0 0 0 3
2 0 0 2
ˆˆ ˆ
y yx xz z
x y z
i j kv vv vv vv i j k
x y z y z x z x yv v v
z i j x k z i x k
u u uu y yx y zv v v
i j kx y z
⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠ ⎝ ⎠
⎛
⎝
part partpart partpart partpart part partnablatimes = = minus minus minus + minuspart part part part part part part part part
= + minus minus + minus = minus
part part partnabla = + + = + + =part part part
nablatimes nablatimes =nabla nablasdot minusnabla
part part part= + +part part part
v
v v v
( ) ( )( ) ( ) ( ) ( )
( )
2 2 22 3
2 2 2ˆˆ ˆ2 2
ˆ ˆˆ ˆ ˆ ˆ 2 2 0 2 0 0 0 0 6 0 0 0
ˆ 2 6
xy x yi z j kx y z
yi xj k y i z j k
x z j
⎛ ⎞⎞⎜ ⎟⎜ ⎟
⎠ ⎝ ⎠⎡ ⎤⎣ ⎦
part part partminus + + minus +part part part
= + + minus + + + + minus + + minus
= +
MingsianMingsian R R BaiBai22
( )v w r w r r wnablatimes =nablatimes times = nablasdot minus nablasdotv v v v v v v ( )r w+ sdotnablav v ( ) constant w
w rminus sdotnablav
v v
Q
Ex Rigid body rotation
MingsianMingsian R R BaiBai23
( )( ) ( )
( ) ( )
ˆˆ ˆ
ˆ ˆˆ ˆ ˆ ˆ 1 1 1 3
ˆˆ ˆ
ˆˆ ˆ
Thus 3 2 is linked directly
x y z
x y z
r xi yj zk
r i j k xi yj zkx y z
w r w w w xi yj zkx y z
w i w j w k w
w r w r w r w w wv
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎝ ⎠
= + +
part part partnablasdot = + + + + = + + =part part part
part part partsdotnabla = + + + +part part part
= + + =
nablatimes times == nablasdot minus sdotnabla = minus =rArrnablatimes
v
v
v v
v
v v v v v v v v v
v spinn to ing
MingsianMingsian R R BaiBai24
Line surface and volume integrals
x
yC
B
A
isΔ
( )ii ηξ
Piecewise smooth curve
Line integral
MingsianMingsian R R BaiBai25
0 1lim ( ) ( ) or ( )
Intepretation of a line integral as an area ( )
i
Bn
i i isA
C
i C
F s F x y ds F x y d
x
s
F y ds
ξ ηΔ rarr =
Δ =sum int
int
int
x
y
z
C
( )z F x y=
MingsianMingsian R R BaiBai26
the area is always on the left of an obs
Positive dierver trave
rection of irsing on the
ntegrationcurve
RC
CCW CW
If is a regular closed curve piecewise smo
( ) ( )
(contou
oth simple ( )
r integral)
x y zC C
x y zC
F dR F F F dx dy dz
F dx F dy
C
F dz
sdot = + + + +
= + +
lowast
int int
int
i j k i j kv v
無交叉
work dW F dR= sdotv v
MingsianMingsian R R BaiBai27
0 1
Surface integral
lim ( ) ( )i
n
i i i iS i S
F S F x y z dSξ η ζΔ rarr =
Δ =sum intint( ) i i iξ η ζ
iSΔ
0 1
mass density
lim ( ) ( )
Ex
Volume integral
( )
i
n
i i i iV i V
V
F V F x y z dV
M x y z dV
ξ η ζ
ρ
Δ rarr =Δ =
=
sum intintint
intintint( ) i i iξ η ζ
iVΔ
MingsianMingsian R R BaiBai28
Ref (1)Hildebrand advanced Calculus for application Prentice-Hall 1976 p306-313(2) Greenberg p383-389 (3) Arfken ch2
1 2 3
1 2 3
1 2 3
1 2 3
(
Orthogonal coordinate systems base vectors are not constant in s
) ( )( )( )( )
ˆˆ ˆ
pacep x y z p u u ux x u u uy y u u uz z u u u
r xi yj zk
primerarr
= +
rarr
===
+v
1uv
2uv3uv
1u
2u
3uP coord curve
xy
z
O
X Curvilinear Coordinates
MingsianMingsian R R BaiBai29
1
11
1 1 1
1
1
1 1 1
11 1
1 1
2 2 2 3 3 3
Tangent vector to the at
where is the arc length
1
where unit vector
Similarly
curv
e
u psr rU
u s us
rs
U h uds rh udu sU h u U h u
partpart part= =part part part
part =part
there4 =part= =part
= =
v vv
vQ
v v
vv
v vv v
1u cood curve
1uv
1du 1dS
drv
P
MingsianMingsian R R BaiBai30
3 31 2 1 21 2 3
1 2 32
curv
In summary scale factor
If is the arc length along a in any directione
k k k
kk k
k k
k k k
U h u
ds rh Udu u
ds h dus
du dudu du du dudr r r r U U Uds u ds u ds u ds ds ds ds
dr drds ds
=
part= = =part
=
part part part= + + = + +part part part
= sdot
v v
v v
v v v v v v v
v v 3 3
1 1
23 3 32 2
21 1 1
2 2 2 2 2 2 2 2 2 21 1 2 2 3 3 1 2 3
1 (for orthogonal coord)
jii j
i j
ji ii j i
i j i
dududr U Uds ds ds
dudu duU U hds ds ds
ds h du h du h du ds ds ds
= =
= = =
⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
= sdot
rarr = sdot =
rarr = + + = + +
sum sum
sumsum sum
v v v
v v
MingsianMingsian R R BaiBai31
1 1 2 2 3 3
1 2 3
are mutually vectors having with arc length
k k k k kds h du U du
U du U du U duds ds ds
= =
rArr perp
v
v v v
1uv
2uv
3uv
1dS
2dS
3dS
dS
MingsianMingsian R R BaiBai32
( ) ( )
( ) ( )
1 1 2 2 3 3 1 2 3 1 2 3 1 2 3
1 2 3 1 2 3
1
1 2 2 3 3 2 2 2 3 3 3 2 3 2 3 1
Differential volume
Differential On const coord surface
area
Gradi t
en
d U du U du U du u u u h h h du du du
d h h h du du du
ud U du U du h u du h u du h h du du u
df
τ
τ
σ
= times sdot = times sdot
=
=
= times = times =
= nabla
v v v v v v
v vv v v v
1 2 31 2 3
1 2 3 1 1 2 2 3 31 2 3
1 1 1 2 2 2 3 3 3
f f ff dr df du du duu u u
r r rdr du du du U du U du U duu u u
h u du h u du h u du
part part partsdot rarr = + +part part part
part part part= + + = + +part part part
= + +
v
v v v v v vv
v v v
=1 (right hand rule)
MingsianMingsian R R BaiBai33
Rectangular Cartesian Coordinates
ij
k
MingsianMingsian R R BaiBai34
Circular Cylindrical Coordinatescossin
xyz z
ρ ϕρ ϕ
===
Ref Arfken
( )P zρ ϕ
MingsianMingsian R R BaiBai35
Polar Spherical Coordinatessin cossin sincos
x ry rz r
θ ϕθ ϕθ
===
Ref Arfken
0
0
0
sin cos sin sin coscos cos cos sin sin
sin cos
θ ϕ θ ϕ θθ ϕ θ ϕ θϕ ϕ
= + +
= + minus= minus +
r i j kθ i j kφ i j
2
2
sinsin
dS r d ddV r dr d d
θ θ ϕ
θ θ ϕ
=
=( )P r θ ϕ
MingsianMingsian R R BaiBai36
Integral TheoremsHeart of this chapter
Interchange of dimensionality Div Curl Grad in integrals High-dimensional integration
rarr
1
Thm1 Divergence theorem Let be a closed region bounded by piecewise smooth orientable surface and let (con
I Divergence theor
tinuous derivatives up to the first order
em (Gauss theorem)
)
VS v Cisin
r
ˆ If denotes the outward unit normal on
the ˆn V S
v dV n v dS
n S
nablasdot = sdotint intr r
MingsianMingsian R R BaiBai37
jS
jP
jV
S
0
ˆBy definition ( ) lim
ˆ( )
ˆ( )
j
j
j
j
Sj V
j
j j S
j j Sj j
n v dSv P
V
v P V n v dS
v P V n v dS
Δ rarr
⎧ ⎫sdot⎪ ⎪nabla sdot ⎨ ⎬Δ⎪ ⎪⎩ ⎭
nabla sdot Δ asymp sdot
nabla sdot Δ = sdot
int
intsum sumint
rr
r r
r r
Pf
MingsianMingsian R R BaiBai38
0
After interior surfaces cancel out only exterior surface (Why Opposite normal vectors)
ˆ( )
volumes exterior surfaces
ˆlim
QE
survive
j
j
j j Sj j
V SV
v p V n v dS
v dV n v dSΔ rarr
nabla sdot Δ = sdot
rArr nablasdot = sdot
sum sumint
int int
r r
r r
D
MingsianMingsian R R BaiBai39
( )
scalar variable constant vectorvector variable
Alternative formsLet be a an arbitrary and
a
ˆ
1 Le
(
ˆt
V
V S
S
V
v ap
v d
v dV vn d
V n v dS
v va
va dV v a v a
S
nablasdot = sdot
= rArr
nablasdot = nabla sdot + nabla
=
sdot
nablaint intint int
int
r
v
r r
r r
r r rQ ( )
( )
ˆ)
2 Let ˆV S
V
V S
dV n va dS
v a p
a p
p dV n p d
d p
S
V a
⎛ ⎞= sdot⎜ ⎟
⎝ ⎠
= times rArr
nablasdot times = sdot nabla
times
times
nabla = times
int int
int int
int
r
r r r
r r v
r
vQ
r
( ) ( ) ( )
( ) ( )
ˆ
ˆV S
V S
a p dV n a p dS
a p dV a n p dS
⎛ ⎞⎡ ⎤minus sdot nablatimes = sdot times⎣ ⎦⎜ ⎟⎜ ⎟⎜ ⎟rArr minus sdot nablatimes = minus sdot times⎜ ⎟⎝ ⎠
int int
int int
r rv v
r rv v
MingsianMingsian R R BaiBai40
( )
( )
From Leibnitzs rule
(1-D) ( )
( ) ( )
B t
A t
d f x t dxdt
f x t dx B ttpart prime= +part
int( )
( )( ( ) ) ( )
B t
A tf B t t A tprimeminusint ( ( ) )
(assume ( ) ( ) )
f A t t
A t B t const= =
ContinuityEx The Equation
rate of increase of mass
mass densityV
dM d dVdt dt
ρ
ρ
= int
flow velocityvv
MingsianMingsian R R BaiBai41
ˆ ˆ ( )
(
Conservation
) ( Divergence theorem)
of mass
( ) 0
V S S
V
V
dV n v dS n v dSt
v dV
v dVt
ρ ρ ρ
ρ
ρ ρ⎡ ⎤⎢ ⎥⎣ ⎦
part = minus sdot = minus sdotpart
= minus nablasdot
partrArr +nablasdot =part
int int int
int
int
v v
vQ
v
ˆOn the other hand rate influof mass through n ox i t
Extended to 3-D for time-independent fixed boundary
(3-D) ( ) ( )
Thus
S
V V
V
S V n v dS
Sd f x y z t dV f x y z t dVdt tdM dVdt t
ρ
ρ
= minus sdot
part=part
part=part
int
int int
intv
Fixed control volume (Eulerian coordinates)Otherwise Raynoldrsquos transport theoremMoving control volume Lagrangian coordinates
MingsianMingsian R R BaiBai42
But the control volime is only arbitrary
In particular for flow
( ) 0
con
(continuity equati
stant
on)
Divergence the
0 incompress
Note is t
ible
orem h
V
v
vt
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
e mother of the other integral theorems
MingsianMingsian R R BaiBai43
Ex Archimedes principle ( ) The buoyant force ( ) on a body with volume immersed in a fluid is equal to the weight of the fluid
ˆ displaced by the object ie In the followin
B
V
gVρ= minusf k
阿基米德原理
浮力
ˆg figure the ˆ with and being the fluid density gravitational
acceleration depth and outward normal unit vector to the surface
fhydraulic pressure g z
g z
S
ρ
ρ
= minusp n
n
1
the modified divergence theorem
ˆ
Hint
scalar fuctionV S
vdV v dS v Cnabla = isinint int n
MingsianMingsian R R BaiBai44
Levelsurfacefluid
V
S
airyx
z
ˆf g zρ= minusp n
ˆPf ( ) ( )
ˆ ˆ ˆ ˆ ˆ (0 0 ) v v
B fS S V
V V
g z g zdS dS dV
g dV g dV gV
ρ ρ
ρ ρ ρ
= = minus = minus nabla
= minus + + = minus = minus
int int int
int int
f p n
i j k k k
k
MingsianMingsian R R BaiBai45
Note This can also be thought of as simple balance of forces (resultant force that acts on and weight of filled with fluid)S V
2
~ a little history about George Green Recall the diveregence theorem
ˆ
Let
II th
e Green
are arbitrary fun
s identiti
ctions
es
V S
v dV n v dS
v u vu v C
nablasdot = sdot
= nablaisin
int intv v
v
ˆ ( ) ( )
ˆ ( )V S
V S
u v dV n u v dS
u v u v dV un v dS
nablasdot nabla = sdot nabla
nabla sdotnabla + nablasdotnabla = sdotnabla
int int
int int
MingsianMingsian R R BaiBai46
2 -( ) -(1)V S
vu v u v dV u dSnpart
nabla sdotnabla + nabla =partint int firs(Greens identt ity)
2
If
( ) --(2)
(1) (2) V S
u vuv u v u dV v dSn
harrpart
nabla sdotnabla + nabla =part
minus
int int
( )2 2
V S
dv duu v v u dV u v dSdn dn
⎛ ⎞nabla minus nabla = minus⎜ ⎟⎝ ⎠int int
second(Greens identity)
MingsianMingsian R R BaiBai47
( )
( )
( ) ( )
1st id ( )
Greens identities in the 1D case integration by parts
b b bb ba a
a a ab
ba
ab
ba
a
v udx v u v u dx u v uv dx uv
u v u v vu dx vu
uv vu dx uv vu
primeprime prime prime prime prime prime primeprime prime= minus rArr + =
prime prime primeprime primeharr + =
primeprime primeprime prime primerArr minus = minus
int int int
int
2
2nd id ( )
Good for any adjoint operators in addition to Good also for and integration by parts Important for theory amp reciprocity
From th
Gree
e 1s
2-
t
ns functi
identity
D 1-
tak 1
D
e
on
u
nablararr
=
int
2
V S
vv dV dSnpartnabla =part
rArr int int2Ex 0 st 0 on the boundary 0 ( 1st id u u unabla = = rArr equiv 以 証)
MingsianMingsian R R BaiBai48
1
Thm1 Stokes theorem Let be defined in Let be a piecewise-smooth orientable surface bounded by a piec
III Stokes theorem and
ewise-smooth simple clos
Greens
ed curve Then
theore
m
v C R S
C
isinv
( )
ˆwhere is a unit normal to according t right hand rulo the e
ˆS C
n S
n v dS v dRsdot nablatimes = sdotint intvv v
(not necessarily planar)S
Cn vnablatimes v vv
MingsianMingsian R R BaiBai49
Pf
( )( )
circulatioWe have shown in deriving thatthe in -direction
ˆor
ˆ
n
z
z
j j j j j
j j j j jj j
vz v dx dy
v dR n v S
v dR n v S
nablatimes
=nablatimes
sdot = sdot nablatimes Δ
sdot = sdot nablatimes Δ
intsum sumint
v
v
vv v
vv v
C
S
jS
jC
jP
ˆ jn
jSΔ
MingsianMingsian R R BaiBai50
( )
( )ext surface
After the interior boundaries cancel out only the exterior boundaries (Why)
ˆ
ˆ0 QED
Alternative forms scalar
survive
va
j
j j j j jC
jC S
v dR n v S
S v dR n v dS
φ
sdot = sdot nablatimes Δ
Δ rarr rArr sdot = sdot nablatimes
sum sumint
int int
vv v
vv v
( )
riable constant vector vector variable
1 Let
2 Let
ˆ
ˆ S C
S C
a p
v n dR dR
n p dS dR p
a
v a p
φ φφ= rArr timesnabla =
timesnabla times= rArr timestimes =
int int
int int
v v
v v
v
v v
vv vv v
MingsianMingsian R R BaiBai51
( ) ( )
( )
( )
( ) ( )
1
Thm2 Greens theorem ( planar version of Stokes theorem)ˆ ˆ Let
pf By Stokes theoremˆ
ˆ ˆ ˆ ˆ ˆ ˆ
S C
S C
S C
v P x y i Q x y j C
n v dS v dR
k Pi Qj dS Pi Qj dx i dy
Q P dS Pdx Qdyx y
⎛ ⎞part partminus = +⎜ ⎟
= + isin
sdot nablatimes = sdot
⎡ ⎤sdot nablatimes + =
part part⎝ ⎠
+ sdot +⎣ ⎦
int int
int
int
int
int
v
vv v
( )
( )
( )
ˆ
ˆˆ ˆ
ˆˆ ˆ ˆ ˆ( ) ( )
0
QEDS C
j
i j kQ PPi Qj i j k
x y z x yP Q
Q P dS Pdx Qdyx y
⎛ ⎞part part part part partnablatimes + = = + + minus⎜ ⎟part part part part part⎝ ⎠
⎛ ⎞part partrArr minus = +⎜ ⎟part part⎝ ⎠int int
L L
S
C
i
j
x
y
MingsianMingsian R R BaiBai52
Potential TheorypotentiGeorge Green coined the term (l
a )
rarr 位能 勢能
例子 重力場 電磁場 流場 聲場 彈性力學
1
2
Thm1 If is in a simply connected region then the following statements are equivalent(i) There exists a scala
Conservative Field
potential functior such that
(ii
n
) 0
v
C
C
v
v R
φφ=nabla
nablatimes
isin
isin
=r
r
r
r in (Irrotational field curl-free)
cf 0 st solenoidal field divergence-free
0
(iii) in (Conservative field)C
Rv w v w
v R Rd
nablasdot = exist nabla
sdot =
= times
int
r r r
r
r
r
MingsianMingsian R R BaiBai53
(iv) is connecting and
independent o
Ex Think
f the pa
of dynamics is the force
thB
A
v dR A B
v
sdotintrr
r
1 2
1
0
Pf
(i) (ii) ( ) 0 (zero identity cg rule) irrotational
ˆ(ii) (iii) ( ) 0 (Stokes theorem)
(iii) (iv) ( ) 0
C S
C C C
C
v
v dR n v dS
v dR v dR
v dR
φ
=
rarr nablatimes = nablatimes nabla =
rarr sdot = sdot nablatimes =
rarr sdot = minus sdot =
rArr sdot =
int int
int int int
int
r
rr
rr r
r rr r
rr r
2
independent of pathC
v dRsdotintr
A
B
1C
2C
MingsianMingsian R R BaiBai54
1 2
(iv) (i) constant ( ) ( )
--(1)
But (2)
(1) (2) QED
Note on application
Check
B
C C A
v dR v dR B A d
v dR d
d dx dy dz dRx y z
v dR dR v R
φ φ φ
φφ φ φφ φ
φ φ
rarr sdot = sdot = = minus =
rArr sdot =part part part
= + + = nabla sdot minus minus minus minus minuspart part part
sdot = nabla sdot rArr = nabla forall
int int intr rr r
rr
r
r r rr r
use simpler path (iv)if 0
find (i)v
φ⎧
nablatimes = ⎨⎩
rr
A
B
x
y
Cz
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai19
2 2 22
2 2 2
0 (dc direct current) solenoidal0 (cg center of gravity) conservative potent
Laplacia
Zero identitie
n
s
ial
x y z
vu
⎧⎪⎨⎪⎩
part part partΔ =nabla nablasdotnabla = + +part part part
nablasdotnablatimes = rarr
nablatimesnabla = rarr
v
v
These differential operators will take more complicated and irregular form for the other coordinate systems
MingsianMingsian R R BaiBai20
2 2 3
2 2
ˆˆ ˆEx 2
ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ 2 0 2
2 0 0 2yx z
u x y v x yi z j k
u i j k u xyi x j k xyi x jx y z
vv vv xy xyx y z
⎛ ⎞⎜ ⎟⎝ ⎠
= = minus +
part part partnabla = + + = + + = +part part part
partpart partnablasdot = + + = + minus =part part part
v
v
MingsianMingsian R R BaiBai21
( ) ( ) ( )
( ) ( )
2 2 2 2
2 2 22
2 2 2
2
ˆˆ ˆ
ˆˆ ˆ
ˆ ˆˆ ˆ ˆ 0 3 0 0 0 3
2 0 0 2
ˆˆ ˆ
y yx xz z
x y z
i j kv vv vv vv i j k
x y z y z x z x yv v v
z i j x k z i x k
u u uu y yx y zv v v
i j kx y z
⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠ ⎝ ⎠
⎛
⎝
part partpart partpart partpart part partnablatimes = = minus minus minus + minuspart part part part part part part part part
= + minus minus + minus = minus
part part partnabla = + + = + + =part part part
nablatimes nablatimes =nabla nablasdot minusnabla
part part part= + +part part part
v
v v v
( ) ( )( ) ( ) ( ) ( )
( )
2 2 22 3
2 2 2ˆˆ ˆ2 2
ˆ ˆˆ ˆ ˆ ˆ 2 2 0 2 0 0 0 0 6 0 0 0
ˆ 2 6
xy x yi z j kx y z
yi xj k y i z j k
x z j
⎛ ⎞⎞⎜ ⎟⎜ ⎟
⎠ ⎝ ⎠⎡ ⎤⎣ ⎦
part part partminus + + minus +part part part
= + + minus + + + + minus + + minus
= +
MingsianMingsian R R BaiBai22
( )v w r w r r wnablatimes =nablatimes times = nablasdot minus nablasdotv v v v v v v ( )r w+ sdotnablav v ( ) constant w
w rminus sdotnablav
v v
Q
Ex Rigid body rotation
MingsianMingsian R R BaiBai23
( )( ) ( )
( ) ( )
ˆˆ ˆ
ˆ ˆˆ ˆ ˆ ˆ 1 1 1 3
ˆˆ ˆ
ˆˆ ˆ
Thus 3 2 is linked directly
x y z
x y z
r xi yj zk
r i j k xi yj zkx y z
w r w w w xi yj zkx y z
w i w j w k w
w r w r w r w w wv
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎝ ⎠
= + +
part part partnablasdot = + + + + = + + =part part part
part part partsdotnabla = + + + +part part part
= + + =
nablatimes times == nablasdot minus sdotnabla = minus =rArrnablatimes
v
v
v v
v
v v v v v v v v v
v spinn to ing
MingsianMingsian R R BaiBai24
Line surface and volume integrals
x
yC
B
A
isΔ
( )ii ηξ
Piecewise smooth curve
Line integral
MingsianMingsian R R BaiBai25
0 1lim ( ) ( ) or ( )
Intepretation of a line integral as an area ( )
i
Bn
i i isA
C
i C
F s F x y ds F x y d
x
s
F y ds
ξ ηΔ rarr =
Δ =sum int
int
int
x
y
z
C
( )z F x y=
MingsianMingsian R R BaiBai26
the area is always on the left of an obs
Positive dierver trave
rection of irsing on the
ntegrationcurve
RC
CCW CW
If is a regular closed curve piecewise smo
( ) ( )
(contou
oth simple ( )
r integral)
x y zC C
x y zC
F dR F F F dx dy dz
F dx F dy
C
F dz
sdot = + + + +
= + +
lowast
int int
int
i j k i j kv v
無交叉
work dW F dR= sdotv v
MingsianMingsian R R BaiBai27
0 1
Surface integral
lim ( ) ( )i
n
i i i iS i S
F S F x y z dSξ η ζΔ rarr =
Δ =sum intint( ) i i iξ η ζ
iSΔ
0 1
mass density
lim ( ) ( )
Ex
Volume integral
( )
i
n
i i i iV i V
V
F V F x y z dV
M x y z dV
ξ η ζ
ρ
Δ rarr =Δ =
=
sum intintint
intintint( ) i i iξ η ζ
iVΔ
MingsianMingsian R R BaiBai28
Ref (1)Hildebrand advanced Calculus for application Prentice-Hall 1976 p306-313(2) Greenberg p383-389 (3) Arfken ch2
1 2 3
1 2 3
1 2 3
1 2 3
(
Orthogonal coordinate systems base vectors are not constant in s
) ( )( )( )( )
ˆˆ ˆ
pacep x y z p u u ux x u u uy y u u uz z u u u
r xi yj zk
primerarr
= +
rarr
===
+v
1uv
2uv3uv
1u
2u
3uP coord curve
xy
z
O
X Curvilinear Coordinates
MingsianMingsian R R BaiBai29
1
11
1 1 1
1
1
1 1 1
11 1
1 1
2 2 2 3 3 3
Tangent vector to the at
where is the arc length
1
where unit vector
Similarly
curv
e
u psr rU
u s us
rs
U h uds rh udu sU h u U h u
partpart part= =part part part
part =part
there4 =part= =part
= =
v vv
vQ
v v
vv
v vv v
1u cood curve
1uv
1du 1dS
drv
P
MingsianMingsian R R BaiBai30
3 31 2 1 21 2 3
1 2 32
curv
In summary scale factor
If is the arc length along a in any directione
k k k
kk k
k k
k k k
U h u
ds rh Udu u
ds h dus
du dudu du du dudr r r r U U Uds u ds u ds u ds ds ds ds
dr drds ds
=
part= = =part
=
part part part= + + = + +part part part
= sdot
v v
v v
v v v v v v v
v v 3 3
1 1
23 3 32 2
21 1 1
2 2 2 2 2 2 2 2 2 21 1 2 2 3 3 1 2 3
1 (for orthogonal coord)
jii j
i j
ji ii j i
i j i
dududr U Uds ds ds
dudu duU U hds ds ds
ds h du h du h du ds ds ds
= =
= = =
⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
= sdot
rarr = sdot =
rarr = + + = + +
sum sum
sumsum sum
v v v
v v
MingsianMingsian R R BaiBai31
1 1 2 2 3 3
1 2 3
are mutually vectors having with arc length
k k k k kds h du U du
U du U du U duds ds ds
= =
rArr perp
v
v v v
1uv
2uv
3uv
1dS
2dS
3dS
dS
MingsianMingsian R R BaiBai32
( ) ( )
( ) ( )
1 1 2 2 3 3 1 2 3 1 2 3 1 2 3
1 2 3 1 2 3
1
1 2 2 3 3 2 2 2 3 3 3 2 3 2 3 1
Differential volume
Differential On const coord surface
area
Gradi t
en
d U du U du U du u u u h h h du du du
d h h h du du du
ud U du U du h u du h u du h h du du u
df
τ
τ
σ
= times sdot = times sdot
=
=
= times = times =
= nabla
v v v v v v
v vv v v v
1 2 31 2 3
1 2 3 1 1 2 2 3 31 2 3
1 1 1 2 2 2 3 3 3
f f ff dr df du du duu u u
r r rdr du du du U du U du U duu u u
h u du h u du h u du
part part partsdot rarr = + +part part part
part part part= + + = + +part part part
= + +
v
v v v v v vv
v v v
=1 (right hand rule)
MingsianMingsian R R BaiBai33
Rectangular Cartesian Coordinates
ij
k
MingsianMingsian R R BaiBai34
Circular Cylindrical Coordinatescossin
xyz z
ρ ϕρ ϕ
===
Ref Arfken
( )P zρ ϕ
MingsianMingsian R R BaiBai35
Polar Spherical Coordinatessin cossin sincos
x ry rz r
θ ϕθ ϕθ
===
Ref Arfken
0
0
0
sin cos sin sin coscos cos cos sin sin
sin cos
θ ϕ θ ϕ θθ ϕ θ ϕ θϕ ϕ
= + +
= + minus= minus +
r i j kθ i j kφ i j
2
2
sinsin
dS r d ddV r dr d d
θ θ ϕ
θ θ ϕ
=
=( )P r θ ϕ
MingsianMingsian R R BaiBai36
Integral TheoremsHeart of this chapter
Interchange of dimensionality Div Curl Grad in integrals High-dimensional integration
rarr
1
Thm1 Divergence theorem Let be a closed region bounded by piecewise smooth orientable surface and let (con
I Divergence theor
tinuous derivatives up to the first order
em (Gauss theorem)
)
VS v Cisin
r
ˆ If denotes the outward unit normal on
the ˆn V S
v dV n v dS
n S
nablasdot = sdotint intr r
MingsianMingsian R R BaiBai37
jS
jP
jV
S
0
ˆBy definition ( ) lim
ˆ( )
ˆ( )
j
j
j
j
Sj V
j
j j S
j j Sj j
n v dSv P
V
v P V n v dS
v P V n v dS
Δ rarr
⎧ ⎫sdot⎪ ⎪nabla sdot ⎨ ⎬Δ⎪ ⎪⎩ ⎭
nabla sdot Δ asymp sdot
nabla sdot Δ = sdot
int
intsum sumint
rr
r r
r r
Pf
MingsianMingsian R R BaiBai38
0
After interior surfaces cancel out only exterior surface (Why Opposite normal vectors)
ˆ( )
volumes exterior surfaces
ˆlim
QE
survive
j
j
j j Sj j
V SV
v p V n v dS
v dV n v dSΔ rarr
nabla sdot Δ = sdot
rArr nablasdot = sdot
sum sumint
int int
r r
r r
D
MingsianMingsian R R BaiBai39
( )
scalar variable constant vectorvector variable
Alternative formsLet be a an arbitrary and
a
ˆ
1 Le
(
ˆt
V
V S
S
V
v ap
v d
v dV vn d
V n v dS
v va
va dV v a v a
S
nablasdot = sdot
= rArr
nablasdot = nabla sdot + nabla
=
sdot
nablaint intint int
int
r
v
r r
r r
r r rQ ( )
( )
ˆ)
2 Let ˆV S
V
V S
dV n va dS
v a p
a p
p dV n p d
d p
S
V a
⎛ ⎞= sdot⎜ ⎟
⎝ ⎠
= times rArr
nablasdot times = sdot nabla
times
times
nabla = times
int int
int int
int
r
r r r
r r v
r
vQ
r
( ) ( ) ( )
( ) ( )
ˆ
ˆV S
V S
a p dV n a p dS
a p dV a n p dS
⎛ ⎞⎡ ⎤minus sdot nablatimes = sdot times⎣ ⎦⎜ ⎟⎜ ⎟⎜ ⎟rArr minus sdot nablatimes = minus sdot times⎜ ⎟⎝ ⎠
int int
int int
r rv v
r rv v
MingsianMingsian R R BaiBai40
( )
( )
From Leibnitzs rule
(1-D) ( )
( ) ( )
B t
A t
d f x t dxdt
f x t dx B ttpart prime= +part
int( )
( )( ( ) ) ( )
B t
A tf B t t A tprimeminusint ( ( ) )
(assume ( ) ( ) )
f A t t
A t B t const= =
ContinuityEx The Equation
rate of increase of mass
mass densityV
dM d dVdt dt
ρ
ρ
= int
flow velocityvv
MingsianMingsian R R BaiBai41
ˆ ˆ ( )
(
Conservation
) ( Divergence theorem)
of mass
( ) 0
V S S
V
V
dV n v dS n v dSt
v dV
v dVt
ρ ρ ρ
ρ
ρ ρ⎡ ⎤⎢ ⎥⎣ ⎦
part = minus sdot = minus sdotpart
= minus nablasdot
partrArr +nablasdot =part
int int int
int
int
v v
vQ
v
ˆOn the other hand rate influof mass through n ox i t
Extended to 3-D for time-independent fixed boundary
(3-D) ( ) ( )
Thus
S
V V
V
S V n v dS
Sd f x y z t dV f x y z t dVdt tdM dVdt t
ρ
ρ
= minus sdot
part=part
part=part
int
int int
intv
Fixed control volume (Eulerian coordinates)Otherwise Raynoldrsquos transport theoremMoving control volume Lagrangian coordinates
MingsianMingsian R R BaiBai42
But the control volime is only arbitrary
In particular for flow
( ) 0
con
(continuity equati
stant
on)
Divergence the
0 incompress
Note is t
ible
orem h
V
v
vt
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
e mother of the other integral theorems
MingsianMingsian R R BaiBai43
Ex Archimedes principle ( ) The buoyant force ( ) on a body with volume immersed in a fluid is equal to the weight of the fluid
ˆ displaced by the object ie In the followin
B
V
gVρ= minusf k
阿基米德原理
浮力
ˆg figure the ˆ with and being the fluid density gravitational
acceleration depth and outward normal unit vector to the surface
fhydraulic pressure g z
g z
S
ρ
ρ
= minusp n
n
1
the modified divergence theorem
ˆ
Hint
scalar fuctionV S
vdV v dS v Cnabla = isinint int n
MingsianMingsian R R BaiBai44
Levelsurfacefluid
V
S
airyx
z
ˆf g zρ= minusp n
ˆPf ( ) ( )
ˆ ˆ ˆ ˆ ˆ (0 0 ) v v
B fS S V
V V
g z g zdS dS dV
g dV g dV gV
ρ ρ
ρ ρ ρ
= = minus = minus nabla
= minus + + = minus = minus
int int int
int int
f p n
i j k k k
k
MingsianMingsian R R BaiBai45
Note This can also be thought of as simple balance of forces (resultant force that acts on and weight of filled with fluid)S V
2
~ a little history about George Green Recall the diveregence theorem
ˆ
Let
II th
e Green
are arbitrary fun
s identiti
ctions
es
V S
v dV n v dS
v u vu v C
nablasdot = sdot
= nablaisin
int intv v
v
ˆ ( ) ( )
ˆ ( )V S
V S
u v dV n u v dS
u v u v dV un v dS
nablasdot nabla = sdot nabla
nabla sdotnabla + nablasdotnabla = sdotnabla
int int
int int
MingsianMingsian R R BaiBai46
2 -( ) -(1)V S
vu v u v dV u dSnpart
nabla sdotnabla + nabla =partint int firs(Greens identt ity)
2
If
( ) --(2)
(1) (2) V S
u vuv u v u dV v dSn
harrpart
nabla sdotnabla + nabla =part
minus
int int
( )2 2
V S
dv duu v v u dV u v dSdn dn
⎛ ⎞nabla minus nabla = minus⎜ ⎟⎝ ⎠int int
second(Greens identity)
MingsianMingsian R R BaiBai47
( )
( )
( ) ( )
1st id ( )
Greens identities in the 1D case integration by parts
b b bb ba a
a a ab
ba
ab
ba
a
v udx v u v u dx u v uv dx uv
u v u v vu dx vu
uv vu dx uv vu
primeprime prime prime prime prime prime primeprime prime= minus rArr + =
prime prime primeprime primeharr + =
primeprime primeprime prime primerArr minus = minus
int int int
int
2
2nd id ( )
Good for any adjoint operators in addition to Good also for and integration by parts Important for theory amp reciprocity
From th
Gree
e 1s
2-
t
ns functi
identity
D 1-
tak 1
D
e
on
u
nablararr
=
int
2
V S
vv dV dSnpartnabla =part
rArr int int2Ex 0 st 0 on the boundary 0 ( 1st id u u unabla = = rArr equiv 以 証)
MingsianMingsian R R BaiBai48
1
Thm1 Stokes theorem Let be defined in Let be a piecewise-smooth orientable surface bounded by a piec
III Stokes theorem and
ewise-smooth simple clos
Greens
ed curve Then
theore
m
v C R S
C
isinv
( )
ˆwhere is a unit normal to according t right hand rulo the e
ˆS C
n S
n v dS v dRsdot nablatimes = sdotint intvv v
(not necessarily planar)S
Cn vnablatimes v vv
MingsianMingsian R R BaiBai49
Pf
( )( )
circulatioWe have shown in deriving thatthe in -direction
ˆor
ˆ
n
z
z
j j j j j
j j j j jj j
vz v dx dy
v dR n v S
v dR n v S
nablatimes
=nablatimes
sdot = sdot nablatimes Δ
sdot = sdot nablatimes Δ
intsum sumint
v
v
vv v
vv v
C
S
jS
jC
jP
ˆ jn
jSΔ
MingsianMingsian R R BaiBai50
( )
( )ext surface
After the interior boundaries cancel out only the exterior boundaries (Why)
ˆ
ˆ0 QED
Alternative forms scalar
survive
va
j
j j j j jC
jC S
v dR n v S
S v dR n v dS
φ
sdot = sdot nablatimes Δ
Δ rarr rArr sdot = sdot nablatimes
sum sumint
int int
vv v
vv v
( )
riable constant vector vector variable
1 Let
2 Let
ˆ
ˆ S C
S C
a p
v n dR dR
n p dS dR p
a
v a p
φ φφ= rArr timesnabla =
timesnabla times= rArr timestimes =
int int
int int
v v
v v
v
v v
vv vv v
MingsianMingsian R R BaiBai51
( ) ( )
( )
( )
( ) ( )
1
Thm2 Greens theorem ( planar version of Stokes theorem)ˆ ˆ Let
pf By Stokes theoremˆ
ˆ ˆ ˆ ˆ ˆ ˆ
S C
S C
S C
v P x y i Q x y j C
n v dS v dR
k Pi Qj dS Pi Qj dx i dy
Q P dS Pdx Qdyx y
⎛ ⎞part partminus = +⎜ ⎟
= + isin
sdot nablatimes = sdot
⎡ ⎤sdot nablatimes + =
part part⎝ ⎠
+ sdot +⎣ ⎦
int int
int
int
int
int
v
vv v
( )
( )
( )
ˆ
ˆˆ ˆ
ˆˆ ˆ ˆ ˆ( ) ( )
0
QEDS C
j
i j kQ PPi Qj i j k
x y z x yP Q
Q P dS Pdx Qdyx y
⎛ ⎞part part part part partnablatimes + = = + + minus⎜ ⎟part part part part part⎝ ⎠
⎛ ⎞part partrArr minus = +⎜ ⎟part part⎝ ⎠int int
L L
S
C
i
j
x
y
MingsianMingsian R R BaiBai52
Potential TheorypotentiGeorge Green coined the term (l
a )
rarr 位能 勢能
例子 重力場 電磁場 流場 聲場 彈性力學
1
2
Thm1 If is in a simply connected region then the following statements are equivalent(i) There exists a scala
Conservative Field
potential functior such that
(ii
n
) 0
v
C
C
v
v R
φφ=nabla
nablatimes
isin
isin
=r
r
r
r in (Irrotational field curl-free)
cf 0 st solenoidal field divergence-free
0
(iii) in (Conservative field)C
Rv w v w
v R Rd
nablasdot = exist nabla
sdot =
= times
int
r r r
r
r
r
MingsianMingsian R R BaiBai53
(iv) is connecting and
independent o
Ex Think
f the pa
of dynamics is the force
thB
A
v dR A B
v
sdotintrr
r
1 2
1
0
Pf
(i) (ii) ( ) 0 (zero identity cg rule) irrotational
ˆ(ii) (iii) ( ) 0 (Stokes theorem)
(iii) (iv) ( ) 0
C S
C C C
C
v
v dR n v dS
v dR v dR
v dR
φ
=
rarr nablatimes = nablatimes nabla =
rarr sdot = sdot nablatimes =
rarr sdot = minus sdot =
rArr sdot =
int int
int int int
int
r
rr
rr r
r rr r
rr r
2
independent of pathC
v dRsdotintr
A
B
1C
2C
MingsianMingsian R R BaiBai54
1 2
(iv) (i) constant ( ) ( )
--(1)
But (2)
(1) (2) QED
Note on application
Check
B
C C A
v dR v dR B A d
v dR d
d dx dy dz dRx y z
v dR dR v R
φ φ φ
φφ φ φφ φ
φ φ
rarr sdot = sdot = = minus =
rArr sdot =part part part
= + + = nabla sdot minus minus minus minus minuspart part part
sdot = nabla sdot rArr = nabla forall
int int intr rr r
rr
r
r r rr r
use simpler path (iv)if 0
find (i)v
φ⎧
nablatimes = ⎨⎩
rr
A
B
x
y
Cz
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai20
2 2 3
2 2
ˆˆ ˆEx 2
ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ 2 0 2
2 0 0 2yx z
u x y v x yi z j k
u i j k u xyi x j k xyi x jx y z
vv vv xy xyx y z
⎛ ⎞⎜ ⎟⎝ ⎠
= = minus +
part part partnabla = + + = + + = +part part part
partpart partnablasdot = + + = + minus =part part part
v
v
MingsianMingsian R R BaiBai21
( ) ( ) ( )
( ) ( )
2 2 2 2
2 2 22
2 2 2
2
ˆˆ ˆ
ˆˆ ˆ
ˆ ˆˆ ˆ ˆ 0 3 0 0 0 3
2 0 0 2
ˆˆ ˆ
y yx xz z
x y z
i j kv vv vv vv i j k
x y z y z x z x yv v v
z i j x k z i x k
u u uu y yx y zv v v
i j kx y z
⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠ ⎝ ⎠
⎛
⎝
part partpart partpart partpart part partnablatimes = = minus minus minus + minuspart part part part part part part part part
= + minus minus + minus = minus
part part partnabla = + + = + + =part part part
nablatimes nablatimes =nabla nablasdot minusnabla
part part part= + +part part part
v
v v v
( ) ( )( ) ( ) ( ) ( )
( )
2 2 22 3
2 2 2ˆˆ ˆ2 2
ˆ ˆˆ ˆ ˆ ˆ 2 2 0 2 0 0 0 0 6 0 0 0
ˆ 2 6
xy x yi z j kx y z
yi xj k y i z j k
x z j
⎛ ⎞⎞⎜ ⎟⎜ ⎟
⎠ ⎝ ⎠⎡ ⎤⎣ ⎦
part part partminus + + minus +part part part
= + + minus + + + + minus + + minus
= +
MingsianMingsian R R BaiBai22
( )v w r w r r wnablatimes =nablatimes times = nablasdot minus nablasdotv v v v v v v ( )r w+ sdotnablav v ( ) constant w
w rminus sdotnablav
v v
Q
Ex Rigid body rotation
MingsianMingsian R R BaiBai23
( )( ) ( )
( ) ( )
ˆˆ ˆ
ˆ ˆˆ ˆ ˆ ˆ 1 1 1 3
ˆˆ ˆ
ˆˆ ˆ
Thus 3 2 is linked directly
x y z
x y z
r xi yj zk
r i j k xi yj zkx y z
w r w w w xi yj zkx y z
w i w j w k w
w r w r w r w w wv
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎝ ⎠
= + +
part part partnablasdot = + + + + = + + =part part part
part part partsdotnabla = + + + +part part part
= + + =
nablatimes times == nablasdot minus sdotnabla = minus =rArrnablatimes
v
v
v v
v
v v v v v v v v v
v spinn to ing
MingsianMingsian R R BaiBai24
Line surface and volume integrals
x
yC
B
A
isΔ
( )ii ηξ
Piecewise smooth curve
Line integral
MingsianMingsian R R BaiBai25
0 1lim ( ) ( ) or ( )
Intepretation of a line integral as an area ( )
i
Bn
i i isA
C
i C
F s F x y ds F x y d
x
s
F y ds
ξ ηΔ rarr =
Δ =sum int
int
int
x
y
z
C
( )z F x y=
MingsianMingsian R R BaiBai26
the area is always on the left of an obs
Positive dierver trave
rection of irsing on the
ntegrationcurve
RC
CCW CW
If is a regular closed curve piecewise smo
( ) ( )
(contou
oth simple ( )
r integral)
x y zC C
x y zC
F dR F F F dx dy dz
F dx F dy
C
F dz
sdot = + + + +
= + +
lowast
int int
int
i j k i j kv v
無交叉
work dW F dR= sdotv v
MingsianMingsian R R BaiBai27
0 1
Surface integral
lim ( ) ( )i
n
i i i iS i S
F S F x y z dSξ η ζΔ rarr =
Δ =sum intint( ) i i iξ η ζ
iSΔ
0 1
mass density
lim ( ) ( )
Ex
Volume integral
( )
i
n
i i i iV i V
V
F V F x y z dV
M x y z dV
ξ η ζ
ρ
Δ rarr =Δ =
=
sum intintint
intintint( ) i i iξ η ζ
iVΔ
MingsianMingsian R R BaiBai28
Ref (1)Hildebrand advanced Calculus for application Prentice-Hall 1976 p306-313(2) Greenberg p383-389 (3) Arfken ch2
1 2 3
1 2 3
1 2 3
1 2 3
(
Orthogonal coordinate systems base vectors are not constant in s
) ( )( )( )( )
ˆˆ ˆ
pacep x y z p u u ux x u u uy y u u uz z u u u
r xi yj zk
primerarr
= +
rarr
===
+v
1uv
2uv3uv
1u
2u
3uP coord curve
xy
z
O
X Curvilinear Coordinates
MingsianMingsian R R BaiBai29
1
11
1 1 1
1
1
1 1 1
11 1
1 1
2 2 2 3 3 3
Tangent vector to the at
where is the arc length
1
where unit vector
Similarly
curv
e
u psr rU
u s us
rs
U h uds rh udu sU h u U h u
partpart part= =part part part
part =part
there4 =part= =part
= =
v vv
vQ
v v
vv
v vv v
1u cood curve
1uv
1du 1dS
drv
P
MingsianMingsian R R BaiBai30
3 31 2 1 21 2 3
1 2 32
curv
In summary scale factor
If is the arc length along a in any directione
k k k
kk k
k k
k k k
U h u
ds rh Udu u
ds h dus
du dudu du du dudr r r r U U Uds u ds u ds u ds ds ds ds
dr drds ds
=
part= = =part
=
part part part= + + = + +part part part
= sdot
v v
v v
v v v v v v v
v v 3 3
1 1
23 3 32 2
21 1 1
2 2 2 2 2 2 2 2 2 21 1 2 2 3 3 1 2 3
1 (for orthogonal coord)
jii j
i j
ji ii j i
i j i
dududr U Uds ds ds
dudu duU U hds ds ds
ds h du h du h du ds ds ds
= =
= = =
⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
= sdot
rarr = sdot =
rarr = + + = + +
sum sum
sumsum sum
v v v
v v
MingsianMingsian R R BaiBai31
1 1 2 2 3 3
1 2 3
are mutually vectors having with arc length
k k k k kds h du U du
U du U du U duds ds ds
= =
rArr perp
v
v v v
1uv
2uv
3uv
1dS
2dS
3dS
dS
MingsianMingsian R R BaiBai32
( ) ( )
( ) ( )
1 1 2 2 3 3 1 2 3 1 2 3 1 2 3
1 2 3 1 2 3
1
1 2 2 3 3 2 2 2 3 3 3 2 3 2 3 1
Differential volume
Differential On const coord surface
area
Gradi t
en
d U du U du U du u u u h h h du du du
d h h h du du du
ud U du U du h u du h u du h h du du u
df
τ
τ
σ
= times sdot = times sdot
=
=
= times = times =
= nabla
v v v v v v
v vv v v v
1 2 31 2 3
1 2 3 1 1 2 2 3 31 2 3
1 1 1 2 2 2 3 3 3
f f ff dr df du du duu u u
r r rdr du du du U du U du U duu u u
h u du h u du h u du
part part partsdot rarr = + +part part part
part part part= + + = + +part part part
= + +
v
v v v v v vv
v v v
=1 (right hand rule)
MingsianMingsian R R BaiBai33
Rectangular Cartesian Coordinates
ij
k
MingsianMingsian R R BaiBai34
Circular Cylindrical Coordinatescossin
xyz z
ρ ϕρ ϕ
===
Ref Arfken
( )P zρ ϕ
MingsianMingsian R R BaiBai35
Polar Spherical Coordinatessin cossin sincos
x ry rz r
θ ϕθ ϕθ
===
Ref Arfken
0
0
0
sin cos sin sin coscos cos cos sin sin
sin cos
θ ϕ θ ϕ θθ ϕ θ ϕ θϕ ϕ
= + +
= + minus= minus +
r i j kθ i j kφ i j
2
2
sinsin
dS r d ddV r dr d d
θ θ ϕ
θ θ ϕ
=
=( )P r θ ϕ
MingsianMingsian R R BaiBai36
Integral TheoremsHeart of this chapter
Interchange of dimensionality Div Curl Grad in integrals High-dimensional integration
rarr
1
Thm1 Divergence theorem Let be a closed region bounded by piecewise smooth orientable surface and let (con
I Divergence theor
tinuous derivatives up to the first order
em (Gauss theorem)
)
VS v Cisin
r
ˆ If denotes the outward unit normal on
the ˆn V S
v dV n v dS
n S
nablasdot = sdotint intr r
MingsianMingsian R R BaiBai37
jS
jP
jV
S
0
ˆBy definition ( ) lim
ˆ( )
ˆ( )
j
j
j
j
Sj V
j
j j S
j j Sj j
n v dSv P
V
v P V n v dS
v P V n v dS
Δ rarr
⎧ ⎫sdot⎪ ⎪nabla sdot ⎨ ⎬Δ⎪ ⎪⎩ ⎭
nabla sdot Δ asymp sdot
nabla sdot Δ = sdot
int
intsum sumint
rr
r r
r r
Pf
MingsianMingsian R R BaiBai38
0
After interior surfaces cancel out only exterior surface (Why Opposite normal vectors)
ˆ( )
volumes exterior surfaces
ˆlim
QE
survive
j
j
j j Sj j
V SV
v p V n v dS
v dV n v dSΔ rarr
nabla sdot Δ = sdot
rArr nablasdot = sdot
sum sumint
int int
r r
r r
D
MingsianMingsian R R BaiBai39
( )
scalar variable constant vectorvector variable
Alternative formsLet be a an arbitrary and
a
ˆ
1 Le
(
ˆt
V
V S
S
V
v ap
v d
v dV vn d
V n v dS
v va
va dV v a v a
S
nablasdot = sdot
= rArr
nablasdot = nabla sdot + nabla
=
sdot
nablaint intint int
int
r
v
r r
r r
r r rQ ( )
( )
ˆ)
2 Let ˆV S
V
V S
dV n va dS
v a p
a p
p dV n p d
d p
S
V a
⎛ ⎞= sdot⎜ ⎟
⎝ ⎠
= times rArr
nablasdot times = sdot nabla
times
times
nabla = times
int int
int int
int
r
r r r
r r v
r
vQ
r
( ) ( ) ( )
( ) ( )
ˆ
ˆV S
V S
a p dV n a p dS
a p dV a n p dS
⎛ ⎞⎡ ⎤minus sdot nablatimes = sdot times⎣ ⎦⎜ ⎟⎜ ⎟⎜ ⎟rArr minus sdot nablatimes = minus sdot times⎜ ⎟⎝ ⎠
int int
int int
r rv v
r rv v
MingsianMingsian R R BaiBai40
( )
( )
From Leibnitzs rule
(1-D) ( )
( ) ( )
B t
A t
d f x t dxdt
f x t dx B ttpart prime= +part
int( )
( )( ( ) ) ( )
B t
A tf B t t A tprimeminusint ( ( ) )
(assume ( ) ( ) )
f A t t
A t B t const= =
ContinuityEx The Equation
rate of increase of mass
mass densityV
dM d dVdt dt
ρ
ρ
= int
flow velocityvv
MingsianMingsian R R BaiBai41
ˆ ˆ ( )
(
Conservation
) ( Divergence theorem)
of mass
( ) 0
V S S
V
V
dV n v dS n v dSt
v dV
v dVt
ρ ρ ρ
ρ
ρ ρ⎡ ⎤⎢ ⎥⎣ ⎦
part = minus sdot = minus sdotpart
= minus nablasdot
partrArr +nablasdot =part
int int int
int
int
v v
vQ
v
ˆOn the other hand rate influof mass through n ox i t
Extended to 3-D for time-independent fixed boundary
(3-D) ( ) ( )
Thus
S
V V
V
S V n v dS
Sd f x y z t dV f x y z t dVdt tdM dVdt t
ρ
ρ
= minus sdot
part=part
part=part
int
int int
intv
Fixed control volume (Eulerian coordinates)Otherwise Raynoldrsquos transport theoremMoving control volume Lagrangian coordinates
MingsianMingsian R R BaiBai42
But the control volime is only arbitrary
In particular for flow
( ) 0
con
(continuity equati
stant
on)
Divergence the
0 incompress
Note is t
ible
orem h
V
v
vt
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
e mother of the other integral theorems
MingsianMingsian R R BaiBai43
Ex Archimedes principle ( ) The buoyant force ( ) on a body with volume immersed in a fluid is equal to the weight of the fluid
ˆ displaced by the object ie In the followin
B
V
gVρ= minusf k
阿基米德原理
浮力
ˆg figure the ˆ with and being the fluid density gravitational
acceleration depth and outward normal unit vector to the surface
fhydraulic pressure g z
g z
S
ρ
ρ
= minusp n
n
1
the modified divergence theorem
ˆ
Hint
scalar fuctionV S
vdV v dS v Cnabla = isinint int n
MingsianMingsian R R BaiBai44
Levelsurfacefluid
V
S
airyx
z
ˆf g zρ= minusp n
ˆPf ( ) ( )
ˆ ˆ ˆ ˆ ˆ (0 0 ) v v
B fS S V
V V
g z g zdS dS dV
g dV g dV gV
ρ ρ
ρ ρ ρ
= = minus = minus nabla
= minus + + = minus = minus
int int int
int int
f p n
i j k k k
k
MingsianMingsian R R BaiBai45
Note This can also be thought of as simple balance of forces (resultant force that acts on and weight of filled with fluid)S V
2
~ a little history about George Green Recall the diveregence theorem
ˆ
Let
II th
e Green
are arbitrary fun
s identiti
ctions
es
V S
v dV n v dS
v u vu v C
nablasdot = sdot
= nablaisin
int intv v
v
ˆ ( ) ( )
ˆ ( )V S
V S
u v dV n u v dS
u v u v dV un v dS
nablasdot nabla = sdot nabla
nabla sdotnabla + nablasdotnabla = sdotnabla
int int
int int
MingsianMingsian R R BaiBai46
2 -( ) -(1)V S
vu v u v dV u dSnpart
nabla sdotnabla + nabla =partint int firs(Greens identt ity)
2
If
( ) --(2)
(1) (2) V S
u vuv u v u dV v dSn
harrpart
nabla sdotnabla + nabla =part
minus
int int
( )2 2
V S
dv duu v v u dV u v dSdn dn
⎛ ⎞nabla minus nabla = minus⎜ ⎟⎝ ⎠int int
second(Greens identity)
MingsianMingsian R R BaiBai47
( )
( )
( ) ( )
1st id ( )
Greens identities in the 1D case integration by parts
b b bb ba a
a a ab
ba
ab
ba
a
v udx v u v u dx u v uv dx uv
u v u v vu dx vu
uv vu dx uv vu
primeprime prime prime prime prime prime primeprime prime= minus rArr + =
prime prime primeprime primeharr + =
primeprime primeprime prime primerArr minus = minus
int int int
int
2
2nd id ( )
Good for any adjoint operators in addition to Good also for and integration by parts Important for theory amp reciprocity
From th
Gree
e 1s
2-
t
ns functi
identity
D 1-
tak 1
D
e
on
u
nablararr
=
int
2
V S
vv dV dSnpartnabla =part
rArr int int2Ex 0 st 0 on the boundary 0 ( 1st id u u unabla = = rArr equiv 以 証)
MingsianMingsian R R BaiBai48
1
Thm1 Stokes theorem Let be defined in Let be a piecewise-smooth orientable surface bounded by a piec
III Stokes theorem and
ewise-smooth simple clos
Greens
ed curve Then
theore
m
v C R S
C
isinv
( )
ˆwhere is a unit normal to according t right hand rulo the e
ˆS C
n S
n v dS v dRsdot nablatimes = sdotint intvv v
(not necessarily planar)S
Cn vnablatimes v vv
MingsianMingsian R R BaiBai49
Pf
( )( )
circulatioWe have shown in deriving thatthe in -direction
ˆor
ˆ
n
z
z
j j j j j
j j j j jj j
vz v dx dy
v dR n v S
v dR n v S
nablatimes
=nablatimes
sdot = sdot nablatimes Δ
sdot = sdot nablatimes Δ
intsum sumint
v
v
vv v
vv v
C
S
jS
jC
jP
ˆ jn
jSΔ
MingsianMingsian R R BaiBai50
( )
( )ext surface
After the interior boundaries cancel out only the exterior boundaries (Why)
ˆ
ˆ0 QED
Alternative forms scalar
survive
va
j
j j j j jC
jC S
v dR n v S
S v dR n v dS
φ
sdot = sdot nablatimes Δ
Δ rarr rArr sdot = sdot nablatimes
sum sumint
int int
vv v
vv v
( )
riable constant vector vector variable
1 Let
2 Let
ˆ
ˆ S C
S C
a p
v n dR dR
n p dS dR p
a
v a p
φ φφ= rArr timesnabla =
timesnabla times= rArr timestimes =
int int
int int
v v
v v
v
v v
vv vv v
MingsianMingsian R R BaiBai51
( ) ( )
( )
( )
( ) ( )
1
Thm2 Greens theorem ( planar version of Stokes theorem)ˆ ˆ Let
pf By Stokes theoremˆ
ˆ ˆ ˆ ˆ ˆ ˆ
S C
S C
S C
v P x y i Q x y j C
n v dS v dR
k Pi Qj dS Pi Qj dx i dy
Q P dS Pdx Qdyx y
⎛ ⎞part partminus = +⎜ ⎟
= + isin
sdot nablatimes = sdot
⎡ ⎤sdot nablatimes + =
part part⎝ ⎠
+ sdot +⎣ ⎦
int int
int
int
int
int
v
vv v
( )
( )
( )
ˆ
ˆˆ ˆ
ˆˆ ˆ ˆ ˆ( ) ( )
0
QEDS C
j
i j kQ PPi Qj i j k
x y z x yP Q
Q P dS Pdx Qdyx y
⎛ ⎞part part part part partnablatimes + = = + + minus⎜ ⎟part part part part part⎝ ⎠
⎛ ⎞part partrArr minus = +⎜ ⎟part part⎝ ⎠int int
L L
S
C
i
j
x
y
MingsianMingsian R R BaiBai52
Potential TheorypotentiGeorge Green coined the term (l
a )
rarr 位能 勢能
例子 重力場 電磁場 流場 聲場 彈性力學
1
2
Thm1 If is in a simply connected region then the following statements are equivalent(i) There exists a scala
Conservative Field
potential functior such that
(ii
n
) 0
v
C
C
v
v R
φφ=nabla
nablatimes
isin
isin
=r
r
r
r in (Irrotational field curl-free)
cf 0 st solenoidal field divergence-free
0
(iii) in (Conservative field)C
Rv w v w
v R Rd
nablasdot = exist nabla
sdot =
= times
int
r r r
r
r
r
MingsianMingsian R R BaiBai53
(iv) is connecting and
independent o
Ex Think
f the pa
of dynamics is the force
thB
A
v dR A B
v
sdotintrr
r
1 2
1
0
Pf
(i) (ii) ( ) 0 (zero identity cg rule) irrotational
ˆ(ii) (iii) ( ) 0 (Stokes theorem)
(iii) (iv) ( ) 0
C S
C C C
C
v
v dR n v dS
v dR v dR
v dR
φ
=
rarr nablatimes = nablatimes nabla =
rarr sdot = sdot nablatimes =
rarr sdot = minus sdot =
rArr sdot =
int int
int int int
int
r
rr
rr r
r rr r
rr r
2
independent of pathC
v dRsdotintr
A
B
1C
2C
MingsianMingsian R R BaiBai54
1 2
(iv) (i) constant ( ) ( )
--(1)
But (2)
(1) (2) QED
Note on application
Check
B
C C A
v dR v dR B A d
v dR d
d dx dy dz dRx y z
v dR dR v R
φ φ φ
φφ φ φφ φ
φ φ
rarr sdot = sdot = = minus =
rArr sdot =part part part
= + + = nabla sdot minus minus minus minus minuspart part part
sdot = nabla sdot rArr = nabla forall
int int intr rr r
rr
r
r r rr r
use simpler path (iv)if 0
find (i)v
φ⎧
nablatimes = ⎨⎩
rr
A
B
x
y
Cz
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai21
( ) ( ) ( )
( ) ( )
2 2 2 2
2 2 22
2 2 2
2
ˆˆ ˆ
ˆˆ ˆ
ˆ ˆˆ ˆ ˆ 0 3 0 0 0 3
2 0 0 2
ˆˆ ˆ
y yx xz z
x y z
i j kv vv vv vv i j k
x y z y z x z x yv v v
z i j x k z i x k
u u uu y yx y zv v v
i j kx y z
⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠ ⎝ ⎠
⎛
⎝
part partpart partpart partpart part partnablatimes = = minus minus minus + minuspart part part part part part part part part
= + minus minus + minus = minus
part part partnabla = + + = + + =part part part
nablatimes nablatimes =nabla nablasdot minusnabla
part part part= + +part part part
v
v v v
( ) ( )( ) ( ) ( ) ( )
( )
2 2 22 3
2 2 2ˆˆ ˆ2 2
ˆ ˆˆ ˆ ˆ ˆ 2 2 0 2 0 0 0 0 6 0 0 0
ˆ 2 6
xy x yi z j kx y z
yi xj k y i z j k
x z j
⎛ ⎞⎞⎜ ⎟⎜ ⎟
⎠ ⎝ ⎠⎡ ⎤⎣ ⎦
part part partminus + + minus +part part part
= + + minus + + + + minus + + minus
= +
MingsianMingsian R R BaiBai22
( )v w r w r r wnablatimes =nablatimes times = nablasdot minus nablasdotv v v v v v v ( )r w+ sdotnablav v ( ) constant w
w rminus sdotnablav
v v
Q
Ex Rigid body rotation
MingsianMingsian R R BaiBai23
( )( ) ( )
( ) ( )
ˆˆ ˆ
ˆ ˆˆ ˆ ˆ ˆ 1 1 1 3
ˆˆ ˆ
ˆˆ ˆ
Thus 3 2 is linked directly
x y z
x y z
r xi yj zk
r i j k xi yj zkx y z
w r w w w xi yj zkx y z
w i w j w k w
w r w r w r w w wv
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎝ ⎠
= + +
part part partnablasdot = + + + + = + + =part part part
part part partsdotnabla = + + + +part part part
= + + =
nablatimes times == nablasdot minus sdotnabla = minus =rArrnablatimes
v
v
v v
v
v v v v v v v v v
v spinn to ing
MingsianMingsian R R BaiBai24
Line surface and volume integrals
x
yC
B
A
isΔ
( )ii ηξ
Piecewise smooth curve
Line integral
MingsianMingsian R R BaiBai25
0 1lim ( ) ( ) or ( )
Intepretation of a line integral as an area ( )
i
Bn
i i isA
C
i C
F s F x y ds F x y d
x
s
F y ds
ξ ηΔ rarr =
Δ =sum int
int
int
x
y
z
C
( )z F x y=
MingsianMingsian R R BaiBai26
the area is always on the left of an obs
Positive dierver trave
rection of irsing on the
ntegrationcurve
RC
CCW CW
If is a regular closed curve piecewise smo
( ) ( )
(contou
oth simple ( )
r integral)
x y zC C
x y zC
F dR F F F dx dy dz
F dx F dy
C
F dz
sdot = + + + +
= + +
lowast
int int
int
i j k i j kv v
無交叉
work dW F dR= sdotv v
MingsianMingsian R R BaiBai27
0 1
Surface integral
lim ( ) ( )i
n
i i i iS i S
F S F x y z dSξ η ζΔ rarr =
Δ =sum intint( ) i i iξ η ζ
iSΔ
0 1
mass density
lim ( ) ( )
Ex
Volume integral
( )
i
n
i i i iV i V
V
F V F x y z dV
M x y z dV
ξ η ζ
ρ
Δ rarr =Δ =
=
sum intintint
intintint( ) i i iξ η ζ
iVΔ
MingsianMingsian R R BaiBai28
Ref (1)Hildebrand advanced Calculus for application Prentice-Hall 1976 p306-313(2) Greenberg p383-389 (3) Arfken ch2
1 2 3
1 2 3
1 2 3
1 2 3
(
Orthogonal coordinate systems base vectors are not constant in s
) ( )( )( )( )
ˆˆ ˆ
pacep x y z p u u ux x u u uy y u u uz z u u u
r xi yj zk
primerarr
= +
rarr
===
+v
1uv
2uv3uv
1u
2u
3uP coord curve
xy
z
O
X Curvilinear Coordinates
MingsianMingsian R R BaiBai29
1
11
1 1 1
1
1
1 1 1
11 1
1 1
2 2 2 3 3 3
Tangent vector to the at
where is the arc length
1
where unit vector
Similarly
curv
e
u psr rU
u s us
rs
U h uds rh udu sU h u U h u
partpart part= =part part part
part =part
there4 =part= =part
= =
v vv
vQ
v v
vv
v vv v
1u cood curve
1uv
1du 1dS
drv
P
MingsianMingsian R R BaiBai30
3 31 2 1 21 2 3
1 2 32
curv
In summary scale factor
If is the arc length along a in any directione
k k k
kk k
k k
k k k
U h u
ds rh Udu u
ds h dus
du dudu du du dudr r r r U U Uds u ds u ds u ds ds ds ds
dr drds ds
=
part= = =part
=
part part part= + + = + +part part part
= sdot
v v
v v
v v v v v v v
v v 3 3
1 1
23 3 32 2
21 1 1
2 2 2 2 2 2 2 2 2 21 1 2 2 3 3 1 2 3
1 (for orthogonal coord)
jii j
i j
ji ii j i
i j i
dududr U Uds ds ds
dudu duU U hds ds ds
ds h du h du h du ds ds ds
= =
= = =
⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
= sdot
rarr = sdot =
rarr = + + = + +
sum sum
sumsum sum
v v v
v v
MingsianMingsian R R BaiBai31
1 1 2 2 3 3
1 2 3
are mutually vectors having with arc length
k k k k kds h du U du
U du U du U duds ds ds
= =
rArr perp
v
v v v
1uv
2uv
3uv
1dS
2dS
3dS
dS
MingsianMingsian R R BaiBai32
( ) ( )
( ) ( )
1 1 2 2 3 3 1 2 3 1 2 3 1 2 3
1 2 3 1 2 3
1
1 2 2 3 3 2 2 2 3 3 3 2 3 2 3 1
Differential volume
Differential On const coord surface
area
Gradi t
en
d U du U du U du u u u h h h du du du
d h h h du du du
ud U du U du h u du h u du h h du du u
df
τ
τ
σ
= times sdot = times sdot
=
=
= times = times =
= nabla
v v v v v v
v vv v v v
1 2 31 2 3
1 2 3 1 1 2 2 3 31 2 3
1 1 1 2 2 2 3 3 3
f f ff dr df du du duu u u
r r rdr du du du U du U du U duu u u
h u du h u du h u du
part part partsdot rarr = + +part part part
part part part= + + = + +part part part
= + +
v
v v v v v vv
v v v
=1 (right hand rule)
MingsianMingsian R R BaiBai33
Rectangular Cartesian Coordinates
ij
k
MingsianMingsian R R BaiBai34
Circular Cylindrical Coordinatescossin
xyz z
ρ ϕρ ϕ
===
Ref Arfken
( )P zρ ϕ
MingsianMingsian R R BaiBai35
Polar Spherical Coordinatessin cossin sincos
x ry rz r
θ ϕθ ϕθ
===
Ref Arfken
0
0
0
sin cos sin sin coscos cos cos sin sin
sin cos
θ ϕ θ ϕ θθ ϕ θ ϕ θϕ ϕ
= + +
= + minus= minus +
r i j kθ i j kφ i j
2
2
sinsin
dS r d ddV r dr d d
θ θ ϕ
θ θ ϕ
=
=( )P r θ ϕ
MingsianMingsian R R BaiBai36
Integral TheoremsHeart of this chapter
Interchange of dimensionality Div Curl Grad in integrals High-dimensional integration
rarr
1
Thm1 Divergence theorem Let be a closed region bounded by piecewise smooth orientable surface and let (con
I Divergence theor
tinuous derivatives up to the first order
em (Gauss theorem)
)
VS v Cisin
r
ˆ If denotes the outward unit normal on
the ˆn V S
v dV n v dS
n S
nablasdot = sdotint intr r
MingsianMingsian R R BaiBai37
jS
jP
jV
S
0
ˆBy definition ( ) lim
ˆ( )
ˆ( )
j
j
j
j
Sj V
j
j j S
j j Sj j
n v dSv P
V
v P V n v dS
v P V n v dS
Δ rarr
⎧ ⎫sdot⎪ ⎪nabla sdot ⎨ ⎬Δ⎪ ⎪⎩ ⎭
nabla sdot Δ asymp sdot
nabla sdot Δ = sdot
int
intsum sumint
rr
r r
r r
Pf
MingsianMingsian R R BaiBai38
0
After interior surfaces cancel out only exterior surface (Why Opposite normal vectors)
ˆ( )
volumes exterior surfaces
ˆlim
QE
survive
j
j
j j Sj j
V SV
v p V n v dS
v dV n v dSΔ rarr
nabla sdot Δ = sdot
rArr nablasdot = sdot
sum sumint
int int
r r
r r
D
MingsianMingsian R R BaiBai39
( )
scalar variable constant vectorvector variable
Alternative formsLet be a an arbitrary and
a
ˆ
1 Le
(
ˆt
V
V S
S
V
v ap
v d
v dV vn d
V n v dS
v va
va dV v a v a
S
nablasdot = sdot
= rArr
nablasdot = nabla sdot + nabla
=
sdot
nablaint intint int
int
r
v
r r
r r
r r rQ ( )
( )
ˆ)
2 Let ˆV S
V
V S
dV n va dS
v a p
a p
p dV n p d
d p
S
V a
⎛ ⎞= sdot⎜ ⎟
⎝ ⎠
= times rArr
nablasdot times = sdot nabla
times
times
nabla = times
int int
int int
int
r
r r r
r r v
r
vQ
r
( ) ( ) ( )
( ) ( )
ˆ
ˆV S
V S
a p dV n a p dS
a p dV a n p dS
⎛ ⎞⎡ ⎤minus sdot nablatimes = sdot times⎣ ⎦⎜ ⎟⎜ ⎟⎜ ⎟rArr minus sdot nablatimes = minus sdot times⎜ ⎟⎝ ⎠
int int
int int
r rv v
r rv v
MingsianMingsian R R BaiBai40
( )
( )
From Leibnitzs rule
(1-D) ( )
( ) ( )
B t
A t
d f x t dxdt
f x t dx B ttpart prime= +part
int( )
( )( ( ) ) ( )
B t
A tf B t t A tprimeminusint ( ( ) )
(assume ( ) ( ) )
f A t t
A t B t const= =
ContinuityEx The Equation
rate of increase of mass
mass densityV
dM d dVdt dt
ρ
ρ
= int
flow velocityvv
MingsianMingsian R R BaiBai41
ˆ ˆ ( )
(
Conservation
) ( Divergence theorem)
of mass
( ) 0
V S S
V
V
dV n v dS n v dSt
v dV
v dVt
ρ ρ ρ
ρ
ρ ρ⎡ ⎤⎢ ⎥⎣ ⎦
part = minus sdot = minus sdotpart
= minus nablasdot
partrArr +nablasdot =part
int int int
int
int
v v
vQ
v
ˆOn the other hand rate influof mass through n ox i t
Extended to 3-D for time-independent fixed boundary
(3-D) ( ) ( )
Thus
S
V V
V
S V n v dS
Sd f x y z t dV f x y z t dVdt tdM dVdt t
ρ
ρ
= minus sdot
part=part
part=part
int
int int
intv
Fixed control volume (Eulerian coordinates)Otherwise Raynoldrsquos transport theoremMoving control volume Lagrangian coordinates
MingsianMingsian R R BaiBai42
But the control volime is only arbitrary
In particular for flow
( ) 0
con
(continuity equati
stant
on)
Divergence the
0 incompress
Note is t
ible
orem h
V
v
vt
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
e mother of the other integral theorems
MingsianMingsian R R BaiBai43
Ex Archimedes principle ( ) The buoyant force ( ) on a body with volume immersed in a fluid is equal to the weight of the fluid
ˆ displaced by the object ie In the followin
B
V
gVρ= minusf k
阿基米德原理
浮力
ˆg figure the ˆ with and being the fluid density gravitational
acceleration depth and outward normal unit vector to the surface
fhydraulic pressure g z
g z
S
ρ
ρ
= minusp n
n
1
the modified divergence theorem
ˆ
Hint
scalar fuctionV S
vdV v dS v Cnabla = isinint int n
MingsianMingsian R R BaiBai44
Levelsurfacefluid
V
S
airyx
z
ˆf g zρ= minusp n
ˆPf ( ) ( )
ˆ ˆ ˆ ˆ ˆ (0 0 ) v v
B fS S V
V V
g z g zdS dS dV
g dV g dV gV
ρ ρ
ρ ρ ρ
= = minus = minus nabla
= minus + + = minus = minus
int int int
int int
f p n
i j k k k
k
MingsianMingsian R R BaiBai45
Note This can also be thought of as simple balance of forces (resultant force that acts on and weight of filled with fluid)S V
2
~ a little history about George Green Recall the diveregence theorem
ˆ
Let
II th
e Green
are arbitrary fun
s identiti
ctions
es
V S
v dV n v dS
v u vu v C
nablasdot = sdot
= nablaisin
int intv v
v
ˆ ( ) ( )
ˆ ( )V S
V S
u v dV n u v dS
u v u v dV un v dS
nablasdot nabla = sdot nabla
nabla sdotnabla + nablasdotnabla = sdotnabla
int int
int int
MingsianMingsian R R BaiBai46
2 -( ) -(1)V S
vu v u v dV u dSnpart
nabla sdotnabla + nabla =partint int firs(Greens identt ity)
2
If
( ) --(2)
(1) (2) V S
u vuv u v u dV v dSn
harrpart
nabla sdotnabla + nabla =part
minus
int int
( )2 2
V S
dv duu v v u dV u v dSdn dn
⎛ ⎞nabla minus nabla = minus⎜ ⎟⎝ ⎠int int
second(Greens identity)
MingsianMingsian R R BaiBai47
( )
( )
( ) ( )
1st id ( )
Greens identities in the 1D case integration by parts
b b bb ba a
a a ab
ba
ab
ba
a
v udx v u v u dx u v uv dx uv
u v u v vu dx vu
uv vu dx uv vu
primeprime prime prime prime prime prime primeprime prime= minus rArr + =
prime prime primeprime primeharr + =
primeprime primeprime prime primerArr minus = minus
int int int
int
2
2nd id ( )
Good for any adjoint operators in addition to Good also for and integration by parts Important for theory amp reciprocity
From th
Gree
e 1s
2-
t
ns functi
identity
D 1-
tak 1
D
e
on
u
nablararr
=
int
2
V S
vv dV dSnpartnabla =part
rArr int int2Ex 0 st 0 on the boundary 0 ( 1st id u u unabla = = rArr equiv 以 証)
MingsianMingsian R R BaiBai48
1
Thm1 Stokes theorem Let be defined in Let be a piecewise-smooth orientable surface bounded by a piec
III Stokes theorem and
ewise-smooth simple clos
Greens
ed curve Then
theore
m
v C R S
C
isinv
( )
ˆwhere is a unit normal to according t right hand rulo the e
ˆS C
n S
n v dS v dRsdot nablatimes = sdotint intvv v
(not necessarily planar)S
Cn vnablatimes v vv
MingsianMingsian R R BaiBai49
Pf
( )( )
circulatioWe have shown in deriving thatthe in -direction
ˆor
ˆ
n
z
z
j j j j j
j j j j jj j
vz v dx dy
v dR n v S
v dR n v S
nablatimes
=nablatimes
sdot = sdot nablatimes Δ
sdot = sdot nablatimes Δ
intsum sumint
v
v
vv v
vv v
C
S
jS
jC
jP
ˆ jn
jSΔ
MingsianMingsian R R BaiBai50
( )
( )ext surface
After the interior boundaries cancel out only the exterior boundaries (Why)
ˆ
ˆ0 QED
Alternative forms scalar
survive
va
j
j j j j jC
jC S
v dR n v S
S v dR n v dS
φ
sdot = sdot nablatimes Δ
Δ rarr rArr sdot = sdot nablatimes
sum sumint
int int
vv v
vv v
( )
riable constant vector vector variable
1 Let
2 Let
ˆ
ˆ S C
S C
a p
v n dR dR
n p dS dR p
a
v a p
φ φφ= rArr timesnabla =
timesnabla times= rArr timestimes =
int int
int int
v v
v v
v
v v
vv vv v
MingsianMingsian R R BaiBai51
( ) ( )
( )
( )
( ) ( )
1
Thm2 Greens theorem ( planar version of Stokes theorem)ˆ ˆ Let
pf By Stokes theoremˆ
ˆ ˆ ˆ ˆ ˆ ˆ
S C
S C
S C
v P x y i Q x y j C
n v dS v dR
k Pi Qj dS Pi Qj dx i dy
Q P dS Pdx Qdyx y
⎛ ⎞part partminus = +⎜ ⎟
= + isin
sdot nablatimes = sdot
⎡ ⎤sdot nablatimes + =
part part⎝ ⎠
+ sdot +⎣ ⎦
int int
int
int
int
int
v
vv v
( )
( )
( )
ˆ
ˆˆ ˆ
ˆˆ ˆ ˆ ˆ( ) ( )
0
QEDS C
j
i j kQ PPi Qj i j k
x y z x yP Q
Q P dS Pdx Qdyx y
⎛ ⎞part part part part partnablatimes + = = + + minus⎜ ⎟part part part part part⎝ ⎠
⎛ ⎞part partrArr minus = +⎜ ⎟part part⎝ ⎠int int
L L
S
C
i
j
x
y
MingsianMingsian R R BaiBai52
Potential TheorypotentiGeorge Green coined the term (l
a )
rarr 位能 勢能
例子 重力場 電磁場 流場 聲場 彈性力學
1
2
Thm1 If is in a simply connected region then the following statements are equivalent(i) There exists a scala
Conservative Field
potential functior such that
(ii
n
) 0
v
C
C
v
v R
φφ=nabla
nablatimes
isin
isin
=r
r
r
r in (Irrotational field curl-free)
cf 0 st solenoidal field divergence-free
0
(iii) in (Conservative field)C
Rv w v w
v R Rd
nablasdot = exist nabla
sdot =
= times
int
r r r
r
r
r
MingsianMingsian R R BaiBai53
(iv) is connecting and
independent o
Ex Think
f the pa
of dynamics is the force
thB
A
v dR A B
v
sdotintrr
r
1 2
1
0
Pf
(i) (ii) ( ) 0 (zero identity cg rule) irrotational
ˆ(ii) (iii) ( ) 0 (Stokes theorem)
(iii) (iv) ( ) 0
C S
C C C
C
v
v dR n v dS
v dR v dR
v dR
φ
=
rarr nablatimes = nablatimes nabla =
rarr sdot = sdot nablatimes =
rarr sdot = minus sdot =
rArr sdot =
int int
int int int
int
r
rr
rr r
r rr r
rr r
2
independent of pathC
v dRsdotintr
A
B
1C
2C
MingsianMingsian R R BaiBai54
1 2
(iv) (i) constant ( ) ( )
--(1)
But (2)
(1) (2) QED
Note on application
Check
B
C C A
v dR v dR B A d
v dR d
d dx dy dz dRx y z
v dR dR v R
φ φ φ
φφ φ φφ φ
φ φ
rarr sdot = sdot = = minus =
rArr sdot =part part part
= + + = nabla sdot minus minus minus minus minuspart part part
sdot = nabla sdot rArr = nabla forall
int int intr rr r
rr
r
r r rr r
use simpler path (iv)if 0
find (i)v
φ⎧
nablatimes = ⎨⎩
rr
A
B
x
y
Cz
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai22
( )v w r w r r wnablatimes =nablatimes times = nablasdot minus nablasdotv v v v v v v ( )r w+ sdotnablav v ( ) constant w
w rminus sdotnablav
v v
Q
Ex Rigid body rotation
MingsianMingsian R R BaiBai23
( )( ) ( )
( ) ( )
ˆˆ ˆ
ˆ ˆˆ ˆ ˆ ˆ 1 1 1 3
ˆˆ ˆ
ˆˆ ˆ
Thus 3 2 is linked directly
x y z
x y z
r xi yj zk
r i j k xi yj zkx y z
w r w w w xi yj zkx y z
w i w j w k w
w r w r w r w w wv
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎝ ⎠
= + +
part part partnablasdot = + + + + = + + =part part part
part part partsdotnabla = + + + +part part part
= + + =
nablatimes times == nablasdot minus sdotnabla = minus =rArrnablatimes
v
v
v v
v
v v v v v v v v v
v spinn to ing
MingsianMingsian R R BaiBai24
Line surface and volume integrals
x
yC
B
A
isΔ
( )ii ηξ
Piecewise smooth curve
Line integral
MingsianMingsian R R BaiBai25
0 1lim ( ) ( ) or ( )
Intepretation of a line integral as an area ( )
i
Bn
i i isA
C
i C
F s F x y ds F x y d
x
s
F y ds
ξ ηΔ rarr =
Δ =sum int
int
int
x
y
z
C
( )z F x y=
MingsianMingsian R R BaiBai26
the area is always on the left of an obs
Positive dierver trave
rection of irsing on the
ntegrationcurve
RC
CCW CW
If is a regular closed curve piecewise smo
( ) ( )
(contou
oth simple ( )
r integral)
x y zC C
x y zC
F dR F F F dx dy dz
F dx F dy
C
F dz
sdot = + + + +
= + +
lowast
int int
int
i j k i j kv v
無交叉
work dW F dR= sdotv v
MingsianMingsian R R BaiBai27
0 1
Surface integral
lim ( ) ( )i
n
i i i iS i S
F S F x y z dSξ η ζΔ rarr =
Δ =sum intint( ) i i iξ η ζ
iSΔ
0 1
mass density
lim ( ) ( )
Ex
Volume integral
( )
i
n
i i i iV i V
V
F V F x y z dV
M x y z dV
ξ η ζ
ρ
Δ rarr =Δ =
=
sum intintint
intintint( ) i i iξ η ζ
iVΔ
MingsianMingsian R R BaiBai28
Ref (1)Hildebrand advanced Calculus for application Prentice-Hall 1976 p306-313(2) Greenberg p383-389 (3) Arfken ch2
1 2 3
1 2 3
1 2 3
1 2 3
(
Orthogonal coordinate systems base vectors are not constant in s
) ( )( )( )( )
ˆˆ ˆ
pacep x y z p u u ux x u u uy y u u uz z u u u
r xi yj zk
primerarr
= +
rarr
===
+v
1uv
2uv3uv
1u
2u
3uP coord curve
xy
z
O
X Curvilinear Coordinates
MingsianMingsian R R BaiBai29
1
11
1 1 1
1
1
1 1 1
11 1
1 1
2 2 2 3 3 3
Tangent vector to the at
where is the arc length
1
where unit vector
Similarly
curv
e
u psr rU
u s us
rs
U h uds rh udu sU h u U h u
partpart part= =part part part
part =part
there4 =part= =part
= =
v vv
vQ
v v
vv
v vv v
1u cood curve
1uv
1du 1dS
drv
P
MingsianMingsian R R BaiBai30
3 31 2 1 21 2 3
1 2 32
curv
In summary scale factor
If is the arc length along a in any directione
k k k
kk k
k k
k k k
U h u
ds rh Udu u
ds h dus
du dudu du du dudr r r r U U Uds u ds u ds u ds ds ds ds
dr drds ds
=
part= = =part
=
part part part= + + = + +part part part
= sdot
v v
v v
v v v v v v v
v v 3 3
1 1
23 3 32 2
21 1 1
2 2 2 2 2 2 2 2 2 21 1 2 2 3 3 1 2 3
1 (for orthogonal coord)
jii j
i j
ji ii j i
i j i
dududr U Uds ds ds
dudu duU U hds ds ds
ds h du h du h du ds ds ds
= =
= = =
⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
= sdot
rarr = sdot =
rarr = + + = + +
sum sum
sumsum sum
v v v
v v
MingsianMingsian R R BaiBai31
1 1 2 2 3 3
1 2 3
are mutually vectors having with arc length
k k k k kds h du U du
U du U du U duds ds ds
= =
rArr perp
v
v v v
1uv
2uv
3uv
1dS
2dS
3dS
dS
MingsianMingsian R R BaiBai32
( ) ( )
( ) ( )
1 1 2 2 3 3 1 2 3 1 2 3 1 2 3
1 2 3 1 2 3
1
1 2 2 3 3 2 2 2 3 3 3 2 3 2 3 1
Differential volume
Differential On const coord surface
area
Gradi t
en
d U du U du U du u u u h h h du du du
d h h h du du du
ud U du U du h u du h u du h h du du u
df
τ
τ
σ
= times sdot = times sdot
=
=
= times = times =
= nabla
v v v v v v
v vv v v v
1 2 31 2 3
1 2 3 1 1 2 2 3 31 2 3
1 1 1 2 2 2 3 3 3
f f ff dr df du du duu u u
r r rdr du du du U du U du U duu u u
h u du h u du h u du
part part partsdot rarr = + +part part part
part part part= + + = + +part part part
= + +
v
v v v v v vv
v v v
=1 (right hand rule)
MingsianMingsian R R BaiBai33
Rectangular Cartesian Coordinates
ij
k
MingsianMingsian R R BaiBai34
Circular Cylindrical Coordinatescossin
xyz z
ρ ϕρ ϕ
===
Ref Arfken
( )P zρ ϕ
MingsianMingsian R R BaiBai35
Polar Spherical Coordinatessin cossin sincos
x ry rz r
θ ϕθ ϕθ
===
Ref Arfken
0
0
0
sin cos sin sin coscos cos cos sin sin
sin cos
θ ϕ θ ϕ θθ ϕ θ ϕ θϕ ϕ
= + +
= + minus= minus +
r i j kθ i j kφ i j
2
2
sinsin
dS r d ddV r dr d d
θ θ ϕ
θ θ ϕ
=
=( )P r θ ϕ
MingsianMingsian R R BaiBai36
Integral TheoremsHeart of this chapter
Interchange of dimensionality Div Curl Grad in integrals High-dimensional integration
rarr
1
Thm1 Divergence theorem Let be a closed region bounded by piecewise smooth orientable surface and let (con
I Divergence theor
tinuous derivatives up to the first order
em (Gauss theorem)
)
VS v Cisin
r
ˆ If denotes the outward unit normal on
the ˆn V S
v dV n v dS
n S
nablasdot = sdotint intr r
MingsianMingsian R R BaiBai37
jS
jP
jV
S
0
ˆBy definition ( ) lim
ˆ( )
ˆ( )
j
j
j
j
Sj V
j
j j S
j j Sj j
n v dSv P
V
v P V n v dS
v P V n v dS
Δ rarr
⎧ ⎫sdot⎪ ⎪nabla sdot ⎨ ⎬Δ⎪ ⎪⎩ ⎭
nabla sdot Δ asymp sdot
nabla sdot Δ = sdot
int
intsum sumint
rr
r r
r r
Pf
MingsianMingsian R R BaiBai38
0
After interior surfaces cancel out only exterior surface (Why Opposite normal vectors)
ˆ( )
volumes exterior surfaces
ˆlim
QE
survive
j
j
j j Sj j
V SV
v p V n v dS
v dV n v dSΔ rarr
nabla sdot Δ = sdot
rArr nablasdot = sdot
sum sumint
int int
r r
r r
D
MingsianMingsian R R BaiBai39
( )
scalar variable constant vectorvector variable
Alternative formsLet be a an arbitrary and
a
ˆ
1 Le
(
ˆt
V
V S
S
V
v ap
v d
v dV vn d
V n v dS
v va
va dV v a v a
S
nablasdot = sdot
= rArr
nablasdot = nabla sdot + nabla
=
sdot
nablaint intint int
int
r
v
r r
r r
r r rQ ( )
( )
ˆ)
2 Let ˆV S
V
V S
dV n va dS
v a p
a p
p dV n p d
d p
S
V a
⎛ ⎞= sdot⎜ ⎟
⎝ ⎠
= times rArr
nablasdot times = sdot nabla
times
times
nabla = times
int int
int int
int
r
r r r
r r v
r
vQ
r
( ) ( ) ( )
( ) ( )
ˆ
ˆV S
V S
a p dV n a p dS
a p dV a n p dS
⎛ ⎞⎡ ⎤minus sdot nablatimes = sdot times⎣ ⎦⎜ ⎟⎜ ⎟⎜ ⎟rArr minus sdot nablatimes = minus sdot times⎜ ⎟⎝ ⎠
int int
int int
r rv v
r rv v
MingsianMingsian R R BaiBai40
( )
( )
From Leibnitzs rule
(1-D) ( )
( ) ( )
B t
A t
d f x t dxdt
f x t dx B ttpart prime= +part
int( )
( )( ( ) ) ( )
B t
A tf B t t A tprimeminusint ( ( ) )
(assume ( ) ( ) )
f A t t
A t B t const= =
ContinuityEx The Equation
rate of increase of mass
mass densityV
dM d dVdt dt
ρ
ρ
= int
flow velocityvv
MingsianMingsian R R BaiBai41
ˆ ˆ ( )
(
Conservation
) ( Divergence theorem)
of mass
( ) 0
V S S
V
V
dV n v dS n v dSt
v dV
v dVt
ρ ρ ρ
ρ
ρ ρ⎡ ⎤⎢ ⎥⎣ ⎦
part = minus sdot = minus sdotpart
= minus nablasdot
partrArr +nablasdot =part
int int int
int
int
v v
vQ
v
ˆOn the other hand rate influof mass through n ox i t
Extended to 3-D for time-independent fixed boundary
(3-D) ( ) ( )
Thus
S
V V
V
S V n v dS
Sd f x y z t dV f x y z t dVdt tdM dVdt t
ρ
ρ
= minus sdot
part=part
part=part
int
int int
intv
Fixed control volume (Eulerian coordinates)Otherwise Raynoldrsquos transport theoremMoving control volume Lagrangian coordinates
MingsianMingsian R R BaiBai42
But the control volime is only arbitrary
In particular for flow
( ) 0
con
(continuity equati
stant
on)
Divergence the
0 incompress
Note is t
ible
orem h
V
v
vt
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
e mother of the other integral theorems
MingsianMingsian R R BaiBai43
Ex Archimedes principle ( ) The buoyant force ( ) on a body with volume immersed in a fluid is equal to the weight of the fluid
ˆ displaced by the object ie In the followin
B
V
gVρ= minusf k
阿基米德原理
浮力
ˆg figure the ˆ with and being the fluid density gravitational
acceleration depth and outward normal unit vector to the surface
fhydraulic pressure g z
g z
S
ρ
ρ
= minusp n
n
1
the modified divergence theorem
ˆ
Hint
scalar fuctionV S
vdV v dS v Cnabla = isinint int n
MingsianMingsian R R BaiBai44
Levelsurfacefluid
V
S
airyx
z
ˆf g zρ= minusp n
ˆPf ( ) ( )
ˆ ˆ ˆ ˆ ˆ (0 0 ) v v
B fS S V
V V
g z g zdS dS dV
g dV g dV gV
ρ ρ
ρ ρ ρ
= = minus = minus nabla
= minus + + = minus = minus
int int int
int int
f p n
i j k k k
k
MingsianMingsian R R BaiBai45
Note This can also be thought of as simple balance of forces (resultant force that acts on and weight of filled with fluid)S V
2
~ a little history about George Green Recall the diveregence theorem
ˆ
Let
II th
e Green
are arbitrary fun
s identiti
ctions
es
V S
v dV n v dS
v u vu v C
nablasdot = sdot
= nablaisin
int intv v
v
ˆ ( ) ( )
ˆ ( )V S
V S
u v dV n u v dS
u v u v dV un v dS
nablasdot nabla = sdot nabla
nabla sdotnabla + nablasdotnabla = sdotnabla
int int
int int
MingsianMingsian R R BaiBai46
2 -( ) -(1)V S
vu v u v dV u dSnpart
nabla sdotnabla + nabla =partint int firs(Greens identt ity)
2
If
( ) --(2)
(1) (2) V S
u vuv u v u dV v dSn
harrpart
nabla sdotnabla + nabla =part
minus
int int
( )2 2
V S
dv duu v v u dV u v dSdn dn
⎛ ⎞nabla minus nabla = minus⎜ ⎟⎝ ⎠int int
second(Greens identity)
MingsianMingsian R R BaiBai47
( )
( )
( ) ( )
1st id ( )
Greens identities in the 1D case integration by parts
b b bb ba a
a a ab
ba
ab
ba
a
v udx v u v u dx u v uv dx uv
u v u v vu dx vu
uv vu dx uv vu
primeprime prime prime prime prime prime primeprime prime= minus rArr + =
prime prime primeprime primeharr + =
primeprime primeprime prime primerArr minus = minus
int int int
int
2
2nd id ( )
Good for any adjoint operators in addition to Good also for and integration by parts Important for theory amp reciprocity
From th
Gree
e 1s
2-
t
ns functi
identity
D 1-
tak 1
D
e
on
u
nablararr
=
int
2
V S
vv dV dSnpartnabla =part
rArr int int2Ex 0 st 0 on the boundary 0 ( 1st id u u unabla = = rArr equiv 以 証)
MingsianMingsian R R BaiBai48
1
Thm1 Stokes theorem Let be defined in Let be a piecewise-smooth orientable surface bounded by a piec
III Stokes theorem and
ewise-smooth simple clos
Greens
ed curve Then
theore
m
v C R S
C
isinv
( )
ˆwhere is a unit normal to according t right hand rulo the e
ˆS C
n S
n v dS v dRsdot nablatimes = sdotint intvv v
(not necessarily planar)S
Cn vnablatimes v vv
MingsianMingsian R R BaiBai49
Pf
( )( )
circulatioWe have shown in deriving thatthe in -direction
ˆor
ˆ
n
z
z
j j j j j
j j j j jj j
vz v dx dy
v dR n v S
v dR n v S
nablatimes
=nablatimes
sdot = sdot nablatimes Δ
sdot = sdot nablatimes Δ
intsum sumint
v
v
vv v
vv v
C
S
jS
jC
jP
ˆ jn
jSΔ
MingsianMingsian R R BaiBai50
( )
( )ext surface
After the interior boundaries cancel out only the exterior boundaries (Why)
ˆ
ˆ0 QED
Alternative forms scalar
survive
va
j
j j j j jC
jC S
v dR n v S
S v dR n v dS
φ
sdot = sdot nablatimes Δ
Δ rarr rArr sdot = sdot nablatimes
sum sumint
int int
vv v
vv v
( )
riable constant vector vector variable
1 Let
2 Let
ˆ
ˆ S C
S C
a p
v n dR dR
n p dS dR p
a
v a p
φ φφ= rArr timesnabla =
timesnabla times= rArr timestimes =
int int
int int
v v
v v
v
v v
vv vv v
MingsianMingsian R R BaiBai51
( ) ( )
( )
( )
( ) ( )
1
Thm2 Greens theorem ( planar version of Stokes theorem)ˆ ˆ Let
pf By Stokes theoremˆ
ˆ ˆ ˆ ˆ ˆ ˆ
S C
S C
S C
v P x y i Q x y j C
n v dS v dR
k Pi Qj dS Pi Qj dx i dy
Q P dS Pdx Qdyx y
⎛ ⎞part partminus = +⎜ ⎟
= + isin
sdot nablatimes = sdot
⎡ ⎤sdot nablatimes + =
part part⎝ ⎠
+ sdot +⎣ ⎦
int int
int
int
int
int
v
vv v
( )
( )
( )
ˆ
ˆˆ ˆ
ˆˆ ˆ ˆ ˆ( ) ( )
0
QEDS C
j
i j kQ PPi Qj i j k
x y z x yP Q
Q P dS Pdx Qdyx y
⎛ ⎞part part part part partnablatimes + = = + + minus⎜ ⎟part part part part part⎝ ⎠
⎛ ⎞part partrArr minus = +⎜ ⎟part part⎝ ⎠int int
L L
S
C
i
j
x
y
MingsianMingsian R R BaiBai52
Potential TheorypotentiGeorge Green coined the term (l
a )
rarr 位能 勢能
例子 重力場 電磁場 流場 聲場 彈性力學
1
2
Thm1 If is in a simply connected region then the following statements are equivalent(i) There exists a scala
Conservative Field
potential functior such that
(ii
n
) 0
v
C
C
v
v R
φφ=nabla
nablatimes
isin
isin
=r
r
r
r in (Irrotational field curl-free)
cf 0 st solenoidal field divergence-free
0
(iii) in (Conservative field)C
Rv w v w
v R Rd
nablasdot = exist nabla
sdot =
= times
int
r r r
r
r
r
MingsianMingsian R R BaiBai53
(iv) is connecting and
independent o
Ex Think
f the pa
of dynamics is the force
thB
A
v dR A B
v
sdotintrr
r
1 2
1
0
Pf
(i) (ii) ( ) 0 (zero identity cg rule) irrotational
ˆ(ii) (iii) ( ) 0 (Stokes theorem)
(iii) (iv) ( ) 0
C S
C C C
C
v
v dR n v dS
v dR v dR
v dR
φ
=
rarr nablatimes = nablatimes nabla =
rarr sdot = sdot nablatimes =
rarr sdot = minus sdot =
rArr sdot =
int int
int int int
int
r
rr
rr r
r rr r
rr r
2
independent of pathC
v dRsdotintr
A
B
1C
2C
MingsianMingsian R R BaiBai54
1 2
(iv) (i) constant ( ) ( )
--(1)
But (2)
(1) (2) QED
Note on application
Check
B
C C A
v dR v dR B A d
v dR d
d dx dy dz dRx y z
v dR dR v R
φ φ φ
φφ φ φφ φ
φ φ
rarr sdot = sdot = = minus =
rArr sdot =part part part
= + + = nabla sdot minus minus minus minus minuspart part part
sdot = nabla sdot rArr = nabla forall
int int intr rr r
rr
r
r r rr r
use simpler path (iv)if 0
find (i)v
φ⎧
nablatimes = ⎨⎩
rr
A
B
x
y
Cz
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai23
( )( ) ( )
( ) ( )
ˆˆ ˆ
ˆ ˆˆ ˆ ˆ ˆ 1 1 1 3
ˆˆ ˆ
ˆˆ ˆ
Thus 3 2 is linked directly
x y z
x y z
r xi yj zk
r i j k xi yj zkx y z
w r w w w xi yj zkx y z
w i w j w k w
w r w r w r w w wv
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎝ ⎠
= + +
part part partnablasdot = + + + + = + + =part part part
part part partsdotnabla = + + + +part part part
= + + =
nablatimes times == nablasdot minus sdotnabla = minus =rArrnablatimes
v
v
v v
v
v v v v v v v v v
v spinn to ing
MingsianMingsian R R BaiBai24
Line surface and volume integrals
x
yC
B
A
isΔ
( )ii ηξ
Piecewise smooth curve
Line integral
MingsianMingsian R R BaiBai25
0 1lim ( ) ( ) or ( )
Intepretation of a line integral as an area ( )
i
Bn
i i isA
C
i C
F s F x y ds F x y d
x
s
F y ds
ξ ηΔ rarr =
Δ =sum int
int
int
x
y
z
C
( )z F x y=
MingsianMingsian R R BaiBai26
the area is always on the left of an obs
Positive dierver trave
rection of irsing on the
ntegrationcurve
RC
CCW CW
If is a regular closed curve piecewise smo
( ) ( )
(contou
oth simple ( )
r integral)
x y zC C
x y zC
F dR F F F dx dy dz
F dx F dy
C
F dz
sdot = + + + +
= + +
lowast
int int
int
i j k i j kv v
無交叉
work dW F dR= sdotv v
MingsianMingsian R R BaiBai27
0 1
Surface integral
lim ( ) ( )i
n
i i i iS i S
F S F x y z dSξ η ζΔ rarr =
Δ =sum intint( ) i i iξ η ζ
iSΔ
0 1
mass density
lim ( ) ( )
Ex
Volume integral
( )
i
n
i i i iV i V
V
F V F x y z dV
M x y z dV
ξ η ζ
ρ
Δ rarr =Δ =
=
sum intintint
intintint( ) i i iξ η ζ
iVΔ
MingsianMingsian R R BaiBai28
Ref (1)Hildebrand advanced Calculus for application Prentice-Hall 1976 p306-313(2) Greenberg p383-389 (3) Arfken ch2
1 2 3
1 2 3
1 2 3
1 2 3
(
Orthogonal coordinate systems base vectors are not constant in s
) ( )( )( )( )
ˆˆ ˆ
pacep x y z p u u ux x u u uy y u u uz z u u u
r xi yj zk
primerarr
= +
rarr
===
+v
1uv
2uv3uv
1u
2u
3uP coord curve
xy
z
O
X Curvilinear Coordinates
MingsianMingsian R R BaiBai29
1
11
1 1 1
1
1
1 1 1
11 1
1 1
2 2 2 3 3 3
Tangent vector to the at
where is the arc length
1
where unit vector
Similarly
curv
e
u psr rU
u s us
rs
U h uds rh udu sU h u U h u
partpart part= =part part part
part =part
there4 =part= =part
= =
v vv
vQ
v v
vv
v vv v
1u cood curve
1uv
1du 1dS
drv
P
MingsianMingsian R R BaiBai30
3 31 2 1 21 2 3
1 2 32
curv
In summary scale factor
If is the arc length along a in any directione
k k k
kk k
k k
k k k
U h u
ds rh Udu u
ds h dus
du dudu du du dudr r r r U U Uds u ds u ds u ds ds ds ds
dr drds ds
=
part= = =part
=
part part part= + + = + +part part part
= sdot
v v
v v
v v v v v v v
v v 3 3
1 1
23 3 32 2
21 1 1
2 2 2 2 2 2 2 2 2 21 1 2 2 3 3 1 2 3
1 (for orthogonal coord)
jii j
i j
ji ii j i
i j i
dududr U Uds ds ds
dudu duU U hds ds ds
ds h du h du h du ds ds ds
= =
= = =
⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
= sdot
rarr = sdot =
rarr = + + = + +
sum sum
sumsum sum
v v v
v v
MingsianMingsian R R BaiBai31
1 1 2 2 3 3
1 2 3
are mutually vectors having with arc length
k k k k kds h du U du
U du U du U duds ds ds
= =
rArr perp
v
v v v
1uv
2uv
3uv
1dS
2dS
3dS
dS
MingsianMingsian R R BaiBai32
( ) ( )
( ) ( )
1 1 2 2 3 3 1 2 3 1 2 3 1 2 3
1 2 3 1 2 3
1
1 2 2 3 3 2 2 2 3 3 3 2 3 2 3 1
Differential volume
Differential On const coord surface
area
Gradi t
en
d U du U du U du u u u h h h du du du
d h h h du du du
ud U du U du h u du h u du h h du du u
df
τ
τ
σ
= times sdot = times sdot
=
=
= times = times =
= nabla
v v v v v v
v vv v v v
1 2 31 2 3
1 2 3 1 1 2 2 3 31 2 3
1 1 1 2 2 2 3 3 3
f f ff dr df du du duu u u
r r rdr du du du U du U du U duu u u
h u du h u du h u du
part part partsdot rarr = + +part part part
part part part= + + = + +part part part
= + +
v
v v v v v vv
v v v
=1 (right hand rule)
MingsianMingsian R R BaiBai33
Rectangular Cartesian Coordinates
ij
k
MingsianMingsian R R BaiBai34
Circular Cylindrical Coordinatescossin
xyz z
ρ ϕρ ϕ
===
Ref Arfken
( )P zρ ϕ
MingsianMingsian R R BaiBai35
Polar Spherical Coordinatessin cossin sincos
x ry rz r
θ ϕθ ϕθ
===
Ref Arfken
0
0
0
sin cos sin sin coscos cos cos sin sin
sin cos
θ ϕ θ ϕ θθ ϕ θ ϕ θϕ ϕ
= + +
= + minus= minus +
r i j kθ i j kφ i j
2
2
sinsin
dS r d ddV r dr d d
θ θ ϕ
θ θ ϕ
=
=( )P r θ ϕ
MingsianMingsian R R BaiBai36
Integral TheoremsHeart of this chapter
Interchange of dimensionality Div Curl Grad in integrals High-dimensional integration
rarr
1
Thm1 Divergence theorem Let be a closed region bounded by piecewise smooth orientable surface and let (con
I Divergence theor
tinuous derivatives up to the first order
em (Gauss theorem)
)
VS v Cisin
r
ˆ If denotes the outward unit normal on
the ˆn V S
v dV n v dS
n S
nablasdot = sdotint intr r
MingsianMingsian R R BaiBai37
jS
jP
jV
S
0
ˆBy definition ( ) lim
ˆ( )
ˆ( )
j
j
j
j
Sj V
j
j j S
j j Sj j
n v dSv P
V
v P V n v dS
v P V n v dS
Δ rarr
⎧ ⎫sdot⎪ ⎪nabla sdot ⎨ ⎬Δ⎪ ⎪⎩ ⎭
nabla sdot Δ asymp sdot
nabla sdot Δ = sdot
int
intsum sumint
rr
r r
r r
Pf
MingsianMingsian R R BaiBai38
0
After interior surfaces cancel out only exterior surface (Why Opposite normal vectors)
ˆ( )
volumes exterior surfaces
ˆlim
QE
survive
j
j
j j Sj j
V SV
v p V n v dS
v dV n v dSΔ rarr
nabla sdot Δ = sdot
rArr nablasdot = sdot
sum sumint
int int
r r
r r
D
MingsianMingsian R R BaiBai39
( )
scalar variable constant vectorvector variable
Alternative formsLet be a an arbitrary and
a
ˆ
1 Le
(
ˆt
V
V S
S
V
v ap
v d
v dV vn d
V n v dS
v va
va dV v a v a
S
nablasdot = sdot
= rArr
nablasdot = nabla sdot + nabla
=
sdot
nablaint intint int
int
r
v
r r
r r
r r rQ ( )
( )
ˆ)
2 Let ˆV S
V
V S
dV n va dS
v a p
a p
p dV n p d
d p
S
V a
⎛ ⎞= sdot⎜ ⎟
⎝ ⎠
= times rArr
nablasdot times = sdot nabla
times
times
nabla = times
int int
int int
int
r
r r r
r r v
r
vQ
r
( ) ( ) ( )
( ) ( )
ˆ
ˆV S
V S
a p dV n a p dS
a p dV a n p dS
⎛ ⎞⎡ ⎤minus sdot nablatimes = sdot times⎣ ⎦⎜ ⎟⎜ ⎟⎜ ⎟rArr minus sdot nablatimes = minus sdot times⎜ ⎟⎝ ⎠
int int
int int
r rv v
r rv v
MingsianMingsian R R BaiBai40
( )
( )
From Leibnitzs rule
(1-D) ( )
( ) ( )
B t
A t
d f x t dxdt
f x t dx B ttpart prime= +part
int( )
( )( ( ) ) ( )
B t
A tf B t t A tprimeminusint ( ( ) )
(assume ( ) ( ) )
f A t t
A t B t const= =
ContinuityEx The Equation
rate of increase of mass
mass densityV
dM d dVdt dt
ρ
ρ
= int
flow velocityvv
MingsianMingsian R R BaiBai41
ˆ ˆ ( )
(
Conservation
) ( Divergence theorem)
of mass
( ) 0
V S S
V
V
dV n v dS n v dSt
v dV
v dVt
ρ ρ ρ
ρ
ρ ρ⎡ ⎤⎢ ⎥⎣ ⎦
part = minus sdot = minus sdotpart
= minus nablasdot
partrArr +nablasdot =part
int int int
int
int
v v
vQ
v
ˆOn the other hand rate influof mass through n ox i t
Extended to 3-D for time-independent fixed boundary
(3-D) ( ) ( )
Thus
S
V V
V
S V n v dS
Sd f x y z t dV f x y z t dVdt tdM dVdt t
ρ
ρ
= minus sdot
part=part
part=part
int
int int
intv
Fixed control volume (Eulerian coordinates)Otherwise Raynoldrsquos transport theoremMoving control volume Lagrangian coordinates
MingsianMingsian R R BaiBai42
But the control volime is only arbitrary
In particular for flow
( ) 0
con
(continuity equati
stant
on)
Divergence the
0 incompress
Note is t
ible
orem h
V
v
vt
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
e mother of the other integral theorems
MingsianMingsian R R BaiBai43
Ex Archimedes principle ( ) The buoyant force ( ) on a body with volume immersed in a fluid is equal to the weight of the fluid
ˆ displaced by the object ie In the followin
B
V
gVρ= minusf k
阿基米德原理
浮力
ˆg figure the ˆ with and being the fluid density gravitational
acceleration depth and outward normal unit vector to the surface
fhydraulic pressure g z
g z
S
ρ
ρ
= minusp n
n
1
the modified divergence theorem
ˆ
Hint
scalar fuctionV S
vdV v dS v Cnabla = isinint int n
MingsianMingsian R R BaiBai44
Levelsurfacefluid
V
S
airyx
z
ˆf g zρ= minusp n
ˆPf ( ) ( )
ˆ ˆ ˆ ˆ ˆ (0 0 ) v v
B fS S V
V V
g z g zdS dS dV
g dV g dV gV
ρ ρ
ρ ρ ρ
= = minus = minus nabla
= minus + + = minus = minus
int int int
int int
f p n
i j k k k
k
MingsianMingsian R R BaiBai45
Note This can also be thought of as simple balance of forces (resultant force that acts on and weight of filled with fluid)S V
2
~ a little history about George Green Recall the diveregence theorem
ˆ
Let
II th
e Green
are arbitrary fun
s identiti
ctions
es
V S
v dV n v dS
v u vu v C
nablasdot = sdot
= nablaisin
int intv v
v
ˆ ( ) ( )
ˆ ( )V S
V S
u v dV n u v dS
u v u v dV un v dS
nablasdot nabla = sdot nabla
nabla sdotnabla + nablasdotnabla = sdotnabla
int int
int int
MingsianMingsian R R BaiBai46
2 -( ) -(1)V S
vu v u v dV u dSnpart
nabla sdotnabla + nabla =partint int firs(Greens identt ity)
2
If
( ) --(2)
(1) (2) V S
u vuv u v u dV v dSn
harrpart
nabla sdotnabla + nabla =part
minus
int int
( )2 2
V S
dv duu v v u dV u v dSdn dn
⎛ ⎞nabla minus nabla = minus⎜ ⎟⎝ ⎠int int
second(Greens identity)
MingsianMingsian R R BaiBai47
( )
( )
( ) ( )
1st id ( )
Greens identities in the 1D case integration by parts
b b bb ba a
a a ab
ba
ab
ba
a
v udx v u v u dx u v uv dx uv
u v u v vu dx vu
uv vu dx uv vu
primeprime prime prime prime prime prime primeprime prime= minus rArr + =
prime prime primeprime primeharr + =
primeprime primeprime prime primerArr minus = minus
int int int
int
2
2nd id ( )
Good for any adjoint operators in addition to Good also for and integration by parts Important for theory amp reciprocity
From th
Gree
e 1s
2-
t
ns functi
identity
D 1-
tak 1
D
e
on
u
nablararr
=
int
2
V S
vv dV dSnpartnabla =part
rArr int int2Ex 0 st 0 on the boundary 0 ( 1st id u u unabla = = rArr equiv 以 証)
MingsianMingsian R R BaiBai48
1
Thm1 Stokes theorem Let be defined in Let be a piecewise-smooth orientable surface bounded by a piec
III Stokes theorem and
ewise-smooth simple clos
Greens
ed curve Then
theore
m
v C R S
C
isinv
( )
ˆwhere is a unit normal to according t right hand rulo the e
ˆS C
n S
n v dS v dRsdot nablatimes = sdotint intvv v
(not necessarily planar)S
Cn vnablatimes v vv
MingsianMingsian R R BaiBai49
Pf
( )( )
circulatioWe have shown in deriving thatthe in -direction
ˆor
ˆ
n
z
z
j j j j j
j j j j jj j
vz v dx dy
v dR n v S
v dR n v S
nablatimes
=nablatimes
sdot = sdot nablatimes Δ
sdot = sdot nablatimes Δ
intsum sumint
v
v
vv v
vv v
C
S
jS
jC
jP
ˆ jn
jSΔ
MingsianMingsian R R BaiBai50
( )
( )ext surface
After the interior boundaries cancel out only the exterior boundaries (Why)
ˆ
ˆ0 QED
Alternative forms scalar
survive
va
j
j j j j jC
jC S
v dR n v S
S v dR n v dS
φ
sdot = sdot nablatimes Δ
Δ rarr rArr sdot = sdot nablatimes
sum sumint
int int
vv v
vv v
( )
riable constant vector vector variable
1 Let
2 Let
ˆ
ˆ S C
S C
a p
v n dR dR
n p dS dR p
a
v a p
φ φφ= rArr timesnabla =
timesnabla times= rArr timestimes =
int int
int int
v v
v v
v
v v
vv vv v
MingsianMingsian R R BaiBai51
( ) ( )
( )
( )
( ) ( )
1
Thm2 Greens theorem ( planar version of Stokes theorem)ˆ ˆ Let
pf By Stokes theoremˆ
ˆ ˆ ˆ ˆ ˆ ˆ
S C
S C
S C
v P x y i Q x y j C
n v dS v dR
k Pi Qj dS Pi Qj dx i dy
Q P dS Pdx Qdyx y
⎛ ⎞part partminus = +⎜ ⎟
= + isin
sdot nablatimes = sdot
⎡ ⎤sdot nablatimes + =
part part⎝ ⎠
+ sdot +⎣ ⎦
int int
int
int
int
int
v
vv v
( )
( )
( )
ˆ
ˆˆ ˆ
ˆˆ ˆ ˆ ˆ( ) ( )
0
QEDS C
j
i j kQ PPi Qj i j k
x y z x yP Q
Q P dS Pdx Qdyx y
⎛ ⎞part part part part partnablatimes + = = + + minus⎜ ⎟part part part part part⎝ ⎠
⎛ ⎞part partrArr minus = +⎜ ⎟part part⎝ ⎠int int
L L
S
C
i
j
x
y
MingsianMingsian R R BaiBai52
Potential TheorypotentiGeorge Green coined the term (l
a )
rarr 位能 勢能
例子 重力場 電磁場 流場 聲場 彈性力學
1
2
Thm1 If is in a simply connected region then the following statements are equivalent(i) There exists a scala
Conservative Field
potential functior such that
(ii
n
) 0
v
C
C
v
v R
φφ=nabla
nablatimes
isin
isin
=r
r
r
r in (Irrotational field curl-free)
cf 0 st solenoidal field divergence-free
0
(iii) in (Conservative field)C
Rv w v w
v R Rd
nablasdot = exist nabla
sdot =
= times
int
r r r
r
r
r
MingsianMingsian R R BaiBai53
(iv) is connecting and
independent o
Ex Think
f the pa
of dynamics is the force
thB
A
v dR A B
v
sdotintrr
r
1 2
1
0
Pf
(i) (ii) ( ) 0 (zero identity cg rule) irrotational
ˆ(ii) (iii) ( ) 0 (Stokes theorem)
(iii) (iv) ( ) 0
C S
C C C
C
v
v dR n v dS
v dR v dR
v dR
φ
=
rarr nablatimes = nablatimes nabla =
rarr sdot = sdot nablatimes =
rarr sdot = minus sdot =
rArr sdot =
int int
int int int
int
r
rr
rr r
r rr r
rr r
2
independent of pathC
v dRsdotintr
A
B
1C
2C
MingsianMingsian R R BaiBai54
1 2
(iv) (i) constant ( ) ( )
--(1)
But (2)
(1) (2) QED
Note on application
Check
B
C C A
v dR v dR B A d
v dR d
d dx dy dz dRx y z
v dR dR v R
φ φ φ
φφ φ φφ φ
φ φ
rarr sdot = sdot = = minus =
rArr sdot =part part part
= + + = nabla sdot minus minus minus minus minuspart part part
sdot = nabla sdot rArr = nabla forall
int int intr rr r
rr
r
r r rr r
use simpler path (iv)if 0
find (i)v
φ⎧
nablatimes = ⎨⎩
rr
A
B
x
y
Cz
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai24
Line surface and volume integrals
x
yC
B
A
isΔ
( )ii ηξ
Piecewise smooth curve
Line integral
MingsianMingsian R R BaiBai25
0 1lim ( ) ( ) or ( )
Intepretation of a line integral as an area ( )
i
Bn
i i isA
C
i C
F s F x y ds F x y d
x
s
F y ds
ξ ηΔ rarr =
Δ =sum int
int
int
x
y
z
C
( )z F x y=
MingsianMingsian R R BaiBai26
the area is always on the left of an obs
Positive dierver trave
rection of irsing on the
ntegrationcurve
RC
CCW CW
If is a regular closed curve piecewise smo
( ) ( )
(contou
oth simple ( )
r integral)
x y zC C
x y zC
F dR F F F dx dy dz
F dx F dy
C
F dz
sdot = + + + +
= + +
lowast
int int
int
i j k i j kv v
無交叉
work dW F dR= sdotv v
MingsianMingsian R R BaiBai27
0 1
Surface integral
lim ( ) ( )i
n
i i i iS i S
F S F x y z dSξ η ζΔ rarr =
Δ =sum intint( ) i i iξ η ζ
iSΔ
0 1
mass density
lim ( ) ( )
Ex
Volume integral
( )
i
n
i i i iV i V
V
F V F x y z dV
M x y z dV
ξ η ζ
ρ
Δ rarr =Δ =
=
sum intintint
intintint( ) i i iξ η ζ
iVΔ
MingsianMingsian R R BaiBai28
Ref (1)Hildebrand advanced Calculus for application Prentice-Hall 1976 p306-313(2) Greenberg p383-389 (3) Arfken ch2
1 2 3
1 2 3
1 2 3
1 2 3
(
Orthogonal coordinate systems base vectors are not constant in s
) ( )( )( )( )
ˆˆ ˆ
pacep x y z p u u ux x u u uy y u u uz z u u u
r xi yj zk
primerarr
= +
rarr
===
+v
1uv
2uv3uv
1u
2u
3uP coord curve
xy
z
O
X Curvilinear Coordinates
MingsianMingsian R R BaiBai29
1
11
1 1 1
1
1
1 1 1
11 1
1 1
2 2 2 3 3 3
Tangent vector to the at
where is the arc length
1
where unit vector
Similarly
curv
e
u psr rU
u s us
rs
U h uds rh udu sU h u U h u
partpart part= =part part part
part =part
there4 =part= =part
= =
v vv
vQ
v v
vv
v vv v
1u cood curve
1uv
1du 1dS
drv
P
MingsianMingsian R R BaiBai30
3 31 2 1 21 2 3
1 2 32
curv
In summary scale factor
If is the arc length along a in any directione
k k k
kk k
k k
k k k
U h u
ds rh Udu u
ds h dus
du dudu du du dudr r r r U U Uds u ds u ds u ds ds ds ds
dr drds ds
=
part= = =part
=
part part part= + + = + +part part part
= sdot
v v
v v
v v v v v v v
v v 3 3
1 1
23 3 32 2
21 1 1
2 2 2 2 2 2 2 2 2 21 1 2 2 3 3 1 2 3
1 (for orthogonal coord)
jii j
i j
ji ii j i
i j i
dududr U Uds ds ds
dudu duU U hds ds ds
ds h du h du h du ds ds ds
= =
= = =
⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
= sdot
rarr = sdot =
rarr = + + = + +
sum sum
sumsum sum
v v v
v v
MingsianMingsian R R BaiBai31
1 1 2 2 3 3
1 2 3
are mutually vectors having with arc length
k k k k kds h du U du
U du U du U duds ds ds
= =
rArr perp
v
v v v
1uv
2uv
3uv
1dS
2dS
3dS
dS
MingsianMingsian R R BaiBai32
( ) ( )
( ) ( )
1 1 2 2 3 3 1 2 3 1 2 3 1 2 3
1 2 3 1 2 3
1
1 2 2 3 3 2 2 2 3 3 3 2 3 2 3 1
Differential volume
Differential On const coord surface
area
Gradi t
en
d U du U du U du u u u h h h du du du
d h h h du du du
ud U du U du h u du h u du h h du du u
df
τ
τ
σ
= times sdot = times sdot
=
=
= times = times =
= nabla
v v v v v v
v vv v v v
1 2 31 2 3
1 2 3 1 1 2 2 3 31 2 3
1 1 1 2 2 2 3 3 3
f f ff dr df du du duu u u
r r rdr du du du U du U du U duu u u
h u du h u du h u du
part part partsdot rarr = + +part part part
part part part= + + = + +part part part
= + +
v
v v v v v vv
v v v
=1 (right hand rule)
MingsianMingsian R R BaiBai33
Rectangular Cartesian Coordinates
ij
k
MingsianMingsian R R BaiBai34
Circular Cylindrical Coordinatescossin
xyz z
ρ ϕρ ϕ
===
Ref Arfken
( )P zρ ϕ
MingsianMingsian R R BaiBai35
Polar Spherical Coordinatessin cossin sincos
x ry rz r
θ ϕθ ϕθ
===
Ref Arfken
0
0
0
sin cos sin sin coscos cos cos sin sin
sin cos
θ ϕ θ ϕ θθ ϕ θ ϕ θϕ ϕ
= + +
= + minus= minus +
r i j kθ i j kφ i j
2
2
sinsin
dS r d ddV r dr d d
θ θ ϕ
θ θ ϕ
=
=( )P r θ ϕ
MingsianMingsian R R BaiBai36
Integral TheoremsHeart of this chapter
Interchange of dimensionality Div Curl Grad in integrals High-dimensional integration
rarr
1
Thm1 Divergence theorem Let be a closed region bounded by piecewise smooth orientable surface and let (con
I Divergence theor
tinuous derivatives up to the first order
em (Gauss theorem)
)
VS v Cisin
r
ˆ If denotes the outward unit normal on
the ˆn V S
v dV n v dS
n S
nablasdot = sdotint intr r
MingsianMingsian R R BaiBai37
jS
jP
jV
S
0
ˆBy definition ( ) lim
ˆ( )
ˆ( )
j
j
j
j
Sj V
j
j j S
j j Sj j
n v dSv P
V
v P V n v dS
v P V n v dS
Δ rarr
⎧ ⎫sdot⎪ ⎪nabla sdot ⎨ ⎬Δ⎪ ⎪⎩ ⎭
nabla sdot Δ asymp sdot
nabla sdot Δ = sdot
int
intsum sumint
rr
r r
r r
Pf
MingsianMingsian R R BaiBai38
0
After interior surfaces cancel out only exterior surface (Why Opposite normal vectors)
ˆ( )
volumes exterior surfaces
ˆlim
QE
survive
j
j
j j Sj j
V SV
v p V n v dS
v dV n v dSΔ rarr
nabla sdot Δ = sdot
rArr nablasdot = sdot
sum sumint
int int
r r
r r
D
MingsianMingsian R R BaiBai39
( )
scalar variable constant vectorvector variable
Alternative formsLet be a an arbitrary and
a
ˆ
1 Le
(
ˆt
V
V S
S
V
v ap
v d
v dV vn d
V n v dS
v va
va dV v a v a
S
nablasdot = sdot
= rArr
nablasdot = nabla sdot + nabla
=
sdot
nablaint intint int
int
r
v
r r
r r
r r rQ ( )
( )
ˆ)
2 Let ˆV S
V
V S
dV n va dS
v a p
a p
p dV n p d
d p
S
V a
⎛ ⎞= sdot⎜ ⎟
⎝ ⎠
= times rArr
nablasdot times = sdot nabla
times
times
nabla = times
int int
int int
int
r
r r r
r r v
r
vQ
r
( ) ( ) ( )
( ) ( )
ˆ
ˆV S
V S
a p dV n a p dS
a p dV a n p dS
⎛ ⎞⎡ ⎤minus sdot nablatimes = sdot times⎣ ⎦⎜ ⎟⎜ ⎟⎜ ⎟rArr minus sdot nablatimes = minus sdot times⎜ ⎟⎝ ⎠
int int
int int
r rv v
r rv v
MingsianMingsian R R BaiBai40
( )
( )
From Leibnitzs rule
(1-D) ( )
( ) ( )
B t
A t
d f x t dxdt
f x t dx B ttpart prime= +part
int( )
( )( ( ) ) ( )
B t
A tf B t t A tprimeminusint ( ( ) )
(assume ( ) ( ) )
f A t t
A t B t const= =
ContinuityEx The Equation
rate of increase of mass
mass densityV
dM d dVdt dt
ρ
ρ
= int
flow velocityvv
MingsianMingsian R R BaiBai41
ˆ ˆ ( )
(
Conservation
) ( Divergence theorem)
of mass
( ) 0
V S S
V
V
dV n v dS n v dSt
v dV
v dVt
ρ ρ ρ
ρ
ρ ρ⎡ ⎤⎢ ⎥⎣ ⎦
part = minus sdot = minus sdotpart
= minus nablasdot
partrArr +nablasdot =part
int int int
int
int
v v
vQ
v
ˆOn the other hand rate influof mass through n ox i t
Extended to 3-D for time-independent fixed boundary
(3-D) ( ) ( )
Thus
S
V V
V
S V n v dS
Sd f x y z t dV f x y z t dVdt tdM dVdt t
ρ
ρ
= minus sdot
part=part
part=part
int
int int
intv
Fixed control volume (Eulerian coordinates)Otherwise Raynoldrsquos transport theoremMoving control volume Lagrangian coordinates
MingsianMingsian R R BaiBai42
But the control volime is only arbitrary
In particular for flow
( ) 0
con
(continuity equati
stant
on)
Divergence the
0 incompress
Note is t
ible
orem h
V
v
vt
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
e mother of the other integral theorems
MingsianMingsian R R BaiBai43
Ex Archimedes principle ( ) The buoyant force ( ) on a body with volume immersed in a fluid is equal to the weight of the fluid
ˆ displaced by the object ie In the followin
B
V
gVρ= minusf k
阿基米德原理
浮力
ˆg figure the ˆ with and being the fluid density gravitational
acceleration depth and outward normal unit vector to the surface
fhydraulic pressure g z
g z
S
ρ
ρ
= minusp n
n
1
the modified divergence theorem
ˆ
Hint
scalar fuctionV S
vdV v dS v Cnabla = isinint int n
MingsianMingsian R R BaiBai44
Levelsurfacefluid
V
S
airyx
z
ˆf g zρ= minusp n
ˆPf ( ) ( )
ˆ ˆ ˆ ˆ ˆ (0 0 ) v v
B fS S V
V V
g z g zdS dS dV
g dV g dV gV
ρ ρ
ρ ρ ρ
= = minus = minus nabla
= minus + + = minus = minus
int int int
int int
f p n
i j k k k
k
MingsianMingsian R R BaiBai45
Note This can also be thought of as simple balance of forces (resultant force that acts on and weight of filled with fluid)S V
2
~ a little history about George Green Recall the diveregence theorem
ˆ
Let
II th
e Green
are arbitrary fun
s identiti
ctions
es
V S
v dV n v dS
v u vu v C
nablasdot = sdot
= nablaisin
int intv v
v
ˆ ( ) ( )
ˆ ( )V S
V S
u v dV n u v dS
u v u v dV un v dS
nablasdot nabla = sdot nabla
nabla sdotnabla + nablasdotnabla = sdotnabla
int int
int int
MingsianMingsian R R BaiBai46
2 -( ) -(1)V S
vu v u v dV u dSnpart
nabla sdotnabla + nabla =partint int firs(Greens identt ity)
2
If
( ) --(2)
(1) (2) V S
u vuv u v u dV v dSn
harrpart
nabla sdotnabla + nabla =part
minus
int int
( )2 2
V S
dv duu v v u dV u v dSdn dn
⎛ ⎞nabla minus nabla = minus⎜ ⎟⎝ ⎠int int
second(Greens identity)
MingsianMingsian R R BaiBai47
( )
( )
( ) ( )
1st id ( )
Greens identities in the 1D case integration by parts
b b bb ba a
a a ab
ba
ab
ba
a
v udx v u v u dx u v uv dx uv
u v u v vu dx vu
uv vu dx uv vu
primeprime prime prime prime prime prime primeprime prime= minus rArr + =
prime prime primeprime primeharr + =
primeprime primeprime prime primerArr minus = minus
int int int
int
2
2nd id ( )
Good for any adjoint operators in addition to Good also for and integration by parts Important for theory amp reciprocity
From th
Gree
e 1s
2-
t
ns functi
identity
D 1-
tak 1
D
e
on
u
nablararr
=
int
2
V S
vv dV dSnpartnabla =part
rArr int int2Ex 0 st 0 on the boundary 0 ( 1st id u u unabla = = rArr equiv 以 証)
MingsianMingsian R R BaiBai48
1
Thm1 Stokes theorem Let be defined in Let be a piecewise-smooth orientable surface bounded by a piec
III Stokes theorem and
ewise-smooth simple clos
Greens
ed curve Then
theore
m
v C R S
C
isinv
( )
ˆwhere is a unit normal to according t right hand rulo the e
ˆS C
n S
n v dS v dRsdot nablatimes = sdotint intvv v
(not necessarily planar)S
Cn vnablatimes v vv
MingsianMingsian R R BaiBai49
Pf
( )( )
circulatioWe have shown in deriving thatthe in -direction
ˆor
ˆ
n
z
z
j j j j j
j j j j jj j
vz v dx dy
v dR n v S
v dR n v S
nablatimes
=nablatimes
sdot = sdot nablatimes Δ
sdot = sdot nablatimes Δ
intsum sumint
v
v
vv v
vv v
C
S
jS
jC
jP
ˆ jn
jSΔ
MingsianMingsian R R BaiBai50
( )
( )ext surface
After the interior boundaries cancel out only the exterior boundaries (Why)
ˆ
ˆ0 QED
Alternative forms scalar
survive
va
j
j j j j jC
jC S
v dR n v S
S v dR n v dS
φ
sdot = sdot nablatimes Δ
Δ rarr rArr sdot = sdot nablatimes
sum sumint
int int
vv v
vv v
( )
riable constant vector vector variable
1 Let
2 Let
ˆ
ˆ S C
S C
a p
v n dR dR
n p dS dR p
a
v a p
φ φφ= rArr timesnabla =
timesnabla times= rArr timestimes =
int int
int int
v v
v v
v
v v
vv vv v
MingsianMingsian R R BaiBai51
( ) ( )
( )
( )
( ) ( )
1
Thm2 Greens theorem ( planar version of Stokes theorem)ˆ ˆ Let
pf By Stokes theoremˆ
ˆ ˆ ˆ ˆ ˆ ˆ
S C
S C
S C
v P x y i Q x y j C
n v dS v dR
k Pi Qj dS Pi Qj dx i dy
Q P dS Pdx Qdyx y
⎛ ⎞part partminus = +⎜ ⎟
= + isin
sdot nablatimes = sdot
⎡ ⎤sdot nablatimes + =
part part⎝ ⎠
+ sdot +⎣ ⎦
int int
int
int
int
int
v
vv v
( )
( )
( )
ˆ
ˆˆ ˆ
ˆˆ ˆ ˆ ˆ( ) ( )
0
QEDS C
j
i j kQ PPi Qj i j k
x y z x yP Q
Q P dS Pdx Qdyx y
⎛ ⎞part part part part partnablatimes + = = + + minus⎜ ⎟part part part part part⎝ ⎠
⎛ ⎞part partrArr minus = +⎜ ⎟part part⎝ ⎠int int
L L
S
C
i
j
x
y
MingsianMingsian R R BaiBai52
Potential TheorypotentiGeorge Green coined the term (l
a )
rarr 位能 勢能
例子 重力場 電磁場 流場 聲場 彈性力學
1
2
Thm1 If is in a simply connected region then the following statements are equivalent(i) There exists a scala
Conservative Field
potential functior such that
(ii
n
) 0
v
C
C
v
v R
φφ=nabla
nablatimes
isin
isin
=r
r
r
r in (Irrotational field curl-free)
cf 0 st solenoidal field divergence-free
0
(iii) in (Conservative field)C
Rv w v w
v R Rd
nablasdot = exist nabla
sdot =
= times
int
r r r
r
r
r
MingsianMingsian R R BaiBai53
(iv) is connecting and
independent o
Ex Think
f the pa
of dynamics is the force
thB
A
v dR A B
v
sdotintrr
r
1 2
1
0
Pf
(i) (ii) ( ) 0 (zero identity cg rule) irrotational
ˆ(ii) (iii) ( ) 0 (Stokes theorem)
(iii) (iv) ( ) 0
C S
C C C
C
v
v dR n v dS
v dR v dR
v dR
φ
=
rarr nablatimes = nablatimes nabla =
rarr sdot = sdot nablatimes =
rarr sdot = minus sdot =
rArr sdot =
int int
int int int
int
r
rr
rr r
r rr r
rr r
2
independent of pathC
v dRsdotintr
A
B
1C
2C
MingsianMingsian R R BaiBai54
1 2
(iv) (i) constant ( ) ( )
--(1)
But (2)
(1) (2) QED
Note on application
Check
B
C C A
v dR v dR B A d
v dR d
d dx dy dz dRx y z
v dR dR v R
φ φ φ
φφ φ φφ φ
φ φ
rarr sdot = sdot = = minus =
rArr sdot =part part part
= + + = nabla sdot minus minus minus minus minuspart part part
sdot = nabla sdot rArr = nabla forall
int int intr rr r
rr
r
r r rr r
use simpler path (iv)if 0
find (i)v
φ⎧
nablatimes = ⎨⎩
rr
A
B
x
y
Cz
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai25
0 1lim ( ) ( ) or ( )
Intepretation of a line integral as an area ( )
i
Bn
i i isA
C
i C
F s F x y ds F x y d
x
s
F y ds
ξ ηΔ rarr =
Δ =sum int
int
int
x
y
z
C
( )z F x y=
MingsianMingsian R R BaiBai26
the area is always on the left of an obs
Positive dierver trave
rection of irsing on the
ntegrationcurve
RC
CCW CW
If is a regular closed curve piecewise smo
( ) ( )
(contou
oth simple ( )
r integral)
x y zC C
x y zC
F dR F F F dx dy dz
F dx F dy
C
F dz
sdot = + + + +
= + +
lowast
int int
int
i j k i j kv v
無交叉
work dW F dR= sdotv v
MingsianMingsian R R BaiBai27
0 1
Surface integral
lim ( ) ( )i
n
i i i iS i S
F S F x y z dSξ η ζΔ rarr =
Δ =sum intint( ) i i iξ η ζ
iSΔ
0 1
mass density
lim ( ) ( )
Ex
Volume integral
( )
i
n
i i i iV i V
V
F V F x y z dV
M x y z dV
ξ η ζ
ρ
Δ rarr =Δ =
=
sum intintint
intintint( ) i i iξ η ζ
iVΔ
MingsianMingsian R R BaiBai28
Ref (1)Hildebrand advanced Calculus for application Prentice-Hall 1976 p306-313(2) Greenberg p383-389 (3) Arfken ch2
1 2 3
1 2 3
1 2 3
1 2 3
(
Orthogonal coordinate systems base vectors are not constant in s
) ( )( )( )( )
ˆˆ ˆ
pacep x y z p u u ux x u u uy y u u uz z u u u
r xi yj zk
primerarr
= +
rarr
===
+v
1uv
2uv3uv
1u
2u
3uP coord curve
xy
z
O
X Curvilinear Coordinates
MingsianMingsian R R BaiBai29
1
11
1 1 1
1
1
1 1 1
11 1
1 1
2 2 2 3 3 3
Tangent vector to the at
where is the arc length
1
where unit vector
Similarly
curv
e
u psr rU
u s us
rs
U h uds rh udu sU h u U h u
partpart part= =part part part
part =part
there4 =part= =part
= =
v vv
vQ
v v
vv
v vv v
1u cood curve
1uv
1du 1dS
drv
P
MingsianMingsian R R BaiBai30
3 31 2 1 21 2 3
1 2 32
curv
In summary scale factor
If is the arc length along a in any directione
k k k
kk k
k k
k k k
U h u
ds rh Udu u
ds h dus
du dudu du du dudr r r r U U Uds u ds u ds u ds ds ds ds
dr drds ds
=
part= = =part
=
part part part= + + = + +part part part
= sdot
v v
v v
v v v v v v v
v v 3 3
1 1
23 3 32 2
21 1 1
2 2 2 2 2 2 2 2 2 21 1 2 2 3 3 1 2 3
1 (for orthogonal coord)
jii j
i j
ji ii j i
i j i
dududr U Uds ds ds
dudu duU U hds ds ds
ds h du h du h du ds ds ds
= =
= = =
⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
= sdot
rarr = sdot =
rarr = + + = + +
sum sum
sumsum sum
v v v
v v
MingsianMingsian R R BaiBai31
1 1 2 2 3 3
1 2 3
are mutually vectors having with arc length
k k k k kds h du U du
U du U du U duds ds ds
= =
rArr perp
v
v v v
1uv
2uv
3uv
1dS
2dS
3dS
dS
MingsianMingsian R R BaiBai32
( ) ( )
( ) ( )
1 1 2 2 3 3 1 2 3 1 2 3 1 2 3
1 2 3 1 2 3
1
1 2 2 3 3 2 2 2 3 3 3 2 3 2 3 1
Differential volume
Differential On const coord surface
area
Gradi t
en
d U du U du U du u u u h h h du du du
d h h h du du du
ud U du U du h u du h u du h h du du u
df
τ
τ
σ
= times sdot = times sdot
=
=
= times = times =
= nabla
v v v v v v
v vv v v v
1 2 31 2 3
1 2 3 1 1 2 2 3 31 2 3
1 1 1 2 2 2 3 3 3
f f ff dr df du du duu u u
r r rdr du du du U du U du U duu u u
h u du h u du h u du
part part partsdot rarr = + +part part part
part part part= + + = + +part part part
= + +
v
v v v v v vv
v v v
=1 (right hand rule)
MingsianMingsian R R BaiBai33
Rectangular Cartesian Coordinates
ij
k
MingsianMingsian R R BaiBai34
Circular Cylindrical Coordinatescossin
xyz z
ρ ϕρ ϕ
===
Ref Arfken
( )P zρ ϕ
MingsianMingsian R R BaiBai35
Polar Spherical Coordinatessin cossin sincos
x ry rz r
θ ϕθ ϕθ
===
Ref Arfken
0
0
0
sin cos sin sin coscos cos cos sin sin
sin cos
θ ϕ θ ϕ θθ ϕ θ ϕ θϕ ϕ
= + +
= + minus= minus +
r i j kθ i j kφ i j
2
2
sinsin
dS r d ddV r dr d d
θ θ ϕ
θ θ ϕ
=
=( )P r θ ϕ
MingsianMingsian R R BaiBai36
Integral TheoremsHeart of this chapter
Interchange of dimensionality Div Curl Grad in integrals High-dimensional integration
rarr
1
Thm1 Divergence theorem Let be a closed region bounded by piecewise smooth orientable surface and let (con
I Divergence theor
tinuous derivatives up to the first order
em (Gauss theorem)
)
VS v Cisin
r
ˆ If denotes the outward unit normal on
the ˆn V S
v dV n v dS
n S
nablasdot = sdotint intr r
MingsianMingsian R R BaiBai37
jS
jP
jV
S
0
ˆBy definition ( ) lim
ˆ( )
ˆ( )
j
j
j
j
Sj V
j
j j S
j j Sj j
n v dSv P
V
v P V n v dS
v P V n v dS
Δ rarr
⎧ ⎫sdot⎪ ⎪nabla sdot ⎨ ⎬Δ⎪ ⎪⎩ ⎭
nabla sdot Δ asymp sdot
nabla sdot Δ = sdot
int
intsum sumint
rr
r r
r r
Pf
MingsianMingsian R R BaiBai38
0
After interior surfaces cancel out only exterior surface (Why Opposite normal vectors)
ˆ( )
volumes exterior surfaces
ˆlim
QE
survive
j
j
j j Sj j
V SV
v p V n v dS
v dV n v dSΔ rarr
nabla sdot Δ = sdot
rArr nablasdot = sdot
sum sumint
int int
r r
r r
D
MingsianMingsian R R BaiBai39
( )
scalar variable constant vectorvector variable
Alternative formsLet be a an arbitrary and
a
ˆ
1 Le
(
ˆt
V
V S
S
V
v ap
v d
v dV vn d
V n v dS
v va
va dV v a v a
S
nablasdot = sdot
= rArr
nablasdot = nabla sdot + nabla
=
sdot
nablaint intint int
int
r
v
r r
r r
r r rQ ( )
( )
ˆ)
2 Let ˆV S
V
V S
dV n va dS
v a p
a p
p dV n p d
d p
S
V a
⎛ ⎞= sdot⎜ ⎟
⎝ ⎠
= times rArr
nablasdot times = sdot nabla
times
times
nabla = times
int int
int int
int
r
r r r
r r v
r
vQ
r
( ) ( ) ( )
( ) ( )
ˆ
ˆV S
V S
a p dV n a p dS
a p dV a n p dS
⎛ ⎞⎡ ⎤minus sdot nablatimes = sdot times⎣ ⎦⎜ ⎟⎜ ⎟⎜ ⎟rArr minus sdot nablatimes = minus sdot times⎜ ⎟⎝ ⎠
int int
int int
r rv v
r rv v
MingsianMingsian R R BaiBai40
( )
( )
From Leibnitzs rule
(1-D) ( )
( ) ( )
B t
A t
d f x t dxdt
f x t dx B ttpart prime= +part
int( )
( )( ( ) ) ( )
B t
A tf B t t A tprimeminusint ( ( ) )
(assume ( ) ( ) )
f A t t
A t B t const= =
ContinuityEx The Equation
rate of increase of mass
mass densityV
dM d dVdt dt
ρ
ρ
= int
flow velocityvv
MingsianMingsian R R BaiBai41
ˆ ˆ ( )
(
Conservation
) ( Divergence theorem)
of mass
( ) 0
V S S
V
V
dV n v dS n v dSt
v dV
v dVt
ρ ρ ρ
ρ
ρ ρ⎡ ⎤⎢ ⎥⎣ ⎦
part = minus sdot = minus sdotpart
= minus nablasdot
partrArr +nablasdot =part
int int int
int
int
v v
vQ
v
ˆOn the other hand rate influof mass through n ox i t
Extended to 3-D for time-independent fixed boundary
(3-D) ( ) ( )
Thus
S
V V
V
S V n v dS
Sd f x y z t dV f x y z t dVdt tdM dVdt t
ρ
ρ
= minus sdot
part=part
part=part
int
int int
intv
Fixed control volume (Eulerian coordinates)Otherwise Raynoldrsquos transport theoremMoving control volume Lagrangian coordinates
MingsianMingsian R R BaiBai42
But the control volime is only arbitrary
In particular for flow
( ) 0
con
(continuity equati
stant
on)
Divergence the
0 incompress
Note is t
ible
orem h
V
v
vt
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
e mother of the other integral theorems
MingsianMingsian R R BaiBai43
Ex Archimedes principle ( ) The buoyant force ( ) on a body with volume immersed in a fluid is equal to the weight of the fluid
ˆ displaced by the object ie In the followin
B
V
gVρ= minusf k
阿基米德原理
浮力
ˆg figure the ˆ with and being the fluid density gravitational
acceleration depth and outward normal unit vector to the surface
fhydraulic pressure g z
g z
S
ρ
ρ
= minusp n
n
1
the modified divergence theorem
ˆ
Hint
scalar fuctionV S
vdV v dS v Cnabla = isinint int n
MingsianMingsian R R BaiBai44
Levelsurfacefluid
V
S
airyx
z
ˆf g zρ= minusp n
ˆPf ( ) ( )
ˆ ˆ ˆ ˆ ˆ (0 0 ) v v
B fS S V
V V
g z g zdS dS dV
g dV g dV gV
ρ ρ
ρ ρ ρ
= = minus = minus nabla
= minus + + = minus = minus
int int int
int int
f p n
i j k k k
k
MingsianMingsian R R BaiBai45
Note This can also be thought of as simple balance of forces (resultant force that acts on and weight of filled with fluid)S V
2
~ a little history about George Green Recall the diveregence theorem
ˆ
Let
II th
e Green
are arbitrary fun
s identiti
ctions
es
V S
v dV n v dS
v u vu v C
nablasdot = sdot
= nablaisin
int intv v
v
ˆ ( ) ( )
ˆ ( )V S
V S
u v dV n u v dS
u v u v dV un v dS
nablasdot nabla = sdot nabla
nabla sdotnabla + nablasdotnabla = sdotnabla
int int
int int
MingsianMingsian R R BaiBai46
2 -( ) -(1)V S
vu v u v dV u dSnpart
nabla sdotnabla + nabla =partint int firs(Greens identt ity)
2
If
( ) --(2)
(1) (2) V S
u vuv u v u dV v dSn
harrpart
nabla sdotnabla + nabla =part
minus
int int
( )2 2
V S
dv duu v v u dV u v dSdn dn
⎛ ⎞nabla minus nabla = minus⎜ ⎟⎝ ⎠int int
second(Greens identity)
MingsianMingsian R R BaiBai47
( )
( )
( ) ( )
1st id ( )
Greens identities in the 1D case integration by parts
b b bb ba a
a a ab
ba
ab
ba
a
v udx v u v u dx u v uv dx uv
u v u v vu dx vu
uv vu dx uv vu
primeprime prime prime prime prime prime primeprime prime= minus rArr + =
prime prime primeprime primeharr + =
primeprime primeprime prime primerArr minus = minus
int int int
int
2
2nd id ( )
Good for any adjoint operators in addition to Good also for and integration by parts Important for theory amp reciprocity
From th
Gree
e 1s
2-
t
ns functi
identity
D 1-
tak 1
D
e
on
u
nablararr
=
int
2
V S
vv dV dSnpartnabla =part
rArr int int2Ex 0 st 0 on the boundary 0 ( 1st id u u unabla = = rArr equiv 以 証)
MingsianMingsian R R BaiBai48
1
Thm1 Stokes theorem Let be defined in Let be a piecewise-smooth orientable surface bounded by a piec
III Stokes theorem and
ewise-smooth simple clos
Greens
ed curve Then
theore
m
v C R S
C
isinv
( )
ˆwhere is a unit normal to according t right hand rulo the e
ˆS C
n S
n v dS v dRsdot nablatimes = sdotint intvv v
(not necessarily planar)S
Cn vnablatimes v vv
MingsianMingsian R R BaiBai49
Pf
( )( )
circulatioWe have shown in deriving thatthe in -direction
ˆor
ˆ
n
z
z
j j j j j
j j j j jj j
vz v dx dy
v dR n v S
v dR n v S
nablatimes
=nablatimes
sdot = sdot nablatimes Δ
sdot = sdot nablatimes Δ
intsum sumint
v
v
vv v
vv v
C
S
jS
jC
jP
ˆ jn
jSΔ
MingsianMingsian R R BaiBai50
( )
( )ext surface
After the interior boundaries cancel out only the exterior boundaries (Why)
ˆ
ˆ0 QED
Alternative forms scalar
survive
va
j
j j j j jC
jC S
v dR n v S
S v dR n v dS
φ
sdot = sdot nablatimes Δ
Δ rarr rArr sdot = sdot nablatimes
sum sumint
int int
vv v
vv v
( )
riable constant vector vector variable
1 Let
2 Let
ˆ
ˆ S C
S C
a p
v n dR dR
n p dS dR p
a
v a p
φ φφ= rArr timesnabla =
timesnabla times= rArr timestimes =
int int
int int
v v
v v
v
v v
vv vv v
MingsianMingsian R R BaiBai51
( ) ( )
( )
( )
( ) ( )
1
Thm2 Greens theorem ( planar version of Stokes theorem)ˆ ˆ Let
pf By Stokes theoremˆ
ˆ ˆ ˆ ˆ ˆ ˆ
S C
S C
S C
v P x y i Q x y j C
n v dS v dR
k Pi Qj dS Pi Qj dx i dy
Q P dS Pdx Qdyx y
⎛ ⎞part partminus = +⎜ ⎟
= + isin
sdot nablatimes = sdot
⎡ ⎤sdot nablatimes + =
part part⎝ ⎠
+ sdot +⎣ ⎦
int int
int
int
int
int
v
vv v
( )
( )
( )
ˆ
ˆˆ ˆ
ˆˆ ˆ ˆ ˆ( ) ( )
0
QEDS C
j
i j kQ PPi Qj i j k
x y z x yP Q
Q P dS Pdx Qdyx y
⎛ ⎞part part part part partnablatimes + = = + + minus⎜ ⎟part part part part part⎝ ⎠
⎛ ⎞part partrArr minus = +⎜ ⎟part part⎝ ⎠int int
L L
S
C
i
j
x
y
MingsianMingsian R R BaiBai52
Potential TheorypotentiGeorge Green coined the term (l
a )
rarr 位能 勢能
例子 重力場 電磁場 流場 聲場 彈性力學
1
2
Thm1 If is in a simply connected region then the following statements are equivalent(i) There exists a scala
Conservative Field
potential functior such that
(ii
n
) 0
v
C
C
v
v R
φφ=nabla
nablatimes
isin
isin
=r
r
r
r in (Irrotational field curl-free)
cf 0 st solenoidal field divergence-free
0
(iii) in (Conservative field)C
Rv w v w
v R Rd
nablasdot = exist nabla
sdot =
= times
int
r r r
r
r
r
MingsianMingsian R R BaiBai53
(iv) is connecting and
independent o
Ex Think
f the pa
of dynamics is the force
thB
A
v dR A B
v
sdotintrr
r
1 2
1
0
Pf
(i) (ii) ( ) 0 (zero identity cg rule) irrotational
ˆ(ii) (iii) ( ) 0 (Stokes theorem)
(iii) (iv) ( ) 0
C S
C C C
C
v
v dR n v dS
v dR v dR
v dR
φ
=
rarr nablatimes = nablatimes nabla =
rarr sdot = sdot nablatimes =
rarr sdot = minus sdot =
rArr sdot =
int int
int int int
int
r
rr
rr r
r rr r
rr r
2
independent of pathC
v dRsdotintr
A
B
1C
2C
MingsianMingsian R R BaiBai54
1 2
(iv) (i) constant ( ) ( )
--(1)
But (2)
(1) (2) QED
Note on application
Check
B
C C A
v dR v dR B A d
v dR d
d dx dy dz dRx y z
v dR dR v R
φ φ φ
φφ φ φφ φ
φ φ
rarr sdot = sdot = = minus =
rArr sdot =part part part
= + + = nabla sdot minus minus minus minus minuspart part part
sdot = nabla sdot rArr = nabla forall
int int intr rr r
rr
r
r r rr r
use simpler path (iv)if 0
find (i)v
φ⎧
nablatimes = ⎨⎩
rr
A
B
x
y
Cz
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai26
the area is always on the left of an obs
Positive dierver trave
rection of irsing on the
ntegrationcurve
RC
CCW CW
If is a regular closed curve piecewise smo
( ) ( )
(contou
oth simple ( )
r integral)
x y zC C
x y zC
F dR F F F dx dy dz
F dx F dy
C
F dz
sdot = + + + +
= + +
lowast
int int
int
i j k i j kv v
無交叉
work dW F dR= sdotv v
MingsianMingsian R R BaiBai27
0 1
Surface integral
lim ( ) ( )i
n
i i i iS i S
F S F x y z dSξ η ζΔ rarr =
Δ =sum intint( ) i i iξ η ζ
iSΔ
0 1
mass density
lim ( ) ( )
Ex
Volume integral
( )
i
n
i i i iV i V
V
F V F x y z dV
M x y z dV
ξ η ζ
ρ
Δ rarr =Δ =
=
sum intintint
intintint( ) i i iξ η ζ
iVΔ
MingsianMingsian R R BaiBai28
Ref (1)Hildebrand advanced Calculus for application Prentice-Hall 1976 p306-313(2) Greenberg p383-389 (3) Arfken ch2
1 2 3
1 2 3
1 2 3
1 2 3
(
Orthogonal coordinate systems base vectors are not constant in s
) ( )( )( )( )
ˆˆ ˆ
pacep x y z p u u ux x u u uy y u u uz z u u u
r xi yj zk
primerarr
= +
rarr
===
+v
1uv
2uv3uv
1u
2u
3uP coord curve
xy
z
O
X Curvilinear Coordinates
MingsianMingsian R R BaiBai29
1
11
1 1 1
1
1
1 1 1
11 1
1 1
2 2 2 3 3 3
Tangent vector to the at
where is the arc length
1
where unit vector
Similarly
curv
e
u psr rU
u s us
rs
U h uds rh udu sU h u U h u
partpart part= =part part part
part =part
there4 =part= =part
= =
v vv
vQ
v v
vv
v vv v
1u cood curve
1uv
1du 1dS
drv
P
MingsianMingsian R R BaiBai30
3 31 2 1 21 2 3
1 2 32
curv
In summary scale factor
If is the arc length along a in any directione
k k k
kk k
k k
k k k
U h u
ds rh Udu u
ds h dus
du dudu du du dudr r r r U U Uds u ds u ds u ds ds ds ds
dr drds ds
=
part= = =part
=
part part part= + + = + +part part part
= sdot
v v
v v
v v v v v v v
v v 3 3
1 1
23 3 32 2
21 1 1
2 2 2 2 2 2 2 2 2 21 1 2 2 3 3 1 2 3
1 (for orthogonal coord)
jii j
i j
ji ii j i
i j i
dududr U Uds ds ds
dudu duU U hds ds ds
ds h du h du h du ds ds ds
= =
= = =
⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
= sdot
rarr = sdot =
rarr = + + = + +
sum sum
sumsum sum
v v v
v v
MingsianMingsian R R BaiBai31
1 1 2 2 3 3
1 2 3
are mutually vectors having with arc length
k k k k kds h du U du
U du U du U duds ds ds
= =
rArr perp
v
v v v
1uv
2uv
3uv
1dS
2dS
3dS
dS
MingsianMingsian R R BaiBai32
( ) ( )
( ) ( )
1 1 2 2 3 3 1 2 3 1 2 3 1 2 3
1 2 3 1 2 3
1
1 2 2 3 3 2 2 2 3 3 3 2 3 2 3 1
Differential volume
Differential On const coord surface
area
Gradi t
en
d U du U du U du u u u h h h du du du
d h h h du du du
ud U du U du h u du h u du h h du du u
df
τ
τ
σ
= times sdot = times sdot
=
=
= times = times =
= nabla
v v v v v v
v vv v v v
1 2 31 2 3
1 2 3 1 1 2 2 3 31 2 3
1 1 1 2 2 2 3 3 3
f f ff dr df du du duu u u
r r rdr du du du U du U du U duu u u
h u du h u du h u du
part part partsdot rarr = + +part part part
part part part= + + = + +part part part
= + +
v
v v v v v vv
v v v
=1 (right hand rule)
MingsianMingsian R R BaiBai33
Rectangular Cartesian Coordinates
ij
k
MingsianMingsian R R BaiBai34
Circular Cylindrical Coordinatescossin
xyz z
ρ ϕρ ϕ
===
Ref Arfken
( )P zρ ϕ
MingsianMingsian R R BaiBai35
Polar Spherical Coordinatessin cossin sincos
x ry rz r
θ ϕθ ϕθ
===
Ref Arfken
0
0
0
sin cos sin sin coscos cos cos sin sin
sin cos
θ ϕ θ ϕ θθ ϕ θ ϕ θϕ ϕ
= + +
= + minus= minus +
r i j kθ i j kφ i j
2
2
sinsin
dS r d ddV r dr d d
θ θ ϕ
θ θ ϕ
=
=( )P r θ ϕ
MingsianMingsian R R BaiBai36
Integral TheoremsHeart of this chapter
Interchange of dimensionality Div Curl Grad in integrals High-dimensional integration
rarr
1
Thm1 Divergence theorem Let be a closed region bounded by piecewise smooth orientable surface and let (con
I Divergence theor
tinuous derivatives up to the first order
em (Gauss theorem)
)
VS v Cisin
r
ˆ If denotes the outward unit normal on
the ˆn V S
v dV n v dS
n S
nablasdot = sdotint intr r
MingsianMingsian R R BaiBai37
jS
jP
jV
S
0
ˆBy definition ( ) lim
ˆ( )
ˆ( )
j
j
j
j
Sj V
j
j j S
j j Sj j
n v dSv P
V
v P V n v dS
v P V n v dS
Δ rarr
⎧ ⎫sdot⎪ ⎪nabla sdot ⎨ ⎬Δ⎪ ⎪⎩ ⎭
nabla sdot Δ asymp sdot
nabla sdot Δ = sdot
int
intsum sumint
rr
r r
r r
Pf
MingsianMingsian R R BaiBai38
0
After interior surfaces cancel out only exterior surface (Why Opposite normal vectors)
ˆ( )
volumes exterior surfaces
ˆlim
QE
survive
j
j
j j Sj j
V SV
v p V n v dS
v dV n v dSΔ rarr
nabla sdot Δ = sdot
rArr nablasdot = sdot
sum sumint
int int
r r
r r
D
MingsianMingsian R R BaiBai39
( )
scalar variable constant vectorvector variable
Alternative formsLet be a an arbitrary and
a
ˆ
1 Le
(
ˆt
V
V S
S
V
v ap
v d
v dV vn d
V n v dS
v va
va dV v a v a
S
nablasdot = sdot
= rArr
nablasdot = nabla sdot + nabla
=
sdot
nablaint intint int
int
r
v
r r
r r
r r rQ ( )
( )
ˆ)
2 Let ˆV S
V
V S
dV n va dS
v a p
a p
p dV n p d
d p
S
V a
⎛ ⎞= sdot⎜ ⎟
⎝ ⎠
= times rArr
nablasdot times = sdot nabla
times
times
nabla = times
int int
int int
int
r
r r r
r r v
r
vQ
r
( ) ( ) ( )
( ) ( )
ˆ
ˆV S
V S
a p dV n a p dS
a p dV a n p dS
⎛ ⎞⎡ ⎤minus sdot nablatimes = sdot times⎣ ⎦⎜ ⎟⎜ ⎟⎜ ⎟rArr minus sdot nablatimes = minus sdot times⎜ ⎟⎝ ⎠
int int
int int
r rv v
r rv v
MingsianMingsian R R BaiBai40
( )
( )
From Leibnitzs rule
(1-D) ( )
( ) ( )
B t
A t
d f x t dxdt
f x t dx B ttpart prime= +part
int( )
( )( ( ) ) ( )
B t
A tf B t t A tprimeminusint ( ( ) )
(assume ( ) ( ) )
f A t t
A t B t const= =
ContinuityEx The Equation
rate of increase of mass
mass densityV
dM d dVdt dt
ρ
ρ
= int
flow velocityvv
MingsianMingsian R R BaiBai41
ˆ ˆ ( )
(
Conservation
) ( Divergence theorem)
of mass
( ) 0
V S S
V
V
dV n v dS n v dSt
v dV
v dVt
ρ ρ ρ
ρ
ρ ρ⎡ ⎤⎢ ⎥⎣ ⎦
part = minus sdot = minus sdotpart
= minus nablasdot
partrArr +nablasdot =part
int int int
int
int
v v
vQ
v
ˆOn the other hand rate influof mass through n ox i t
Extended to 3-D for time-independent fixed boundary
(3-D) ( ) ( )
Thus
S
V V
V
S V n v dS
Sd f x y z t dV f x y z t dVdt tdM dVdt t
ρ
ρ
= minus sdot
part=part
part=part
int
int int
intv
Fixed control volume (Eulerian coordinates)Otherwise Raynoldrsquos transport theoremMoving control volume Lagrangian coordinates
MingsianMingsian R R BaiBai42
But the control volime is only arbitrary
In particular for flow
( ) 0
con
(continuity equati
stant
on)
Divergence the
0 incompress
Note is t
ible
orem h
V
v
vt
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
e mother of the other integral theorems
MingsianMingsian R R BaiBai43
Ex Archimedes principle ( ) The buoyant force ( ) on a body with volume immersed in a fluid is equal to the weight of the fluid
ˆ displaced by the object ie In the followin
B
V
gVρ= minusf k
阿基米德原理
浮力
ˆg figure the ˆ with and being the fluid density gravitational
acceleration depth and outward normal unit vector to the surface
fhydraulic pressure g z
g z
S
ρ
ρ
= minusp n
n
1
the modified divergence theorem
ˆ
Hint
scalar fuctionV S
vdV v dS v Cnabla = isinint int n
MingsianMingsian R R BaiBai44
Levelsurfacefluid
V
S
airyx
z
ˆf g zρ= minusp n
ˆPf ( ) ( )
ˆ ˆ ˆ ˆ ˆ (0 0 ) v v
B fS S V
V V
g z g zdS dS dV
g dV g dV gV
ρ ρ
ρ ρ ρ
= = minus = minus nabla
= minus + + = minus = minus
int int int
int int
f p n
i j k k k
k
MingsianMingsian R R BaiBai45
Note This can also be thought of as simple balance of forces (resultant force that acts on and weight of filled with fluid)S V
2
~ a little history about George Green Recall the diveregence theorem
ˆ
Let
II th
e Green
are arbitrary fun
s identiti
ctions
es
V S
v dV n v dS
v u vu v C
nablasdot = sdot
= nablaisin
int intv v
v
ˆ ( ) ( )
ˆ ( )V S
V S
u v dV n u v dS
u v u v dV un v dS
nablasdot nabla = sdot nabla
nabla sdotnabla + nablasdotnabla = sdotnabla
int int
int int
MingsianMingsian R R BaiBai46
2 -( ) -(1)V S
vu v u v dV u dSnpart
nabla sdotnabla + nabla =partint int firs(Greens identt ity)
2
If
( ) --(2)
(1) (2) V S
u vuv u v u dV v dSn
harrpart
nabla sdotnabla + nabla =part
minus
int int
( )2 2
V S
dv duu v v u dV u v dSdn dn
⎛ ⎞nabla minus nabla = minus⎜ ⎟⎝ ⎠int int
second(Greens identity)
MingsianMingsian R R BaiBai47
( )
( )
( ) ( )
1st id ( )
Greens identities in the 1D case integration by parts
b b bb ba a
a a ab
ba
ab
ba
a
v udx v u v u dx u v uv dx uv
u v u v vu dx vu
uv vu dx uv vu
primeprime prime prime prime prime prime primeprime prime= minus rArr + =
prime prime primeprime primeharr + =
primeprime primeprime prime primerArr minus = minus
int int int
int
2
2nd id ( )
Good for any adjoint operators in addition to Good also for and integration by parts Important for theory amp reciprocity
From th
Gree
e 1s
2-
t
ns functi
identity
D 1-
tak 1
D
e
on
u
nablararr
=
int
2
V S
vv dV dSnpartnabla =part
rArr int int2Ex 0 st 0 on the boundary 0 ( 1st id u u unabla = = rArr equiv 以 証)
MingsianMingsian R R BaiBai48
1
Thm1 Stokes theorem Let be defined in Let be a piecewise-smooth orientable surface bounded by a piec
III Stokes theorem and
ewise-smooth simple clos
Greens
ed curve Then
theore
m
v C R S
C
isinv
( )
ˆwhere is a unit normal to according t right hand rulo the e
ˆS C
n S
n v dS v dRsdot nablatimes = sdotint intvv v
(not necessarily planar)S
Cn vnablatimes v vv
MingsianMingsian R R BaiBai49
Pf
( )( )
circulatioWe have shown in deriving thatthe in -direction
ˆor
ˆ
n
z
z
j j j j j
j j j j jj j
vz v dx dy
v dR n v S
v dR n v S
nablatimes
=nablatimes
sdot = sdot nablatimes Δ
sdot = sdot nablatimes Δ
intsum sumint
v
v
vv v
vv v
C
S
jS
jC
jP
ˆ jn
jSΔ
MingsianMingsian R R BaiBai50
( )
( )ext surface
After the interior boundaries cancel out only the exterior boundaries (Why)
ˆ
ˆ0 QED
Alternative forms scalar
survive
va
j
j j j j jC
jC S
v dR n v S
S v dR n v dS
φ
sdot = sdot nablatimes Δ
Δ rarr rArr sdot = sdot nablatimes
sum sumint
int int
vv v
vv v
( )
riable constant vector vector variable
1 Let
2 Let
ˆ
ˆ S C
S C
a p
v n dR dR
n p dS dR p
a
v a p
φ φφ= rArr timesnabla =
timesnabla times= rArr timestimes =
int int
int int
v v
v v
v
v v
vv vv v
MingsianMingsian R R BaiBai51
( ) ( )
( )
( )
( ) ( )
1
Thm2 Greens theorem ( planar version of Stokes theorem)ˆ ˆ Let
pf By Stokes theoremˆ
ˆ ˆ ˆ ˆ ˆ ˆ
S C
S C
S C
v P x y i Q x y j C
n v dS v dR
k Pi Qj dS Pi Qj dx i dy
Q P dS Pdx Qdyx y
⎛ ⎞part partminus = +⎜ ⎟
= + isin
sdot nablatimes = sdot
⎡ ⎤sdot nablatimes + =
part part⎝ ⎠
+ sdot +⎣ ⎦
int int
int
int
int
int
v
vv v
( )
( )
( )
ˆ
ˆˆ ˆ
ˆˆ ˆ ˆ ˆ( ) ( )
0
QEDS C
j
i j kQ PPi Qj i j k
x y z x yP Q
Q P dS Pdx Qdyx y
⎛ ⎞part part part part partnablatimes + = = + + minus⎜ ⎟part part part part part⎝ ⎠
⎛ ⎞part partrArr minus = +⎜ ⎟part part⎝ ⎠int int
L L
S
C
i
j
x
y
MingsianMingsian R R BaiBai52
Potential TheorypotentiGeorge Green coined the term (l
a )
rarr 位能 勢能
例子 重力場 電磁場 流場 聲場 彈性力學
1
2
Thm1 If is in a simply connected region then the following statements are equivalent(i) There exists a scala
Conservative Field
potential functior such that
(ii
n
) 0
v
C
C
v
v R
φφ=nabla
nablatimes
isin
isin
=r
r
r
r in (Irrotational field curl-free)
cf 0 st solenoidal field divergence-free
0
(iii) in (Conservative field)C
Rv w v w
v R Rd
nablasdot = exist nabla
sdot =
= times
int
r r r
r
r
r
MingsianMingsian R R BaiBai53
(iv) is connecting and
independent o
Ex Think
f the pa
of dynamics is the force
thB
A
v dR A B
v
sdotintrr
r
1 2
1
0
Pf
(i) (ii) ( ) 0 (zero identity cg rule) irrotational
ˆ(ii) (iii) ( ) 0 (Stokes theorem)
(iii) (iv) ( ) 0
C S
C C C
C
v
v dR n v dS
v dR v dR
v dR
φ
=
rarr nablatimes = nablatimes nabla =
rarr sdot = sdot nablatimes =
rarr sdot = minus sdot =
rArr sdot =
int int
int int int
int
r
rr
rr r
r rr r
rr r
2
independent of pathC
v dRsdotintr
A
B
1C
2C
MingsianMingsian R R BaiBai54
1 2
(iv) (i) constant ( ) ( )
--(1)
But (2)
(1) (2) QED
Note on application
Check
B
C C A
v dR v dR B A d
v dR d
d dx dy dz dRx y z
v dR dR v R
φ φ φ
φφ φ φφ φ
φ φ
rarr sdot = sdot = = minus =
rArr sdot =part part part
= + + = nabla sdot minus minus minus minus minuspart part part
sdot = nabla sdot rArr = nabla forall
int int intr rr r
rr
r
r r rr r
use simpler path (iv)if 0
find (i)v
φ⎧
nablatimes = ⎨⎩
rr
A
B
x
y
Cz
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai27
0 1
Surface integral
lim ( ) ( )i
n
i i i iS i S
F S F x y z dSξ η ζΔ rarr =
Δ =sum intint( ) i i iξ η ζ
iSΔ
0 1
mass density
lim ( ) ( )
Ex
Volume integral
( )
i
n
i i i iV i V
V
F V F x y z dV
M x y z dV
ξ η ζ
ρ
Δ rarr =Δ =
=
sum intintint
intintint( ) i i iξ η ζ
iVΔ
MingsianMingsian R R BaiBai28
Ref (1)Hildebrand advanced Calculus for application Prentice-Hall 1976 p306-313(2) Greenberg p383-389 (3) Arfken ch2
1 2 3
1 2 3
1 2 3
1 2 3
(
Orthogonal coordinate systems base vectors are not constant in s
) ( )( )( )( )
ˆˆ ˆ
pacep x y z p u u ux x u u uy y u u uz z u u u
r xi yj zk
primerarr
= +
rarr
===
+v
1uv
2uv3uv
1u
2u
3uP coord curve
xy
z
O
X Curvilinear Coordinates
MingsianMingsian R R BaiBai29
1
11
1 1 1
1
1
1 1 1
11 1
1 1
2 2 2 3 3 3
Tangent vector to the at
where is the arc length
1
where unit vector
Similarly
curv
e
u psr rU
u s us
rs
U h uds rh udu sU h u U h u
partpart part= =part part part
part =part
there4 =part= =part
= =
v vv
vQ
v v
vv
v vv v
1u cood curve
1uv
1du 1dS
drv
P
MingsianMingsian R R BaiBai30
3 31 2 1 21 2 3
1 2 32
curv
In summary scale factor
If is the arc length along a in any directione
k k k
kk k
k k
k k k
U h u
ds rh Udu u
ds h dus
du dudu du du dudr r r r U U Uds u ds u ds u ds ds ds ds
dr drds ds
=
part= = =part
=
part part part= + + = + +part part part
= sdot
v v
v v
v v v v v v v
v v 3 3
1 1
23 3 32 2
21 1 1
2 2 2 2 2 2 2 2 2 21 1 2 2 3 3 1 2 3
1 (for orthogonal coord)
jii j
i j
ji ii j i
i j i
dududr U Uds ds ds
dudu duU U hds ds ds
ds h du h du h du ds ds ds
= =
= = =
⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
= sdot
rarr = sdot =
rarr = + + = + +
sum sum
sumsum sum
v v v
v v
MingsianMingsian R R BaiBai31
1 1 2 2 3 3
1 2 3
are mutually vectors having with arc length
k k k k kds h du U du
U du U du U duds ds ds
= =
rArr perp
v
v v v
1uv
2uv
3uv
1dS
2dS
3dS
dS
MingsianMingsian R R BaiBai32
( ) ( )
( ) ( )
1 1 2 2 3 3 1 2 3 1 2 3 1 2 3
1 2 3 1 2 3
1
1 2 2 3 3 2 2 2 3 3 3 2 3 2 3 1
Differential volume
Differential On const coord surface
area
Gradi t
en
d U du U du U du u u u h h h du du du
d h h h du du du
ud U du U du h u du h u du h h du du u
df
τ
τ
σ
= times sdot = times sdot
=
=
= times = times =
= nabla
v v v v v v
v vv v v v
1 2 31 2 3
1 2 3 1 1 2 2 3 31 2 3
1 1 1 2 2 2 3 3 3
f f ff dr df du du duu u u
r r rdr du du du U du U du U duu u u
h u du h u du h u du
part part partsdot rarr = + +part part part
part part part= + + = + +part part part
= + +
v
v v v v v vv
v v v
=1 (right hand rule)
MingsianMingsian R R BaiBai33
Rectangular Cartesian Coordinates
ij
k
MingsianMingsian R R BaiBai34
Circular Cylindrical Coordinatescossin
xyz z
ρ ϕρ ϕ
===
Ref Arfken
( )P zρ ϕ
MingsianMingsian R R BaiBai35
Polar Spherical Coordinatessin cossin sincos
x ry rz r
θ ϕθ ϕθ
===
Ref Arfken
0
0
0
sin cos sin sin coscos cos cos sin sin
sin cos
θ ϕ θ ϕ θθ ϕ θ ϕ θϕ ϕ
= + +
= + minus= minus +
r i j kθ i j kφ i j
2
2
sinsin
dS r d ddV r dr d d
θ θ ϕ
θ θ ϕ
=
=( )P r θ ϕ
MingsianMingsian R R BaiBai36
Integral TheoremsHeart of this chapter
Interchange of dimensionality Div Curl Grad in integrals High-dimensional integration
rarr
1
Thm1 Divergence theorem Let be a closed region bounded by piecewise smooth orientable surface and let (con
I Divergence theor
tinuous derivatives up to the first order
em (Gauss theorem)
)
VS v Cisin
r
ˆ If denotes the outward unit normal on
the ˆn V S
v dV n v dS
n S
nablasdot = sdotint intr r
MingsianMingsian R R BaiBai37
jS
jP
jV
S
0
ˆBy definition ( ) lim
ˆ( )
ˆ( )
j
j
j
j
Sj V
j
j j S
j j Sj j
n v dSv P
V
v P V n v dS
v P V n v dS
Δ rarr
⎧ ⎫sdot⎪ ⎪nabla sdot ⎨ ⎬Δ⎪ ⎪⎩ ⎭
nabla sdot Δ asymp sdot
nabla sdot Δ = sdot
int
intsum sumint
rr
r r
r r
Pf
MingsianMingsian R R BaiBai38
0
After interior surfaces cancel out only exterior surface (Why Opposite normal vectors)
ˆ( )
volumes exterior surfaces
ˆlim
QE
survive
j
j
j j Sj j
V SV
v p V n v dS
v dV n v dSΔ rarr
nabla sdot Δ = sdot
rArr nablasdot = sdot
sum sumint
int int
r r
r r
D
MingsianMingsian R R BaiBai39
( )
scalar variable constant vectorvector variable
Alternative formsLet be a an arbitrary and
a
ˆ
1 Le
(
ˆt
V
V S
S
V
v ap
v d
v dV vn d
V n v dS
v va
va dV v a v a
S
nablasdot = sdot
= rArr
nablasdot = nabla sdot + nabla
=
sdot
nablaint intint int
int
r
v
r r
r r
r r rQ ( )
( )
ˆ)
2 Let ˆV S
V
V S
dV n va dS
v a p
a p
p dV n p d
d p
S
V a
⎛ ⎞= sdot⎜ ⎟
⎝ ⎠
= times rArr
nablasdot times = sdot nabla
times
times
nabla = times
int int
int int
int
r
r r r
r r v
r
vQ
r
( ) ( ) ( )
( ) ( )
ˆ
ˆV S
V S
a p dV n a p dS
a p dV a n p dS
⎛ ⎞⎡ ⎤minus sdot nablatimes = sdot times⎣ ⎦⎜ ⎟⎜ ⎟⎜ ⎟rArr minus sdot nablatimes = minus sdot times⎜ ⎟⎝ ⎠
int int
int int
r rv v
r rv v
MingsianMingsian R R BaiBai40
( )
( )
From Leibnitzs rule
(1-D) ( )
( ) ( )
B t
A t
d f x t dxdt
f x t dx B ttpart prime= +part
int( )
( )( ( ) ) ( )
B t
A tf B t t A tprimeminusint ( ( ) )
(assume ( ) ( ) )
f A t t
A t B t const= =
ContinuityEx The Equation
rate of increase of mass
mass densityV
dM d dVdt dt
ρ
ρ
= int
flow velocityvv
MingsianMingsian R R BaiBai41
ˆ ˆ ( )
(
Conservation
) ( Divergence theorem)
of mass
( ) 0
V S S
V
V
dV n v dS n v dSt
v dV
v dVt
ρ ρ ρ
ρ
ρ ρ⎡ ⎤⎢ ⎥⎣ ⎦
part = minus sdot = minus sdotpart
= minus nablasdot
partrArr +nablasdot =part
int int int
int
int
v v
vQ
v
ˆOn the other hand rate influof mass through n ox i t
Extended to 3-D for time-independent fixed boundary
(3-D) ( ) ( )
Thus
S
V V
V
S V n v dS
Sd f x y z t dV f x y z t dVdt tdM dVdt t
ρ
ρ
= minus sdot
part=part
part=part
int
int int
intv
Fixed control volume (Eulerian coordinates)Otherwise Raynoldrsquos transport theoremMoving control volume Lagrangian coordinates
MingsianMingsian R R BaiBai42
But the control volime is only arbitrary
In particular for flow
( ) 0
con
(continuity equati
stant
on)
Divergence the
0 incompress
Note is t
ible
orem h
V
v
vt
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
e mother of the other integral theorems
MingsianMingsian R R BaiBai43
Ex Archimedes principle ( ) The buoyant force ( ) on a body with volume immersed in a fluid is equal to the weight of the fluid
ˆ displaced by the object ie In the followin
B
V
gVρ= minusf k
阿基米德原理
浮力
ˆg figure the ˆ with and being the fluid density gravitational
acceleration depth and outward normal unit vector to the surface
fhydraulic pressure g z
g z
S
ρ
ρ
= minusp n
n
1
the modified divergence theorem
ˆ
Hint
scalar fuctionV S
vdV v dS v Cnabla = isinint int n
MingsianMingsian R R BaiBai44
Levelsurfacefluid
V
S
airyx
z
ˆf g zρ= minusp n
ˆPf ( ) ( )
ˆ ˆ ˆ ˆ ˆ (0 0 ) v v
B fS S V
V V
g z g zdS dS dV
g dV g dV gV
ρ ρ
ρ ρ ρ
= = minus = minus nabla
= minus + + = minus = minus
int int int
int int
f p n
i j k k k
k
MingsianMingsian R R BaiBai45
Note This can also be thought of as simple balance of forces (resultant force that acts on and weight of filled with fluid)S V
2
~ a little history about George Green Recall the diveregence theorem
ˆ
Let
II th
e Green
are arbitrary fun
s identiti
ctions
es
V S
v dV n v dS
v u vu v C
nablasdot = sdot
= nablaisin
int intv v
v
ˆ ( ) ( )
ˆ ( )V S
V S
u v dV n u v dS
u v u v dV un v dS
nablasdot nabla = sdot nabla
nabla sdotnabla + nablasdotnabla = sdotnabla
int int
int int
MingsianMingsian R R BaiBai46
2 -( ) -(1)V S
vu v u v dV u dSnpart
nabla sdotnabla + nabla =partint int firs(Greens identt ity)
2
If
( ) --(2)
(1) (2) V S
u vuv u v u dV v dSn
harrpart
nabla sdotnabla + nabla =part
minus
int int
( )2 2
V S
dv duu v v u dV u v dSdn dn
⎛ ⎞nabla minus nabla = minus⎜ ⎟⎝ ⎠int int
second(Greens identity)
MingsianMingsian R R BaiBai47
( )
( )
( ) ( )
1st id ( )
Greens identities in the 1D case integration by parts
b b bb ba a
a a ab
ba
ab
ba
a
v udx v u v u dx u v uv dx uv
u v u v vu dx vu
uv vu dx uv vu
primeprime prime prime prime prime prime primeprime prime= minus rArr + =
prime prime primeprime primeharr + =
primeprime primeprime prime primerArr minus = minus
int int int
int
2
2nd id ( )
Good for any adjoint operators in addition to Good also for and integration by parts Important for theory amp reciprocity
From th
Gree
e 1s
2-
t
ns functi
identity
D 1-
tak 1
D
e
on
u
nablararr
=
int
2
V S
vv dV dSnpartnabla =part
rArr int int2Ex 0 st 0 on the boundary 0 ( 1st id u u unabla = = rArr equiv 以 証)
MingsianMingsian R R BaiBai48
1
Thm1 Stokes theorem Let be defined in Let be a piecewise-smooth orientable surface bounded by a piec
III Stokes theorem and
ewise-smooth simple clos
Greens
ed curve Then
theore
m
v C R S
C
isinv
( )
ˆwhere is a unit normal to according t right hand rulo the e
ˆS C
n S
n v dS v dRsdot nablatimes = sdotint intvv v
(not necessarily planar)S
Cn vnablatimes v vv
MingsianMingsian R R BaiBai49
Pf
( )( )
circulatioWe have shown in deriving thatthe in -direction
ˆor
ˆ
n
z
z
j j j j j
j j j j jj j
vz v dx dy
v dR n v S
v dR n v S
nablatimes
=nablatimes
sdot = sdot nablatimes Δ
sdot = sdot nablatimes Δ
intsum sumint
v
v
vv v
vv v
C
S
jS
jC
jP
ˆ jn
jSΔ
MingsianMingsian R R BaiBai50
( )
( )ext surface
After the interior boundaries cancel out only the exterior boundaries (Why)
ˆ
ˆ0 QED
Alternative forms scalar
survive
va
j
j j j j jC
jC S
v dR n v S
S v dR n v dS
φ
sdot = sdot nablatimes Δ
Δ rarr rArr sdot = sdot nablatimes
sum sumint
int int
vv v
vv v
( )
riable constant vector vector variable
1 Let
2 Let
ˆ
ˆ S C
S C
a p
v n dR dR
n p dS dR p
a
v a p
φ φφ= rArr timesnabla =
timesnabla times= rArr timestimes =
int int
int int
v v
v v
v
v v
vv vv v
MingsianMingsian R R BaiBai51
( ) ( )
( )
( )
( ) ( )
1
Thm2 Greens theorem ( planar version of Stokes theorem)ˆ ˆ Let
pf By Stokes theoremˆ
ˆ ˆ ˆ ˆ ˆ ˆ
S C
S C
S C
v P x y i Q x y j C
n v dS v dR
k Pi Qj dS Pi Qj dx i dy
Q P dS Pdx Qdyx y
⎛ ⎞part partminus = +⎜ ⎟
= + isin
sdot nablatimes = sdot
⎡ ⎤sdot nablatimes + =
part part⎝ ⎠
+ sdot +⎣ ⎦
int int
int
int
int
int
v
vv v
( )
( )
( )
ˆ
ˆˆ ˆ
ˆˆ ˆ ˆ ˆ( ) ( )
0
QEDS C
j
i j kQ PPi Qj i j k
x y z x yP Q
Q P dS Pdx Qdyx y
⎛ ⎞part part part part partnablatimes + = = + + minus⎜ ⎟part part part part part⎝ ⎠
⎛ ⎞part partrArr minus = +⎜ ⎟part part⎝ ⎠int int
L L
S
C
i
j
x
y
MingsianMingsian R R BaiBai52
Potential TheorypotentiGeorge Green coined the term (l
a )
rarr 位能 勢能
例子 重力場 電磁場 流場 聲場 彈性力學
1
2
Thm1 If is in a simply connected region then the following statements are equivalent(i) There exists a scala
Conservative Field
potential functior such that
(ii
n
) 0
v
C
C
v
v R
φφ=nabla
nablatimes
isin
isin
=r
r
r
r in (Irrotational field curl-free)
cf 0 st solenoidal field divergence-free
0
(iii) in (Conservative field)C
Rv w v w
v R Rd
nablasdot = exist nabla
sdot =
= times
int
r r r
r
r
r
MingsianMingsian R R BaiBai53
(iv) is connecting and
independent o
Ex Think
f the pa
of dynamics is the force
thB
A
v dR A B
v
sdotintrr
r
1 2
1
0
Pf
(i) (ii) ( ) 0 (zero identity cg rule) irrotational
ˆ(ii) (iii) ( ) 0 (Stokes theorem)
(iii) (iv) ( ) 0
C S
C C C
C
v
v dR n v dS
v dR v dR
v dR
φ
=
rarr nablatimes = nablatimes nabla =
rarr sdot = sdot nablatimes =
rarr sdot = minus sdot =
rArr sdot =
int int
int int int
int
r
rr
rr r
r rr r
rr r
2
independent of pathC
v dRsdotintr
A
B
1C
2C
MingsianMingsian R R BaiBai54
1 2
(iv) (i) constant ( ) ( )
--(1)
But (2)
(1) (2) QED
Note on application
Check
B
C C A
v dR v dR B A d
v dR d
d dx dy dz dRx y z
v dR dR v R
φ φ φ
φφ φ φφ φ
φ φ
rarr sdot = sdot = = minus =
rArr sdot =part part part
= + + = nabla sdot minus minus minus minus minuspart part part
sdot = nabla sdot rArr = nabla forall
int int intr rr r
rr
r
r r rr r
use simpler path (iv)if 0
find (i)v
φ⎧
nablatimes = ⎨⎩
rr
A
B
x
y
Cz
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai28
Ref (1)Hildebrand advanced Calculus for application Prentice-Hall 1976 p306-313(2) Greenberg p383-389 (3) Arfken ch2
1 2 3
1 2 3
1 2 3
1 2 3
(
Orthogonal coordinate systems base vectors are not constant in s
) ( )( )( )( )
ˆˆ ˆ
pacep x y z p u u ux x u u uy y u u uz z u u u
r xi yj zk
primerarr
= +
rarr
===
+v
1uv
2uv3uv
1u
2u
3uP coord curve
xy
z
O
X Curvilinear Coordinates
MingsianMingsian R R BaiBai29
1
11
1 1 1
1
1
1 1 1
11 1
1 1
2 2 2 3 3 3
Tangent vector to the at
where is the arc length
1
where unit vector
Similarly
curv
e
u psr rU
u s us
rs
U h uds rh udu sU h u U h u
partpart part= =part part part
part =part
there4 =part= =part
= =
v vv
vQ
v v
vv
v vv v
1u cood curve
1uv
1du 1dS
drv
P
MingsianMingsian R R BaiBai30
3 31 2 1 21 2 3
1 2 32
curv
In summary scale factor
If is the arc length along a in any directione
k k k
kk k
k k
k k k
U h u
ds rh Udu u
ds h dus
du dudu du du dudr r r r U U Uds u ds u ds u ds ds ds ds
dr drds ds
=
part= = =part
=
part part part= + + = + +part part part
= sdot
v v
v v
v v v v v v v
v v 3 3
1 1
23 3 32 2
21 1 1
2 2 2 2 2 2 2 2 2 21 1 2 2 3 3 1 2 3
1 (for orthogonal coord)
jii j
i j
ji ii j i
i j i
dududr U Uds ds ds
dudu duU U hds ds ds
ds h du h du h du ds ds ds
= =
= = =
⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
= sdot
rarr = sdot =
rarr = + + = + +
sum sum
sumsum sum
v v v
v v
MingsianMingsian R R BaiBai31
1 1 2 2 3 3
1 2 3
are mutually vectors having with arc length
k k k k kds h du U du
U du U du U duds ds ds
= =
rArr perp
v
v v v
1uv
2uv
3uv
1dS
2dS
3dS
dS
MingsianMingsian R R BaiBai32
( ) ( )
( ) ( )
1 1 2 2 3 3 1 2 3 1 2 3 1 2 3
1 2 3 1 2 3
1
1 2 2 3 3 2 2 2 3 3 3 2 3 2 3 1
Differential volume
Differential On const coord surface
area
Gradi t
en
d U du U du U du u u u h h h du du du
d h h h du du du
ud U du U du h u du h u du h h du du u
df
τ
τ
σ
= times sdot = times sdot
=
=
= times = times =
= nabla
v v v v v v
v vv v v v
1 2 31 2 3
1 2 3 1 1 2 2 3 31 2 3
1 1 1 2 2 2 3 3 3
f f ff dr df du du duu u u
r r rdr du du du U du U du U duu u u
h u du h u du h u du
part part partsdot rarr = + +part part part
part part part= + + = + +part part part
= + +
v
v v v v v vv
v v v
=1 (right hand rule)
MingsianMingsian R R BaiBai33
Rectangular Cartesian Coordinates
ij
k
MingsianMingsian R R BaiBai34
Circular Cylindrical Coordinatescossin
xyz z
ρ ϕρ ϕ
===
Ref Arfken
( )P zρ ϕ
MingsianMingsian R R BaiBai35
Polar Spherical Coordinatessin cossin sincos
x ry rz r
θ ϕθ ϕθ
===
Ref Arfken
0
0
0
sin cos sin sin coscos cos cos sin sin
sin cos
θ ϕ θ ϕ θθ ϕ θ ϕ θϕ ϕ
= + +
= + minus= minus +
r i j kθ i j kφ i j
2
2
sinsin
dS r d ddV r dr d d
θ θ ϕ
θ θ ϕ
=
=( )P r θ ϕ
MingsianMingsian R R BaiBai36
Integral TheoremsHeart of this chapter
Interchange of dimensionality Div Curl Grad in integrals High-dimensional integration
rarr
1
Thm1 Divergence theorem Let be a closed region bounded by piecewise smooth orientable surface and let (con
I Divergence theor
tinuous derivatives up to the first order
em (Gauss theorem)
)
VS v Cisin
r
ˆ If denotes the outward unit normal on
the ˆn V S
v dV n v dS
n S
nablasdot = sdotint intr r
MingsianMingsian R R BaiBai37
jS
jP
jV
S
0
ˆBy definition ( ) lim
ˆ( )
ˆ( )
j
j
j
j
Sj V
j
j j S
j j Sj j
n v dSv P
V
v P V n v dS
v P V n v dS
Δ rarr
⎧ ⎫sdot⎪ ⎪nabla sdot ⎨ ⎬Δ⎪ ⎪⎩ ⎭
nabla sdot Δ asymp sdot
nabla sdot Δ = sdot
int
intsum sumint
rr
r r
r r
Pf
MingsianMingsian R R BaiBai38
0
After interior surfaces cancel out only exterior surface (Why Opposite normal vectors)
ˆ( )
volumes exterior surfaces
ˆlim
QE
survive
j
j
j j Sj j
V SV
v p V n v dS
v dV n v dSΔ rarr
nabla sdot Δ = sdot
rArr nablasdot = sdot
sum sumint
int int
r r
r r
D
MingsianMingsian R R BaiBai39
( )
scalar variable constant vectorvector variable
Alternative formsLet be a an arbitrary and
a
ˆ
1 Le
(
ˆt
V
V S
S
V
v ap
v d
v dV vn d
V n v dS
v va
va dV v a v a
S
nablasdot = sdot
= rArr
nablasdot = nabla sdot + nabla
=
sdot
nablaint intint int
int
r
v
r r
r r
r r rQ ( )
( )
ˆ)
2 Let ˆV S
V
V S
dV n va dS
v a p
a p
p dV n p d
d p
S
V a
⎛ ⎞= sdot⎜ ⎟
⎝ ⎠
= times rArr
nablasdot times = sdot nabla
times
times
nabla = times
int int
int int
int
r
r r r
r r v
r
vQ
r
( ) ( ) ( )
( ) ( )
ˆ
ˆV S
V S
a p dV n a p dS
a p dV a n p dS
⎛ ⎞⎡ ⎤minus sdot nablatimes = sdot times⎣ ⎦⎜ ⎟⎜ ⎟⎜ ⎟rArr minus sdot nablatimes = minus sdot times⎜ ⎟⎝ ⎠
int int
int int
r rv v
r rv v
MingsianMingsian R R BaiBai40
( )
( )
From Leibnitzs rule
(1-D) ( )
( ) ( )
B t
A t
d f x t dxdt
f x t dx B ttpart prime= +part
int( )
( )( ( ) ) ( )
B t
A tf B t t A tprimeminusint ( ( ) )
(assume ( ) ( ) )
f A t t
A t B t const= =
ContinuityEx The Equation
rate of increase of mass
mass densityV
dM d dVdt dt
ρ
ρ
= int
flow velocityvv
MingsianMingsian R R BaiBai41
ˆ ˆ ( )
(
Conservation
) ( Divergence theorem)
of mass
( ) 0
V S S
V
V
dV n v dS n v dSt
v dV
v dVt
ρ ρ ρ
ρ
ρ ρ⎡ ⎤⎢ ⎥⎣ ⎦
part = minus sdot = minus sdotpart
= minus nablasdot
partrArr +nablasdot =part
int int int
int
int
v v
vQ
v
ˆOn the other hand rate influof mass through n ox i t
Extended to 3-D for time-independent fixed boundary
(3-D) ( ) ( )
Thus
S
V V
V
S V n v dS
Sd f x y z t dV f x y z t dVdt tdM dVdt t
ρ
ρ
= minus sdot
part=part
part=part
int
int int
intv
Fixed control volume (Eulerian coordinates)Otherwise Raynoldrsquos transport theoremMoving control volume Lagrangian coordinates
MingsianMingsian R R BaiBai42
But the control volime is only arbitrary
In particular for flow
( ) 0
con
(continuity equati
stant
on)
Divergence the
0 incompress
Note is t
ible
orem h
V
v
vt
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
e mother of the other integral theorems
MingsianMingsian R R BaiBai43
Ex Archimedes principle ( ) The buoyant force ( ) on a body with volume immersed in a fluid is equal to the weight of the fluid
ˆ displaced by the object ie In the followin
B
V
gVρ= minusf k
阿基米德原理
浮力
ˆg figure the ˆ with and being the fluid density gravitational
acceleration depth and outward normal unit vector to the surface
fhydraulic pressure g z
g z
S
ρ
ρ
= minusp n
n
1
the modified divergence theorem
ˆ
Hint
scalar fuctionV S
vdV v dS v Cnabla = isinint int n
MingsianMingsian R R BaiBai44
Levelsurfacefluid
V
S
airyx
z
ˆf g zρ= minusp n
ˆPf ( ) ( )
ˆ ˆ ˆ ˆ ˆ (0 0 ) v v
B fS S V
V V
g z g zdS dS dV
g dV g dV gV
ρ ρ
ρ ρ ρ
= = minus = minus nabla
= minus + + = minus = minus
int int int
int int
f p n
i j k k k
k
MingsianMingsian R R BaiBai45
Note This can also be thought of as simple balance of forces (resultant force that acts on and weight of filled with fluid)S V
2
~ a little history about George Green Recall the diveregence theorem
ˆ
Let
II th
e Green
are arbitrary fun
s identiti
ctions
es
V S
v dV n v dS
v u vu v C
nablasdot = sdot
= nablaisin
int intv v
v
ˆ ( ) ( )
ˆ ( )V S
V S
u v dV n u v dS
u v u v dV un v dS
nablasdot nabla = sdot nabla
nabla sdotnabla + nablasdotnabla = sdotnabla
int int
int int
MingsianMingsian R R BaiBai46
2 -( ) -(1)V S
vu v u v dV u dSnpart
nabla sdotnabla + nabla =partint int firs(Greens identt ity)
2
If
( ) --(2)
(1) (2) V S
u vuv u v u dV v dSn
harrpart
nabla sdotnabla + nabla =part
minus
int int
( )2 2
V S
dv duu v v u dV u v dSdn dn
⎛ ⎞nabla minus nabla = minus⎜ ⎟⎝ ⎠int int
second(Greens identity)
MingsianMingsian R R BaiBai47
( )
( )
( ) ( )
1st id ( )
Greens identities in the 1D case integration by parts
b b bb ba a
a a ab
ba
ab
ba
a
v udx v u v u dx u v uv dx uv
u v u v vu dx vu
uv vu dx uv vu
primeprime prime prime prime prime prime primeprime prime= minus rArr + =
prime prime primeprime primeharr + =
primeprime primeprime prime primerArr minus = minus
int int int
int
2
2nd id ( )
Good for any adjoint operators in addition to Good also for and integration by parts Important for theory amp reciprocity
From th
Gree
e 1s
2-
t
ns functi
identity
D 1-
tak 1
D
e
on
u
nablararr
=
int
2
V S
vv dV dSnpartnabla =part
rArr int int2Ex 0 st 0 on the boundary 0 ( 1st id u u unabla = = rArr equiv 以 証)
MingsianMingsian R R BaiBai48
1
Thm1 Stokes theorem Let be defined in Let be a piecewise-smooth orientable surface bounded by a piec
III Stokes theorem and
ewise-smooth simple clos
Greens
ed curve Then
theore
m
v C R S
C
isinv
( )
ˆwhere is a unit normal to according t right hand rulo the e
ˆS C
n S
n v dS v dRsdot nablatimes = sdotint intvv v
(not necessarily planar)S
Cn vnablatimes v vv
MingsianMingsian R R BaiBai49
Pf
( )( )
circulatioWe have shown in deriving thatthe in -direction
ˆor
ˆ
n
z
z
j j j j j
j j j j jj j
vz v dx dy
v dR n v S
v dR n v S
nablatimes
=nablatimes
sdot = sdot nablatimes Δ
sdot = sdot nablatimes Δ
intsum sumint
v
v
vv v
vv v
C
S
jS
jC
jP
ˆ jn
jSΔ
MingsianMingsian R R BaiBai50
( )
( )ext surface
After the interior boundaries cancel out only the exterior boundaries (Why)
ˆ
ˆ0 QED
Alternative forms scalar
survive
va
j
j j j j jC
jC S
v dR n v S
S v dR n v dS
φ
sdot = sdot nablatimes Δ
Δ rarr rArr sdot = sdot nablatimes
sum sumint
int int
vv v
vv v
( )
riable constant vector vector variable
1 Let
2 Let
ˆ
ˆ S C
S C
a p
v n dR dR
n p dS dR p
a
v a p
φ φφ= rArr timesnabla =
timesnabla times= rArr timestimes =
int int
int int
v v
v v
v
v v
vv vv v
MingsianMingsian R R BaiBai51
( ) ( )
( )
( )
( ) ( )
1
Thm2 Greens theorem ( planar version of Stokes theorem)ˆ ˆ Let
pf By Stokes theoremˆ
ˆ ˆ ˆ ˆ ˆ ˆ
S C
S C
S C
v P x y i Q x y j C
n v dS v dR
k Pi Qj dS Pi Qj dx i dy
Q P dS Pdx Qdyx y
⎛ ⎞part partminus = +⎜ ⎟
= + isin
sdot nablatimes = sdot
⎡ ⎤sdot nablatimes + =
part part⎝ ⎠
+ sdot +⎣ ⎦
int int
int
int
int
int
v
vv v
( )
( )
( )
ˆ
ˆˆ ˆ
ˆˆ ˆ ˆ ˆ( ) ( )
0
QEDS C
j
i j kQ PPi Qj i j k
x y z x yP Q
Q P dS Pdx Qdyx y
⎛ ⎞part part part part partnablatimes + = = + + minus⎜ ⎟part part part part part⎝ ⎠
⎛ ⎞part partrArr minus = +⎜ ⎟part part⎝ ⎠int int
L L
S
C
i
j
x
y
MingsianMingsian R R BaiBai52
Potential TheorypotentiGeorge Green coined the term (l
a )
rarr 位能 勢能
例子 重力場 電磁場 流場 聲場 彈性力學
1
2
Thm1 If is in a simply connected region then the following statements are equivalent(i) There exists a scala
Conservative Field
potential functior such that
(ii
n
) 0
v
C
C
v
v R
φφ=nabla
nablatimes
isin
isin
=r
r
r
r in (Irrotational field curl-free)
cf 0 st solenoidal field divergence-free
0
(iii) in (Conservative field)C
Rv w v w
v R Rd
nablasdot = exist nabla
sdot =
= times
int
r r r
r
r
r
MingsianMingsian R R BaiBai53
(iv) is connecting and
independent o
Ex Think
f the pa
of dynamics is the force
thB
A
v dR A B
v
sdotintrr
r
1 2
1
0
Pf
(i) (ii) ( ) 0 (zero identity cg rule) irrotational
ˆ(ii) (iii) ( ) 0 (Stokes theorem)
(iii) (iv) ( ) 0
C S
C C C
C
v
v dR n v dS
v dR v dR
v dR
φ
=
rarr nablatimes = nablatimes nabla =
rarr sdot = sdot nablatimes =
rarr sdot = minus sdot =
rArr sdot =
int int
int int int
int
r
rr
rr r
r rr r
rr r
2
independent of pathC
v dRsdotintr
A
B
1C
2C
MingsianMingsian R R BaiBai54
1 2
(iv) (i) constant ( ) ( )
--(1)
But (2)
(1) (2) QED
Note on application
Check
B
C C A
v dR v dR B A d
v dR d
d dx dy dz dRx y z
v dR dR v R
φ φ φ
φφ φ φφ φ
φ φ
rarr sdot = sdot = = minus =
rArr sdot =part part part
= + + = nabla sdot minus minus minus minus minuspart part part
sdot = nabla sdot rArr = nabla forall
int int intr rr r
rr
r
r r rr r
use simpler path (iv)if 0
find (i)v
φ⎧
nablatimes = ⎨⎩
rr
A
B
x
y
Cz
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai29
1
11
1 1 1
1
1
1 1 1
11 1
1 1
2 2 2 3 3 3
Tangent vector to the at
where is the arc length
1
where unit vector
Similarly
curv
e
u psr rU
u s us
rs
U h uds rh udu sU h u U h u
partpart part= =part part part
part =part
there4 =part= =part
= =
v vv
vQ
v v
vv
v vv v
1u cood curve
1uv
1du 1dS
drv
P
MingsianMingsian R R BaiBai30
3 31 2 1 21 2 3
1 2 32
curv
In summary scale factor
If is the arc length along a in any directione
k k k
kk k
k k
k k k
U h u
ds rh Udu u
ds h dus
du dudu du du dudr r r r U U Uds u ds u ds u ds ds ds ds
dr drds ds
=
part= = =part
=
part part part= + + = + +part part part
= sdot
v v
v v
v v v v v v v
v v 3 3
1 1
23 3 32 2
21 1 1
2 2 2 2 2 2 2 2 2 21 1 2 2 3 3 1 2 3
1 (for orthogonal coord)
jii j
i j
ji ii j i
i j i
dududr U Uds ds ds
dudu duU U hds ds ds
ds h du h du h du ds ds ds
= =
= = =
⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
= sdot
rarr = sdot =
rarr = + + = + +
sum sum
sumsum sum
v v v
v v
MingsianMingsian R R BaiBai31
1 1 2 2 3 3
1 2 3
are mutually vectors having with arc length
k k k k kds h du U du
U du U du U duds ds ds
= =
rArr perp
v
v v v
1uv
2uv
3uv
1dS
2dS
3dS
dS
MingsianMingsian R R BaiBai32
( ) ( )
( ) ( )
1 1 2 2 3 3 1 2 3 1 2 3 1 2 3
1 2 3 1 2 3
1
1 2 2 3 3 2 2 2 3 3 3 2 3 2 3 1
Differential volume
Differential On const coord surface
area
Gradi t
en
d U du U du U du u u u h h h du du du
d h h h du du du
ud U du U du h u du h u du h h du du u
df
τ
τ
σ
= times sdot = times sdot
=
=
= times = times =
= nabla
v v v v v v
v vv v v v
1 2 31 2 3
1 2 3 1 1 2 2 3 31 2 3
1 1 1 2 2 2 3 3 3
f f ff dr df du du duu u u
r r rdr du du du U du U du U duu u u
h u du h u du h u du
part part partsdot rarr = + +part part part
part part part= + + = + +part part part
= + +
v
v v v v v vv
v v v
=1 (right hand rule)
MingsianMingsian R R BaiBai33
Rectangular Cartesian Coordinates
ij
k
MingsianMingsian R R BaiBai34
Circular Cylindrical Coordinatescossin
xyz z
ρ ϕρ ϕ
===
Ref Arfken
( )P zρ ϕ
MingsianMingsian R R BaiBai35
Polar Spherical Coordinatessin cossin sincos
x ry rz r
θ ϕθ ϕθ
===
Ref Arfken
0
0
0
sin cos sin sin coscos cos cos sin sin
sin cos
θ ϕ θ ϕ θθ ϕ θ ϕ θϕ ϕ
= + +
= + minus= minus +
r i j kθ i j kφ i j
2
2
sinsin
dS r d ddV r dr d d
θ θ ϕ
θ θ ϕ
=
=( )P r θ ϕ
MingsianMingsian R R BaiBai36
Integral TheoremsHeart of this chapter
Interchange of dimensionality Div Curl Grad in integrals High-dimensional integration
rarr
1
Thm1 Divergence theorem Let be a closed region bounded by piecewise smooth orientable surface and let (con
I Divergence theor
tinuous derivatives up to the first order
em (Gauss theorem)
)
VS v Cisin
r
ˆ If denotes the outward unit normal on
the ˆn V S
v dV n v dS
n S
nablasdot = sdotint intr r
MingsianMingsian R R BaiBai37
jS
jP
jV
S
0
ˆBy definition ( ) lim
ˆ( )
ˆ( )
j
j
j
j
Sj V
j
j j S
j j Sj j
n v dSv P
V
v P V n v dS
v P V n v dS
Δ rarr
⎧ ⎫sdot⎪ ⎪nabla sdot ⎨ ⎬Δ⎪ ⎪⎩ ⎭
nabla sdot Δ asymp sdot
nabla sdot Δ = sdot
int
intsum sumint
rr
r r
r r
Pf
MingsianMingsian R R BaiBai38
0
After interior surfaces cancel out only exterior surface (Why Opposite normal vectors)
ˆ( )
volumes exterior surfaces
ˆlim
QE
survive
j
j
j j Sj j
V SV
v p V n v dS
v dV n v dSΔ rarr
nabla sdot Δ = sdot
rArr nablasdot = sdot
sum sumint
int int
r r
r r
D
MingsianMingsian R R BaiBai39
( )
scalar variable constant vectorvector variable
Alternative formsLet be a an arbitrary and
a
ˆ
1 Le
(
ˆt
V
V S
S
V
v ap
v d
v dV vn d
V n v dS
v va
va dV v a v a
S
nablasdot = sdot
= rArr
nablasdot = nabla sdot + nabla
=
sdot
nablaint intint int
int
r
v
r r
r r
r r rQ ( )
( )
ˆ)
2 Let ˆV S
V
V S
dV n va dS
v a p
a p
p dV n p d
d p
S
V a
⎛ ⎞= sdot⎜ ⎟
⎝ ⎠
= times rArr
nablasdot times = sdot nabla
times
times
nabla = times
int int
int int
int
r
r r r
r r v
r
vQ
r
( ) ( ) ( )
( ) ( )
ˆ
ˆV S
V S
a p dV n a p dS
a p dV a n p dS
⎛ ⎞⎡ ⎤minus sdot nablatimes = sdot times⎣ ⎦⎜ ⎟⎜ ⎟⎜ ⎟rArr minus sdot nablatimes = minus sdot times⎜ ⎟⎝ ⎠
int int
int int
r rv v
r rv v
MingsianMingsian R R BaiBai40
( )
( )
From Leibnitzs rule
(1-D) ( )
( ) ( )
B t
A t
d f x t dxdt
f x t dx B ttpart prime= +part
int( )
( )( ( ) ) ( )
B t
A tf B t t A tprimeminusint ( ( ) )
(assume ( ) ( ) )
f A t t
A t B t const= =
ContinuityEx The Equation
rate of increase of mass
mass densityV
dM d dVdt dt
ρ
ρ
= int
flow velocityvv
MingsianMingsian R R BaiBai41
ˆ ˆ ( )
(
Conservation
) ( Divergence theorem)
of mass
( ) 0
V S S
V
V
dV n v dS n v dSt
v dV
v dVt
ρ ρ ρ
ρ
ρ ρ⎡ ⎤⎢ ⎥⎣ ⎦
part = minus sdot = minus sdotpart
= minus nablasdot
partrArr +nablasdot =part
int int int
int
int
v v
vQ
v
ˆOn the other hand rate influof mass through n ox i t
Extended to 3-D for time-independent fixed boundary
(3-D) ( ) ( )
Thus
S
V V
V
S V n v dS
Sd f x y z t dV f x y z t dVdt tdM dVdt t
ρ
ρ
= minus sdot
part=part
part=part
int
int int
intv
Fixed control volume (Eulerian coordinates)Otherwise Raynoldrsquos transport theoremMoving control volume Lagrangian coordinates
MingsianMingsian R R BaiBai42
But the control volime is only arbitrary
In particular for flow
( ) 0
con
(continuity equati
stant
on)
Divergence the
0 incompress
Note is t
ible
orem h
V
v
vt
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
e mother of the other integral theorems
MingsianMingsian R R BaiBai43
Ex Archimedes principle ( ) The buoyant force ( ) on a body with volume immersed in a fluid is equal to the weight of the fluid
ˆ displaced by the object ie In the followin
B
V
gVρ= minusf k
阿基米德原理
浮力
ˆg figure the ˆ with and being the fluid density gravitational
acceleration depth and outward normal unit vector to the surface
fhydraulic pressure g z
g z
S
ρ
ρ
= minusp n
n
1
the modified divergence theorem
ˆ
Hint
scalar fuctionV S
vdV v dS v Cnabla = isinint int n
MingsianMingsian R R BaiBai44
Levelsurfacefluid
V
S
airyx
z
ˆf g zρ= minusp n
ˆPf ( ) ( )
ˆ ˆ ˆ ˆ ˆ (0 0 ) v v
B fS S V
V V
g z g zdS dS dV
g dV g dV gV
ρ ρ
ρ ρ ρ
= = minus = minus nabla
= minus + + = minus = minus
int int int
int int
f p n
i j k k k
k
MingsianMingsian R R BaiBai45
Note This can also be thought of as simple balance of forces (resultant force that acts on and weight of filled with fluid)S V
2
~ a little history about George Green Recall the diveregence theorem
ˆ
Let
II th
e Green
are arbitrary fun
s identiti
ctions
es
V S
v dV n v dS
v u vu v C
nablasdot = sdot
= nablaisin
int intv v
v
ˆ ( ) ( )
ˆ ( )V S
V S
u v dV n u v dS
u v u v dV un v dS
nablasdot nabla = sdot nabla
nabla sdotnabla + nablasdotnabla = sdotnabla
int int
int int
MingsianMingsian R R BaiBai46
2 -( ) -(1)V S
vu v u v dV u dSnpart
nabla sdotnabla + nabla =partint int firs(Greens identt ity)
2
If
( ) --(2)
(1) (2) V S
u vuv u v u dV v dSn
harrpart
nabla sdotnabla + nabla =part
minus
int int
( )2 2
V S
dv duu v v u dV u v dSdn dn
⎛ ⎞nabla minus nabla = minus⎜ ⎟⎝ ⎠int int
second(Greens identity)
MingsianMingsian R R BaiBai47
( )
( )
( ) ( )
1st id ( )
Greens identities in the 1D case integration by parts
b b bb ba a
a a ab
ba
ab
ba
a
v udx v u v u dx u v uv dx uv
u v u v vu dx vu
uv vu dx uv vu
primeprime prime prime prime prime prime primeprime prime= minus rArr + =
prime prime primeprime primeharr + =
primeprime primeprime prime primerArr minus = minus
int int int
int
2
2nd id ( )
Good for any adjoint operators in addition to Good also for and integration by parts Important for theory amp reciprocity
From th
Gree
e 1s
2-
t
ns functi
identity
D 1-
tak 1
D
e
on
u
nablararr
=
int
2
V S
vv dV dSnpartnabla =part
rArr int int2Ex 0 st 0 on the boundary 0 ( 1st id u u unabla = = rArr equiv 以 証)
MingsianMingsian R R BaiBai48
1
Thm1 Stokes theorem Let be defined in Let be a piecewise-smooth orientable surface bounded by a piec
III Stokes theorem and
ewise-smooth simple clos
Greens
ed curve Then
theore
m
v C R S
C
isinv
( )
ˆwhere is a unit normal to according t right hand rulo the e
ˆS C
n S
n v dS v dRsdot nablatimes = sdotint intvv v
(not necessarily planar)S
Cn vnablatimes v vv
MingsianMingsian R R BaiBai49
Pf
( )( )
circulatioWe have shown in deriving thatthe in -direction
ˆor
ˆ
n
z
z
j j j j j
j j j j jj j
vz v dx dy
v dR n v S
v dR n v S
nablatimes
=nablatimes
sdot = sdot nablatimes Δ
sdot = sdot nablatimes Δ
intsum sumint
v
v
vv v
vv v
C
S
jS
jC
jP
ˆ jn
jSΔ
MingsianMingsian R R BaiBai50
( )
( )ext surface
After the interior boundaries cancel out only the exterior boundaries (Why)
ˆ
ˆ0 QED
Alternative forms scalar
survive
va
j
j j j j jC
jC S
v dR n v S
S v dR n v dS
φ
sdot = sdot nablatimes Δ
Δ rarr rArr sdot = sdot nablatimes
sum sumint
int int
vv v
vv v
( )
riable constant vector vector variable
1 Let
2 Let
ˆ
ˆ S C
S C
a p
v n dR dR
n p dS dR p
a
v a p
φ φφ= rArr timesnabla =
timesnabla times= rArr timestimes =
int int
int int
v v
v v
v
v v
vv vv v
MingsianMingsian R R BaiBai51
( ) ( )
( )
( )
( ) ( )
1
Thm2 Greens theorem ( planar version of Stokes theorem)ˆ ˆ Let
pf By Stokes theoremˆ
ˆ ˆ ˆ ˆ ˆ ˆ
S C
S C
S C
v P x y i Q x y j C
n v dS v dR
k Pi Qj dS Pi Qj dx i dy
Q P dS Pdx Qdyx y
⎛ ⎞part partminus = +⎜ ⎟
= + isin
sdot nablatimes = sdot
⎡ ⎤sdot nablatimes + =
part part⎝ ⎠
+ sdot +⎣ ⎦
int int
int
int
int
int
v
vv v
( )
( )
( )
ˆ
ˆˆ ˆ
ˆˆ ˆ ˆ ˆ( ) ( )
0
QEDS C
j
i j kQ PPi Qj i j k
x y z x yP Q
Q P dS Pdx Qdyx y
⎛ ⎞part part part part partnablatimes + = = + + minus⎜ ⎟part part part part part⎝ ⎠
⎛ ⎞part partrArr minus = +⎜ ⎟part part⎝ ⎠int int
L L
S
C
i
j
x
y
MingsianMingsian R R BaiBai52
Potential TheorypotentiGeorge Green coined the term (l
a )
rarr 位能 勢能
例子 重力場 電磁場 流場 聲場 彈性力學
1
2
Thm1 If is in a simply connected region then the following statements are equivalent(i) There exists a scala
Conservative Field
potential functior such that
(ii
n
) 0
v
C
C
v
v R
φφ=nabla
nablatimes
isin
isin
=r
r
r
r in (Irrotational field curl-free)
cf 0 st solenoidal field divergence-free
0
(iii) in (Conservative field)C
Rv w v w
v R Rd
nablasdot = exist nabla
sdot =
= times
int
r r r
r
r
r
MingsianMingsian R R BaiBai53
(iv) is connecting and
independent o
Ex Think
f the pa
of dynamics is the force
thB
A
v dR A B
v
sdotintrr
r
1 2
1
0
Pf
(i) (ii) ( ) 0 (zero identity cg rule) irrotational
ˆ(ii) (iii) ( ) 0 (Stokes theorem)
(iii) (iv) ( ) 0
C S
C C C
C
v
v dR n v dS
v dR v dR
v dR
φ
=
rarr nablatimes = nablatimes nabla =
rarr sdot = sdot nablatimes =
rarr sdot = minus sdot =
rArr sdot =
int int
int int int
int
r
rr
rr r
r rr r
rr r
2
independent of pathC
v dRsdotintr
A
B
1C
2C
MingsianMingsian R R BaiBai54
1 2
(iv) (i) constant ( ) ( )
--(1)
But (2)
(1) (2) QED
Note on application
Check
B
C C A
v dR v dR B A d
v dR d
d dx dy dz dRx y z
v dR dR v R
φ φ φ
φφ φ φφ φ
φ φ
rarr sdot = sdot = = minus =
rArr sdot =part part part
= + + = nabla sdot minus minus minus minus minuspart part part
sdot = nabla sdot rArr = nabla forall
int int intr rr r
rr
r
r r rr r
use simpler path (iv)if 0
find (i)v
φ⎧
nablatimes = ⎨⎩
rr
A
B
x
y
Cz
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai30
3 31 2 1 21 2 3
1 2 32
curv
In summary scale factor
If is the arc length along a in any directione
k k k
kk k
k k
k k k
U h u
ds rh Udu u
ds h dus
du dudu du du dudr r r r U U Uds u ds u ds u ds ds ds ds
dr drds ds
=
part= = =part
=
part part part= + + = + +part part part
= sdot
v v
v v
v v v v v v v
v v 3 3
1 1
23 3 32 2
21 1 1
2 2 2 2 2 2 2 2 2 21 1 2 2 3 3 1 2 3
1 (for orthogonal coord)
jii j
i j
ji ii j i
i j i
dududr U Uds ds ds
dudu duU U hds ds ds
ds h du h du h du ds ds ds
= =
= = =
⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
= sdot
rarr = sdot =
rarr = + + = + +
sum sum
sumsum sum
v v v
v v
MingsianMingsian R R BaiBai31
1 1 2 2 3 3
1 2 3
are mutually vectors having with arc length
k k k k kds h du U du
U du U du U duds ds ds
= =
rArr perp
v
v v v
1uv
2uv
3uv
1dS
2dS
3dS
dS
MingsianMingsian R R BaiBai32
( ) ( )
( ) ( )
1 1 2 2 3 3 1 2 3 1 2 3 1 2 3
1 2 3 1 2 3
1
1 2 2 3 3 2 2 2 3 3 3 2 3 2 3 1
Differential volume
Differential On const coord surface
area
Gradi t
en
d U du U du U du u u u h h h du du du
d h h h du du du
ud U du U du h u du h u du h h du du u
df
τ
τ
σ
= times sdot = times sdot
=
=
= times = times =
= nabla
v v v v v v
v vv v v v
1 2 31 2 3
1 2 3 1 1 2 2 3 31 2 3
1 1 1 2 2 2 3 3 3
f f ff dr df du du duu u u
r r rdr du du du U du U du U duu u u
h u du h u du h u du
part part partsdot rarr = + +part part part
part part part= + + = + +part part part
= + +
v
v v v v v vv
v v v
=1 (right hand rule)
MingsianMingsian R R BaiBai33
Rectangular Cartesian Coordinates
ij
k
MingsianMingsian R R BaiBai34
Circular Cylindrical Coordinatescossin
xyz z
ρ ϕρ ϕ
===
Ref Arfken
( )P zρ ϕ
MingsianMingsian R R BaiBai35
Polar Spherical Coordinatessin cossin sincos
x ry rz r
θ ϕθ ϕθ
===
Ref Arfken
0
0
0
sin cos sin sin coscos cos cos sin sin
sin cos
θ ϕ θ ϕ θθ ϕ θ ϕ θϕ ϕ
= + +
= + minus= minus +
r i j kθ i j kφ i j
2
2
sinsin
dS r d ddV r dr d d
θ θ ϕ
θ θ ϕ
=
=( )P r θ ϕ
MingsianMingsian R R BaiBai36
Integral TheoremsHeart of this chapter
Interchange of dimensionality Div Curl Grad in integrals High-dimensional integration
rarr
1
Thm1 Divergence theorem Let be a closed region bounded by piecewise smooth orientable surface and let (con
I Divergence theor
tinuous derivatives up to the first order
em (Gauss theorem)
)
VS v Cisin
r
ˆ If denotes the outward unit normal on
the ˆn V S
v dV n v dS
n S
nablasdot = sdotint intr r
MingsianMingsian R R BaiBai37
jS
jP
jV
S
0
ˆBy definition ( ) lim
ˆ( )
ˆ( )
j
j
j
j
Sj V
j
j j S
j j Sj j
n v dSv P
V
v P V n v dS
v P V n v dS
Δ rarr
⎧ ⎫sdot⎪ ⎪nabla sdot ⎨ ⎬Δ⎪ ⎪⎩ ⎭
nabla sdot Δ asymp sdot
nabla sdot Δ = sdot
int
intsum sumint
rr
r r
r r
Pf
MingsianMingsian R R BaiBai38
0
After interior surfaces cancel out only exterior surface (Why Opposite normal vectors)
ˆ( )
volumes exterior surfaces
ˆlim
QE
survive
j
j
j j Sj j
V SV
v p V n v dS
v dV n v dSΔ rarr
nabla sdot Δ = sdot
rArr nablasdot = sdot
sum sumint
int int
r r
r r
D
MingsianMingsian R R BaiBai39
( )
scalar variable constant vectorvector variable
Alternative formsLet be a an arbitrary and
a
ˆ
1 Le
(
ˆt
V
V S
S
V
v ap
v d
v dV vn d
V n v dS
v va
va dV v a v a
S
nablasdot = sdot
= rArr
nablasdot = nabla sdot + nabla
=
sdot
nablaint intint int
int
r
v
r r
r r
r r rQ ( )
( )
ˆ)
2 Let ˆV S
V
V S
dV n va dS
v a p
a p
p dV n p d
d p
S
V a
⎛ ⎞= sdot⎜ ⎟
⎝ ⎠
= times rArr
nablasdot times = sdot nabla
times
times
nabla = times
int int
int int
int
r
r r r
r r v
r
vQ
r
( ) ( ) ( )
( ) ( )
ˆ
ˆV S
V S
a p dV n a p dS
a p dV a n p dS
⎛ ⎞⎡ ⎤minus sdot nablatimes = sdot times⎣ ⎦⎜ ⎟⎜ ⎟⎜ ⎟rArr minus sdot nablatimes = minus sdot times⎜ ⎟⎝ ⎠
int int
int int
r rv v
r rv v
MingsianMingsian R R BaiBai40
( )
( )
From Leibnitzs rule
(1-D) ( )
( ) ( )
B t
A t
d f x t dxdt
f x t dx B ttpart prime= +part
int( )
( )( ( ) ) ( )
B t
A tf B t t A tprimeminusint ( ( ) )
(assume ( ) ( ) )
f A t t
A t B t const= =
ContinuityEx The Equation
rate of increase of mass
mass densityV
dM d dVdt dt
ρ
ρ
= int
flow velocityvv
MingsianMingsian R R BaiBai41
ˆ ˆ ( )
(
Conservation
) ( Divergence theorem)
of mass
( ) 0
V S S
V
V
dV n v dS n v dSt
v dV
v dVt
ρ ρ ρ
ρ
ρ ρ⎡ ⎤⎢ ⎥⎣ ⎦
part = minus sdot = minus sdotpart
= minus nablasdot
partrArr +nablasdot =part
int int int
int
int
v v
vQ
v
ˆOn the other hand rate influof mass through n ox i t
Extended to 3-D for time-independent fixed boundary
(3-D) ( ) ( )
Thus
S
V V
V
S V n v dS
Sd f x y z t dV f x y z t dVdt tdM dVdt t
ρ
ρ
= minus sdot
part=part
part=part
int
int int
intv
Fixed control volume (Eulerian coordinates)Otherwise Raynoldrsquos transport theoremMoving control volume Lagrangian coordinates
MingsianMingsian R R BaiBai42
But the control volime is only arbitrary
In particular for flow
( ) 0
con
(continuity equati
stant
on)
Divergence the
0 incompress
Note is t
ible
orem h
V
v
vt
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
e mother of the other integral theorems
MingsianMingsian R R BaiBai43
Ex Archimedes principle ( ) The buoyant force ( ) on a body with volume immersed in a fluid is equal to the weight of the fluid
ˆ displaced by the object ie In the followin
B
V
gVρ= minusf k
阿基米德原理
浮力
ˆg figure the ˆ with and being the fluid density gravitational
acceleration depth and outward normal unit vector to the surface
fhydraulic pressure g z
g z
S
ρ
ρ
= minusp n
n
1
the modified divergence theorem
ˆ
Hint
scalar fuctionV S
vdV v dS v Cnabla = isinint int n
MingsianMingsian R R BaiBai44
Levelsurfacefluid
V
S
airyx
z
ˆf g zρ= minusp n
ˆPf ( ) ( )
ˆ ˆ ˆ ˆ ˆ (0 0 ) v v
B fS S V
V V
g z g zdS dS dV
g dV g dV gV
ρ ρ
ρ ρ ρ
= = minus = minus nabla
= minus + + = minus = minus
int int int
int int
f p n
i j k k k
k
MingsianMingsian R R BaiBai45
Note This can also be thought of as simple balance of forces (resultant force that acts on and weight of filled with fluid)S V
2
~ a little history about George Green Recall the diveregence theorem
ˆ
Let
II th
e Green
are arbitrary fun
s identiti
ctions
es
V S
v dV n v dS
v u vu v C
nablasdot = sdot
= nablaisin
int intv v
v
ˆ ( ) ( )
ˆ ( )V S
V S
u v dV n u v dS
u v u v dV un v dS
nablasdot nabla = sdot nabla
nabla sdotnabla + nablasdotnabla = sdotnabla
int int
int int
MingsianMingsian R R BaiBai46
2 -( ) -(1)V S
vu v u v dV u dSnpart
nabla sdotnabla + nabla =partint int firs(Greens identt ity)
2
If
( ) --(2)
(1) (2) V S
u vuv u v u dV v dSn
harrpart
nabla sdotnabla + nabla =part
minus
int int
( )2 2
V S
dv duu v v u dV u v dSdn dn
⎛ ⎞nabla minus nabla = minus⎜ ⎟⎝ ⎠int int
second(Greens identity)
MingsianMingsian R R BaiBai47
( )
( )
( ) ( )
1st id ( )
Greens identities in the 1D case integration by parts
b b bb ba a
a a ab
ba
ab
ba
a
v udx v u v u dx u v uv dx uv
u v u v vu dx vu
uv vu dx uv vu
primeprime prime prime prime prime prime primeprime prime= minus rArr + =
prime prime primeprime primeharr + =
primeprime primeprime prime primerArr minus = minus
int int int
int
2
2nd id ( )
Good for any adjoint operators in addition to Good also for and integration by parts Important for theory amp reciprocity
From th
Gree
e 1s
2-
t
ns functi
identity
D 1-
tak 1
D
e
on
u
nablararr
=
int
2
V S
vv dV dSnpartnabla =part
rArr int int2Ex 0 st 0 on the boundary 0 ( 1st id u u unabla = = rArr equiv 以 証)
MingsianMingsian R R BaiBai48
1
Thm1 Stokes theorem Let be defined in Let be a piecewise-smooth orientable surface bounded by a piec
III Stokes theorem and
ewise-smooth simple clos
Greens
ed curve Then
theore
m
v C R S
C
isinv
( )
ˆwhere is a unit normal to according t right hand rulo the e
ˆS C
n S
n v dS v dRsdot nablatimes = sdotint intvv v
(not necessarily planar)S
Cn vnablatimes v vv
MingsianMingsian R R BaiBai49
Pf
( )( )
circulatioWe have shown in deriving thatthe in -direction
ˆor
ˆ
n
z
z
j j j j j
j j j j jj j
vz v dx dy
v dR n v S
v dR n v S
nablatimes
=nablatimes
sdot = sdot nablatimes Δ
sdot = sdot nablatimes Δ
intsum sumint
v
v
vv v
vv v
C
S
jS
jC
jP
ˆ jn
jSΔ
MingsianMingsian R R BaiBai50
( )
( )ext surface
After the interior boundaries cancel out only the exterior boundaries (Why)
ˆ
ˆ0 QED
Alternative forms scalar
survive
va
j
j j j j jC
jC S
v dR n v S
S v dR n v dS
φ
sdot = sdot nablatimes Δ
Δ rarr rArr sdot = sdot nablatimes
sum sumint
int int
vv v
vv v
( )
riable constant vector vector variable
1 Let
2 Let
ˆ
ˆ S C
S C
a p
v n dR dR
n p dS dR p
a
v a p
φ φφ= rArr timesnabla =
timesnabla times= rArr timestimes =
int int
int int
v v
v v
v
v v
vv vv v
MingsianMingsian R R BaiBai51
( ) ( )
( )
( )
( ) ( )
1
Thm2 Greens theorem ( planar version of Stokes theorem)ˆ ˆ Let
pf By Stokes theoremˆ
ˆ ˆ ˆ ˆ ˆ ˆ
S C
S C
S C
v P x y i Q x y j C
n v dS v dR
k Pi Qj dS Pi Qj dx i dy
Q P dS Pdx Qdyx y
⎛ ⎞part partminus = +⎜ ⎟
= + isin
sdot nablatimes = sdot
⎡ ⎤sdot nablatimes + =
part part⎝ ⎠
+ sdot +⎣ ⎦
int int
int
int
int
int
v
vv v
( )
( )
( )
ˆ
ˆˆ ˆ
ˆˆ ˆ ˆ ˆ( ) ( )
0
QEDS C
j
i j kQ PPi Qj i j k
x y z x yP Q
Q P dS Pdx Qdyx y
⎛ ⎞part part part part partnablatimes + = = + + minus⎜ ⎟part part part part part⎝ ⎠
⎛ ⎞part partrArr minus = +⎜ ⎟part part⎝ ⎠int int
L L
S
C
i
j
x
y
MingsianMingsian R R BaiBai52
Potential TheorypotentiGeorge Green coined the term (l
a )
rarr 位能 勢能
例子 重力場 電磁場 流場 聲場 彈性力學
1
2
Thm1 If is in a simply connected region then the following statements are equivalent(i) There exists a scala
Conservative Field
potential functior such that
(ii
n
) 0
v
C
C
v
v R
φφ=nabla
nablatimes
isin
isin
=r
r
r
r in (Irrotational field curl-free)
cf 0 st solenoidal field divergence-free
0
(iii) in (Conservative field)C
Rv w v w
v R Rd
nablasdot = exist nabla
sdot =
= times
int
r r r
r
r
r
MingsianMingsian R R BaiBai53
(iv) is connecting and
independent o
Ex Think
f the pa
of dynamics is the force
thB
A
v dR A B
v
sdotintrr
r
1 2
1
0
Pf
(i) (ii) ( ) 0 (zero identity cg rule) irrotational
ˆ(ii) (iii) ( ) 0 (Stokes theorem)
(iii) (iv) ( ) 0
C S
C C C
C
v
v dR n v dS
v dR v dR
v dR
φ
=
rarr nablatimes = nablatimes nabla =
rarr sdot = sdot nablatimes =
rarr sdot = minus sdot =
rArr sdot =
int int
int int int
int
r
rr
rr r
r rr r
rr r
2
independent of pathC
v dRsdotintr
A
B
1C
2C
MingsianMingsian R R BaiBai54
1 2
(iv) (i) constant ( ) ( )
--(1)
But (2)
(1) (2) QED
Note on application
Check
B
C C A
v dR v dR B A d
v dR d
d dx dy dz dRx y z
v dR dR v R
φ φ φ
φφ φ φφ φ
φ φ
rarr sdot = sdot = = minus =
rArr sdot =part part part
= + + = nabla sdot minus minus minus minus minuspart part part
sdot = nabla sdot rArr = nabla forall
int int intr rr r
rr
r
r r rr r
use simpler path (iv)if 0
find (i)v
φ⎧
nablatimes = ⎨⎩
rr
A
B
x
y
Cz
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai31
1 1 2 2 3 3
1 2 3
are mutually vectors having with arc length
k k k k kds h du U du
U du U du U duds ds ds
= =
rArr perp
v
v v v
1uv
2uv
3uv
1dS
2dS
3dS
dS
MingsianMingsian R R BaiBai32
( ) ( )
( ) ( )
1 1 2 2 3 3 1 2 3 1 2 3 1 2 3
1 2 3 1 2 3
1
1 2 2 3 3 2 2 2 3 3 3 2 3 2 3 1
Differential volume
Differential On const coord surface
area
Gradi t
en
d U du U du U du u u u h h h du du du
d h h h du du du
ud U du U du h u du h u du h h du du u
df
τ
τ
σ
= times sdot = times sdot
=
=
= times = times =
= nabla
v v v v v v
v vv v v v
1 2 31 2 3
1 2 3 1 1 2 2 3 31 2 3
1 1 1 2 2 2 3 3 3
f f ff dr df du du duu u u
r r rdr du du du U du U du U duu u u
h u du h u du h u du
part part partsdot rarr = + +part part part
part part part= + + = + +part part part
= + +
v
v v v v v vv
v v v
=1 (right hand rule)
MingsianMingsian R R BaiBai33
Rectangular Cartesian Coordinates
ij
k
MingsianMingsian R R BaiBai34
Circular Cylindrical Coordinatescossin
xyz z
ρ ϕρ ϕ
===
Ref Arfken
( )P zρ ϕ
MingsianMingsian R R BaiBai35
Polar Spherical Coordinatessin cossin sincos
x ry rz r
θ ϕθ ϕθ
===
Ref Arfken
0
0
0
sin cos sin sin coscos cos cos sin sin
sin cos
θ ϕ θ ϕ θθ ϕ θ ϕ θϕ ϕ
= + +
= + minus= minus +
r i j kθ i j kφ i j
2
2
sinsin
dS r d ddV r dr d d
θ θ ϕ
θ θ ϕ
=
=( )P r θ ϕ
MingsianMingsian R R BaiBai36
Integral TheoremsHeart of this chapter
Interchange of dimensionality Div Curl Grad in integrals High-dimensional integration
rarr
1
Thm1 Divergence theorem Let be a closed region bounded by piecewise smooth orientable surface and let (con
I Divergence theor
tinuous derivatives up to the first order
em (Gauss theorem)
)
VS v Cisin
r
ˆ If denotes the outward unit normal on
the ˆn V S
v dV n v dS
n S
nablasdot = sdotint intr r
MingsianMingsian R R BaiBai37
jS
jP
jV
S
0
ˆBy definition ( ) lim
ˆ( )
ˆ( )
j
j
j
j
Sj V
j
j j S
j j Sj j
n v dSv P
V
v P V n v dS
v P V n v dS
Δ rarr
⎧ ⎫sdot⎪ ⎪nabla sdot ⎨ ⎬Δ⎪ ⎪⎩ ⎭
nabla sdot Δ asymp sdot
nabla sdot Δ = sdot
int
intsum sumint
rr
r r
r r
Pf
MingsianMingsian R R BaiBai38
0
After interior surfaces cancel out only exterior surface (Why Opposite normal vectors)
ˆ( )
volumes exterior surfaces
ˆlim
QE
survive
j
j
j j Sj j
V SV
v p V n v dS
v dV n v dSΔ rarr
nabla sdot Δ = sdot
rArr nablasdot = sdot
sum sumint
int int
r r
r r
D
MingsianMingsian R R BaiBai39
( )
scalar variable constant vectorvector variable
Alternative formsLet be a an arbitrary and
a
ˆ
1 Le
(
ˆt
V
V S
S
V
v ap
v d
v dV vn d
V n v dS
v va
va dV v a v a
S
nablasdot = sdot
= rArr
nablasdot = nabla sdot + nabla
=
sdot
nablaint intint int
int
r
v
r r
r r
r r rQ ( )
( )
ˆ)
2 Let ˆV S
V
V S
dV n va dS
v a p
a p
p dV n p d
d p
S
V a
⎛ ⎞= sdot⎜ ⎟
⎝ ⎠
= times rArr
nablasdot times = sdot nabla
times
times
nabla = times
int int
int int
int
r
r r r
r r v
r
vQ
r
( ) ( ) ( )
( ) ( )
ˆ
ˆV S
V S
a p dV n a p dS
a p dV a n p dS
⎛ ⎞⎡ ⎤minus sdot nablatimes = sdot times⎣ ⎦⎜ ⎟⎜ ⎟⎜ ⎟rArr minus sdot nablatimes = minus sdot times⎜ ⎟⎝ ⎠
int int
int int
r rv v
r rv v
MingsianMingsian R R BaiBai40
( )
( )
From Leibnitzs rule
(1-D) ( )
( ) ( )
B t
A t
d f x t dxdt
f x t dx B ttpart prime= +part
int( )
( )( ( ) ) ( )
B t
A tf B t t A tprimeminusint ( ( ) )
(assume ( ) ( ) )
f A t t
A t B t const= =
ContinuityEx The Equation
rate of increase of mass
mass densityV
dM d dVdt dt
ρ
ρ
= int
flow velocityvv
MingsianMingsian R R BaiBai41
ˆ ˆ ( )
(
Conservation
) ( Divergence theorem)
of mass
( ) 0
V S S
V
V
dV n v dS n v dSt
v dV
v dVt
ρ ρ ρ
ρ
ρ ρ⎡ ⎤⎢ ⎥⎣ ⎦
part = minus sdot = minus sdotpart
= minus nablasdot
partrArr +nablasdot =part
int int int
int
int
v v
vQ
v
ˆOn the other hand rate influof mass through n ox i t
Extended to 3-D for time-independent fixed boundary
(3-D) ( ) ( )
Thus
S
V V
V
S V n v dS
Sd f x y z t dV f x y z t dVdt tdM dVdt t
ρ
ρ
= minus sdot
part=part
part=part
int
int int
intv
Fixed control volume (Eulerian coordinates)Otherwise Raynoldrsquos transport theoremMoving control volume Lagrangian coordinates
MingsianMingsian R R BaiBai42
But the control volime is only arbitrary
In particular for flow
( ) 0
con
(continuity equati
stant
on)
Divergence the
0 incompress
Note is t
ible
orem h
V
v
vt
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
e mother of the other integral theorems
MingsianMingsian R R BaiBai43
Ex Archimedes principle ( ) The buoyant force ( ) on a body with volume immersed in a fluid is equal to the weight of the fluid
ˆ displaced by the object ie In the followin
B
V
gVρ= minusf k
阿基米德原理
浮力
ˆg figure the ˆ with and being the fluid density gravitational
acceleration depth and outward normal unit vector to the surface
fhydraulic pressure g z
g z
S
ρ
ρ
= minusp n
n
1
the modified divergence theorem
ˆ
Hint
scalar fuctionV S
vdV v dS v Cnabla = isinint int n
MingsianMingsian R R BaiBai44
Levelsurfacefluid
V
S
airyx
z
ˆf g zρ= minusp n
ˆPf ( ) ( )
ˆ ˆ ˆ ˆ ˆ (0 0 ) v v
B fS S V
V V
g z g zdS dS dV
g dV g dV gV
ρ ρ
ρ ρ ρ
= = minus = minus nabla
= minus + + = minus = minus
int int int
int int
f p n
i j k k k
k
MingsianMingsian R R BaiBai45
Note This can also be thought of as simple balance of forces (resultant force that acts on and weight of filled with fluid)S V
2
~ a little history about George Green Recall the diveregence theorem
ˆ
Let
II th
e Green
are arbitrary fun
s identiti
ctions
es
V S
v dV n v dS
v u vu v C
nablasdot = sdot
= nablaisin
int intv v
v
ˆ ( ) ( )
ˆ ( )V S
V S
u v dV n u v dS
u v u v dV un v dS
nablasdot nabla = sdot nabla
nabla sdotnabla + nablasdotnabla = sdotnabla
int int
int int
MingsianMingsian R R BaiBai46
2 -( ) -(1)V S
vu v u v dV u dSnpart
nabla sdotnabla + nabla =partint int firs(Greens identt ity)
2
If
( ) --(2)
(1) (2) V S
u vuv u v u dV v dSn
harrpart
nabla sdotnabla + nabla =part
minus
int int
( )2 2
V S
dv duu v v u dV u v dSdn dn
⎛ ⎞nabla minus nabla = minus⎜ ⎟⎝ ⎠int int
second(Greens identity)
MingsianMingsian R R BaiBai47
( )
( )
( ) ( )
1st id ( )
Greens identities in the 1D case integration by parts
b b bb ba a
a a ab
ba
ab
ba
a
v udx v u v u dx u v uv dx uv
u v u v vu dx vu
uv vu dx uv vu
primeprime prime prime prime prime prime primeprime prime= minus rArr + =
prime prime primeprime primeharr + =
primeprime primeprime prime primerArr minus = minus
int int int
int
2
2nd id ( )
Good for any adjoint operators in addition to Good also for and integration by parts Important for theory amp reciprocity
From th
Gree
e 1s
2-
t
ns functi
identity
D 1-
tak 1
D
e
on
u
nablararr
=
int
2
V S
vv dV dSnpartnabla =part
rArr int int2Ex 0 st 0 on the boundary 0 ( 1st id u u unabla = = rArr equiv 以 証)
MingsianMingsian R R BaiBai48
1
Thm1 Stokes theorem Let be defined in Let be a piecewise-smooth orientable surface bounded by a piec
III Stokes theorem and
ewise-smooth simple clos
Greens
ed curve Then
theore
m
v C R S
C
isinv
( )
ˆwhere is a unit normal to according t right hand rulo the e
ˆS C
n S
n v dS v dRsdot nablatimes = sdotint intvv v
(not necessarily planar)S
Cn vnablatimes v vv
MingsianMingsian R R BaiBai49
Pf
( )( )
circulatioWe have shown in deriving thatthe in -direction
ˆor
ˆ
n
z
z
j j j j j
j j j j jj j
vz v dx dy
v dR n v S
v dR n v S
nablatimes
=nablatimes
sdot = sdot nablatimes Δ
sdot = sdot nablatimes Δ
intsum sumint
v
v
vv v
vv v
C
S
jS
jC
jP
ˆ jn
jSΔ
MingsianMingsian R R BaiBai50
( )
( )ext surface
After the interior boundaries cancel out only the exterior boundaries (Why)
ˆ
ˆ0 QED
Alternative forms scalar
survive
va
j
j j j j jC
jC S
v dR n v S
S v dR n v dS
φ
sdot = sdot nablatimes Δ
Δ rarr rArr sdot = sdot nablatimes
sum sumint
int int
vv v
vv v
( )
riable constant vector vector variable
1 Let
2 Let
ˆ
ˆ S C
S C
a p
v n dR dR
n p dS dR p
a
v a p
φ φφ= rArr timesnabla =
timesnabla times= rArr timestimes =
int int
int int
v v
v v
v
v v
vv vv v
MingsianMingsian R R BaiBai51
( ) ( )
( )
( )
( ) ( )
1
Thm2 Greens theorem ( planar version of Stokes theorem)ˆ ˆ Let
pf By Stokes theoremˆ
ˆ ˆ ˆ ˆ ˆ ˆ
S C
S C
S C
v P x y i Q x y j C
n v dS v dR
k Pi Qj dS Pi Qj dx i dy
Q P dS Pdx Qdyx y
⎛ ⎞part partminus = +⎜ ⎟
= + isin
sdot nablatimes = sdot
⎡ ⎤sdot nablatimes + =
part part⎝ ⎠
+ sdot +⎣ ⎦
int int
int
int
int
int
v
vv v
( )
( )
( )
ˆ
ˆˆ ˆ
ˆˆ ˆ ˆ ˆ( ) ( )
0
QEDS C
j
i j kQ PPi Qj i j k
x y z x yP Q
Q P dS Pdx Qdyx y
⎛ ⎞part part part part partnablatimes + = = + + minus⎜ ⎟part part part part part⎝ ⎠
⎛ ⎞part partrArr minus = +⎜ ⎟part part⎝ ⎠int int
L L
S
C
i
j
x
y
MingsianMingsian R R BaiBai52
Potential TheorypotentiGeorge Green coined the term (l
a )
rarr 位能 勢能
例子 重力場 電磁場 流場 聲場 彈性力學
1
2
Thm1 If is in a simply connected region then the following statements are equivalent(i) There exists a scala
Conservative Field
potential functior such that
(ii
n
) 0
v
C
C
v
v R
φφ=nabla
nablatimes
isin
isin
=r
r
r
r in (Irrotational field curl-free)
cf 0 st solenoidal field divergence-free
0
(iii) in (Conservative field)C
Rv w v w
v R Rd
nablasdot = exist nabla
sdot =
= times
int
r r r
r
r
r
MingsianMingsian R R BaiBai53
(iv) is connecting and
independent o
Ex Think
f the pa
of dynamics is the force
thB
A
v dR A B
v
sdotintrr
r
1 2
1
0
Pf
(i) (ii) ( ) 0 (zero identity cg rule) irrotational
ˆ(ii) (iii) ( ) 0 (Stokes theorem)
(iii) (iv) ( ) 0
C S
C C C
C
v
v dR n v dS
v dR v dR
v dR
φ
=
rarr nablatimes = nablatimes nabla =
rarr sdot = sdot nablatimes =
rarr sdot = minus sdot =
rArr sdot =
int int
int int int
int
r
rr
rr r
r rr r
rr r
2
independent of pathC
v dRsdotintr
A
B
1C
2C
MingsianMingsian R R BaiBai54
1 2
(iv) (i) constant ( ) ( )
--(1)
But (2)
(1) (2) QED
Note on application
Check
B
C C A
v dR v dR B A d
v dR d
d dx dy dz dRx y z
v dR dR v R
φ φ φ
φφ φ φφ φ
φ φ
rarr sdot = sdot = = minus =
rArr sdot =part part part
= + + = nabla sdot minus minus minus minus minuspart part part
sdot = nabla sdot rArr = nabla forall
int int intr rr r
rr
r
r r rr r
use simpler path (iv)if 0
find (i)v
φ⎧
nablatimes = ⎨⎩
rr
A
B
x
y
Cz
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai32
( ) ( )
( ) ( )
1 1 2 2 3 3 1 2 3 1 2 3 1 2 3
1 2 3 1 2 3
1
1 2 2 3 3 2 2 2 3 3 3 2 3 2 3 1
Differential volume
Differential On const coord surface
area
Gradi t
en
d U du U du U du u u u h h h du du du
d h h h du du du
ud U du U du h u du h u du h h du du u
df
τ
τ
σ
= times sdot = times sdot
=
=
= times = times =
= nabla
v v v v v v
v vv v v v
1 2 31 2 3
1 2 3 1 1 2 2 3 31 2 3
1 1 1 2 2 2 3 3 3
f f ff dr df du du duu u u
r r rdr du du du U du U du U duu u u
h u du h u du h u du
part part partsdot rarr = + +part part part
part part part= + + = + +part part part
= + +
v
v v v v v vv
v v v
=1 (right hand rule)
MingsianMingsian R R BaiBai33
Rectangular Cartesian Coordinates
ij
k
MingsianMingsian R R BaiBai34
Circular Cylindrical Coordinatescossin
xyz z
ρ ϕρ ϕ
===
Ref Arfken
( )P zρ ϕ
MingsianMingsian R R BaiBai35
Polar Spherical Coordinatessin cossin sincos
x ry rz r
θ ϕθ ϕθ
===
Ref Arfken
0
0
0
sin cos sin sin coscos cos cos sin sin
sin cos
θ ϕ θ ϕ θθ ϕ θ ϕ θϕ ϕ
= + +
= + minus= minus +
r i j kθ i j kφ i j
2
2
sinsin
dS r d ddV r dr d d
θ θ ϕ
θ θ ϕ
=
=( )P r θ ϕ
MingsianMingsian R R BaiBai36
Integral TheoremsHeart of this chapter
Interchange of dimensionality Div Curl Grad in integrals High-dimensional integration
rarr
1
Thm1 Divergence theorem Let be a closed region bounded by piecewise smooth orientable surface and let (con
I Divergence theor
tinuous derivatives up to the first order
em (Gauss theorem)
)
VS v Cisin
r
ˆ If denotes the outward unit normal on
the ˆn V S
v dV n v dS
n S
nablasdot = sdotint intr r
MingsianMingsian R R BaiBai37
jS
jP
jV
S
0
ˆBy definition ( ) lim
ˆ( )
ˆ( )
j
j
j
j
Sj V
j
j j S
j j Sj j
n v dSv P
V
v P V n v dS
v P V n v dS
Δ rarr
⎧ ⎫sdot⎪ ⎪nabla sdot ⎨ ⎬Δ⎪ ⎪⎩ ⎭
nabla sdot Δ asymp sdot
nabla sdot Δ = sdot
int
intsum sumint
rr
r r
r r
Pf
MingsianMingsian R R BaiBai38
0
After interior surfaces cancel out only exterior surface (Why Opposite normal vectors)
ˆ( )
volumes exterior surfaces
ˆlim
QE
survive
j
j
j j Sj j
V SV
v p V n v dS
v dV n v dSΔ rarr
nabla sdot Δ = sdot
rArr nablasdot = sdot
sum sumint
int int
r r
r r
D
MingsianMingsian R R BaiBai39
( )
scalar variable constant vectorvector variable
Alternative formsLet be a an arbitrary and
a
ˆ
1 Le
(
ˆt
V
V S
S
V
v ap
v d
v dV vn d
V n v dS
v va
va dV v a v a
S
nablasdot = sdot
= rArr
nablasdot = nabla sdot + nabla
=
sdot
nablaint intint int
int
r
v
r r
r r
r r rQ ( )
( )
ˆ)
2 Let ˆV S
V
V S
dV n va dS
v a p
a p
p dV n p d
d p
S
V a
⎛ ⎞= sdot⎜ ⎟
⎝ ⎠
= times rArr
nablasdot times = sdot nabla
times
times
nabla = times
int int
int int
int
r
r r r
r r v
r
vQ
r
( ) ( ) ( )
( ) ( )
ˆ
ˆV S
V S
a p dV n a p dS
a p dV a n p dS
⎛ ⎞⎡ ⎤minus sdot nablatimes = sdot times⎣ ⎦⎜ ⎟⎜ ⎟⎜ ⎟rArr minus sdot nablatimes = minus sdot times⎜ ⎟⎝ ⎠
int int
int int
r rv v
r rv v
MingsianMingsian R R BaiBai40
( )
( )
From Leibnitzs rule
(1-D) ( )
( ) ( )
B t
A t
d f x t dxdt
f x t dx B ttpart prime= +part
int( )
( )( ( ) ) ( )
B t
A tf B t t A tprimeminusint ( ( ) )
(assume ( ) ( ) )
f A t t
A t B t const= =
ContinuityEx The Equation
rate of increase of mass
mass densityV
dM d dVdt dt
ρ
ρ
= int
flow velocityvv
MingsianMingsian R R BaiBai41
ˆ ˆ ( )
(
Conservation
) ( Divergence theorem)
of mass
( ) 0
V S S
V
V
dV n v dS n v dSt
v dV
v dVt
ρ ρ ρ
ρ
ρ ρ⎡ ⎤⎢ ⎥⎣ ⎦
part = minus sdot = minus sdotpart
= minus nablasdot
partrArr +nablasdot =part
int int int
int
int
v v
vQ
v
ˆOn the other hand rate influof mass through n ox i t
Extended to 3-D for time-independent fixed boundary
(3-D) ( ) ( )
Thus
S
V V
V
S V n v dS
Sd f x y z t dV f x y z t dVdt tdM dVdt t
ρ
ρ
= minus sdot
part=part
part=part
int
int int
intv
Fixed control volume (Eulerian coordinates)Otherwise Raynoldrsquos transport theoremMoving control volume Lagrangian coordinates
MingsianMingsian R R BaiBai42
But the control volime is only arbitrary
In particular for flow
( ) 0
con
(continuity equati
stant
on)
Divergence the
0 incompress
Note is t
ible
orem h
V
v
vt
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
e mother of the other integral theorems
MingsianMingsian R R BaiBai43
Ex Archimedes principle ( ) The buoyant force ( ) on a body with volume immersed in a fluid is equal to the weight of the fluid
ˆ displaced by the object ie In the followin
B
V
gVρ= minusf k
阿基米德原理
浮力
ˆg figure the ˆ with and being the fluid density gravitational
acceleration depth and outward normal unit vector to the surface
fhydraulic pressure g z
g z
S
ρ
ρ
= minusp n
n
1
the modified divergence theorem
ˆ
Hint
scalar fuctionV S
vdV v dS v Cnabla = isinint int n
MingsianMingsian R R BaiBai44
Levelsurfacefluid
V
S
airyx
z
ˆf g zρ= minusp n
ˆPf ( ) ( )
ˆ ˆ ˆ ˆ ˆ (0 0 ) v v
B fS S V
V V
g z g zdS dS dV
g dV g dV gV
ρ ρ
ρ ρ ρ
= = minus = minus nabla
= minus + + = minus = minus
int int int
int int
f p n
i j k k k
k
MingsianMingsian R R BaiBai45
Note This can also be thought of as simple balance of forces (resultant force that acts on and weight of filled with fluid)S V
2
~ a little history about George Green Recall the diveregence theorem
ˆ
Let
II th
e Green
are arbitrary fun
s identiti
ctions
es
V S
v dV n v dS
v u vu v C
nablasdot = sdot
= nablaisin
int intv v
v
ˆ ( ) ( )
ˆ ( )V S
V S
u v dV n u v dS
u v u v dV un v dS
nablasdot nabla = sdot nabla
nabla sdotnabla + nablasdotnabla = sdotnabla
int int
int int
MingsianMingsian R R BaiBai46
2 -( ) -(1)V S
vu v u v dV u dSnpart
nabla sdotnabla + nabla =partint int firs(Greens identt ity)
2
If
( ) --(2)
(1) (2) V S
u vuv u v u dV v dSn
harrpart
nabla sdotnabla + nabla =part
minus
int int
( )2 2
V S
dv duu v v u dV u v dSdn dn
⎛ ⎞nabla minus nabla = minus⎜ ⎟⎝ ⎠int int
second(Greens identity)
MingsianMingsian R R BaiBai47
( )
( )
( ) ( )
1st id ( )
Greens identities in the 1D case integration by parts
b b bb ba a
a a ab
ba
ab
ba
a
v udx v u v u dx u v uv dx uv
u v u v vu dx vu
uv vu dx uv vu
primeprime prime prime prime prime prime primeprime prime= minus rArr + =
prime prime primeprime primeharr + =
primeprime primeprime prime primerArr minus = minus
int int int
int
2
2nd id ( )
Good for any adjoint operators in addition to Good also for and integration by parts Important for theory amp reciprocity
From th
Gree
e 1s
2-
t
ns functi
identity
D 1-
tak 1
D
e
on
u
nablararr
=
int
2
V S
vv dV dSnpartnabla =part
rArr int int2Ex 0 st 0 on the boundary 0 ( 1st id u u unabla = = rArr equiv 以 証)
MingsianMingsian R R BaiBai48
1
Thm1 Stokes theorem Let be defined in Let be a piecewise-smooth orientable surface bounded by a piec
III Stokes theorem and
ewise-smooth simple clos
Greens
ed curve Then
theore
m
v C R S
C
isinv
( )
ˆwhere is a unit normal to according t right hand rulo the e
ˆS C
n S
n v dS v dRsdot nablatimes = sdotint intvv v
(not necessarily planar)S
Cn vnablatimes v vv
MingsianMingsian R R BaiBai49
Pf
( )( )
circulatioWe have shown in deriving thatthe in -direction
ˆor
ˆ
n
z
z
j j j j j
j j j j jj j
vz v dx dy
v dR n v S
v dR n v S
nablatimes
=nablatimes
sdot = sdot nablatimes Δ
sdot = sdot nablatimes Δ
intsum sumint
v
v
vv v
vv v
C
S
jS
jC
jP
ˆ jn
jSΔ
MingsianMingsian R R BaiBai50
( )
( )ext surface
After the interior boundaries cancel out only the exterior boundaries (Why)
ˆ
ˆ0 QED
Alternative forms scalar
survive
va
j
j j j j jC
jC S
v dR n v S
S v dR n v dS
φ
sdot = sdot nablatimes Δ
Δ rarr rArr sdot = sdot nablatimes
sum sumint
int int
vv v
vv v
( )
riable constant vector vector variable
1 Let
2 Let
ˆ
ˆ S C
S C
a p
v n dR dR
n p dS dR p
a
v a p
φ φφ= rArr timesnabla =
timesnabla times= rArr timestimes =
int int
int int
v v
v v
v
v v
vv vv v
MingsianMingsian R R BaiBai51
( ) ( )
( )
( )
( ) ( )
1
Thm2 Greens theorem ( planar version of Stokes theorem)ˆ ˆ Let
pf By Stokes theoremˆ
ˆ ˆ ˆ ˆ ˆ ˆ
S C
S C
S C
v P x y i Q x y j C
n v dS v dR
k Pi Qj dS Pi Qj dx i dy
Q P dS Pdx Qdyx y
⎛ ⎞part partminus = +⎜ ⎟
= + isin
sdot nablatimes = sdot
⎡ ⎤sdot nablatimes + =
part part⎝ ⎠
+ sdot +⎣ ⎦
int int
int
int
int
int
v
vv v
( )
( )
( )
ˆ
ˆˆ ˆ
ˆˆ ˆ ˆ ˆ( ) ( )
0
QEDS C
j
i j kQ PPi Qj i j k
x y z x yP Q
Q P dS Pdx Qdyx y
⎛ ⎞part part part part partnablatimes + = = + + minus⎜ ⎟part part part part part⎝ ⎠
⎛ ⎞part partrArr minus = +⎜ ⎟part part⎝ ⎠int int
L L
S
C
i
j
x
y
MingsianMingsian R R BaiBai52
Potential TheorypotentiGeorge Green coined the term (l
a )
rarr 位能 勢能
例子 重力場 電磁場 流場 聲場 彈性力學
1
2
Thm1 If is in a simply connected region then the following statements are equivalent(i) There exists a scala
Conservative Field
potential functior such that
(ii
n
) 0
v
C
C
v
v R
φφ=nabla
nablatimes
isin
isin
=r
r
r
r in (Irrotational field curl-free)
cf 0 st solenoidal field divergence-free
0
(iii) in (Conservative field)C
Rv w v w
v R Rd
nablasdot = exist nabla
sdot =
= times
int
r r r
r
r
r
MingsianMingsian R R BaiBai53
(iv) is connecting and
independent o
Ex Think
f the pa
of dynamics is the force
thB
A
v dR A B
v
sdotintrr
r
1 2
1
0
Pf
(i) (ii) ( ) 0 (zero identity cg rule) irrotational
ˆ(ii) (iii) ( ) 0 (Stokes theorem)
(iii) (iv) ( ) 0
C S
C C C
C
v
v dR n v dS
v dR v dR
v dR
φ
=
rarr nablatimes = nablatimes nabla =
rarr sdot = sdot nablatimes =
rarr sdot = minus sdot =
rArr sdot =
int int
int int int
int
r
rr
rr r
r rr r
rr r
2
independent of pathC
v dRsdotintr
A
B
1C
2C
MingsianMingsian R R BaiBai54
1 2
(iv) (i) constant ( ) ( )
--(1)
But (2)
(1) (2) QED
Note on application
Check
B
C C A
v dR v dR B A d
v dR d
d dx dy dz dRx y z
v dR dR v R
φ φ φ
φφ φ φφ φ
φ φ
rarr sdot = sdot = = minus =
rArr sdot =part part part
= + + = nabla sdot minus minus minus minus minuspart part part
sdot = nabla sdot rArr = nabla forall
int int intr rr r
rr
r
r r rr r
use simpler path (iv)if 0
find (i)v
φ⎧
nablatimes = ⎨⎩
rr
A
B
x
y
Cz
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai33
Rectangular Cartesian Coordinates
ij
k
MingsianMingsian R R BaiBai34
Circular Cylindrical Coordinatescossin
xyz z
ρ ϕρ ϕ
===
Ref Arfken
( )P zρ ϕ
MingsianMingsian R R BaiBai35
Polar Spherical Coordinatessin cossin sincos
x ry rz r
θ ϕθ ϕθ
===
Ref Arfken
0
0
0
sin cos sin sin coscos cos cos sin sin
sin cos
θ ϕ θ ϕ θθ ϕ θ ϕ θϕ ϕ
= + +
= + minus= minus +
r i j kθ i j kφ i j
2
2
sinsin
dS r d ddV r dr d d
θ θ ϕ
θ θ ϕ
=
=( )P r θ ϕ
MingsianMingsian R R BaiBai36
Integral TheoremsHeart of this chapter
Interchange of dimensionality Div Curl Grad in integrals High-dimensional integration
rarr
1
Thm1 Divergence theorem Let be a closed region bounded by piecewise smooth orientable surface and let (con
I Divergence theor
tinuous derivatives up to the first order
em (Gauss theorem)
)
VS v Cisin
r
ˆ If denotes the outward unit normal on
the ˆn V S
v dV n v dS
n S
nablasdot = sdotint intr r
MingsianMingsian R R BaiBai37
jS
jP
jV
S
0
ˆBy definition ( ) lim
ˆ( )
ˆ( )
j
j
j
j
Sj V
j
j j S
j j Sj j
n v dSv P
V
v P V n v dS
v P V n v dS
Δ rarr
⎧ ⎫sdot⎪ ⎪nabla sdot ⎨ ⎬Δ⎪ ⎪⎩ ⎭
nabla sdot Δ asymp sdot
nabla sdot Δ = sdot
int
intsum sumint
rr
r r
r r
Pf
MingsianMingsian R R BaiBai38
0
After interior surfaces cancel out only exterior surface (Why Opposite normal vectors)
ˆ( )
volumes exterior surfaces
ˆlim
QE
survive
j
j
j j Sj j
V SV
v p V n v dS
v dV n v dSΔ rarr
nabla sdot Δ = sdot
rArr nablasdot = sdot
sum sumint
int int
r r
r r
D
MingsianMingsian R R BaiBai39
( )
scalar variable constant vectorvector variable
Alternative formsLet be a an arbitrary and
a
ˆ
1 Le
(
ˆt
V
V S
S
V
v ap
v d
v dV vn d
V n v dS
v va
va dV v a v a
S
nablasdot = sdot
= rArr
nablasdot = nabla sdot + nabla
=
sdot
nablaint intint int
int
r
v
r r
r r
r r rQ ( )
( )
ˆ)
2 Let ˆV S
V
V S
dV n va dS
v a p
a p
p dV n p d
d p
S
V a
⎛ ⎞= sdot⎜ ⎟
⎝ ⎠
= times rArr
nablasdot times = sdot nabla
times
times
nabla = times
int int
int int
int
r
r r r
r r v
r
vQ
r
( ) ( ) ( )
( ) ( )
ˆ
ˆV S
V S
a p dV n a p dS
a p dV a n p dS
⎛ ⎞⎡ ⎤minus sdot nablatimes = sdot times⎣ ⎦⎜ ⎟⎜ ⎟⎜ ⎟rArr minus sdot nablatimes = minus sdot times⎜ ⎟⎝ ⎠
int int
int int
r rv v
r rv v
MingsianMingsian R R BaiBai40
( )
( )
From Leibnitzs rule
(1-D) ( )
( ) ( )
B t
A t
d f x t dxdt
f x t dx B ttpart prime= +part
int( )
( )( ( ) ) ( )
B t
A tf B t t A tprimeminusint ( ( ) )
(assume ( ) ( ) )
f A t t
A t B t const= =
ContinuityEx The Equation
rate of increase of mass
mass densityV
dM d dVdt dt
ρ
ρ
= int
flow velocityvv
MingsianMingsian R R BaiBai41
ˆ ˆ ( )
(
Conservation
) ( Divergence theorem)
of mass
( ) 0
V S S
V
V
dV n v dS n v dSt
v dV
v dVt
ρ ρ ρ
ρ
ρ ρ⎡ ⎤⎢ ⎥⎣ ⎦
part = minus sdot = minus sdotpart
= minus nablasdot
partrArr +nablasdot =part
int int int
int
int
v v
vQ
v
ˆOn the other hand rate influof mass through n ox i t
Extended to 3-D for time-independent fixed boundary
(3-D) ( ) ( )
Thus
S
V V
V
S V n v dS
Sd f x y z t dV f x y z t dVdt tdM dVdt t
ρ
ρ
= minus sdot
part=part
part=part
int
int int
intv
Fixed control volume (Eulerian coordinates)Otherwise Raynoldrsquos transport theoremMoving control volume Lagrangian coordinates
MingsianMingsian R R BaiBai42
But the control volime is only arbitrary
In particular for flow
( ) 0
con
(continuity equati
stant
on)
Divergence the
0 incompress
Note is t
ible
orem h
V
v
vt
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
e mother of the other integral theorems
MingsianMingsian R R BaiBai43
Ex Archimedes principle ( ) The buoyant force ( ) on a body with volume immersed in a fluid is equal to the weight of the fluid
ˆ displaced by the object ie In the followin
B
V
gVρ= minusf k
阿基米德原理
浮力
ˆg figure the ˆ with and being the fluid density gravitational
acceleration depth and outward normal unit vector to the surface
fhydraulic pressure g z
g z
S
ρ
ρ
= minusp n
n
1
the modified divergence theorem
ˆ
Hint
scalar fuctionV S
vdV v dS v Cnabla = isinint int n
MingsianMingsian R R BaiBai44
Levelsurfacefluid
V
S
airyx
z
ˆf g zρ= minusp n
ˆPf ( ) ( )
ˆ ˆ ˆ ˆ ˆ (0 0 ) v v
B fS S V
V V
g z g zdS dS dV
g dV g dV gV
ρ ρ
ρ ρ ρ
= = minus = minus nabla
= minus + + = minus = minus
int int int
int int
f p n
i j k k k
k
MingsianMingsian R R BaiBai45
Note This can also be thought of as simple balance of forces (resultant force that acts on and weight of filled with fluid)S V
2
~ a little history about George Green Recall the diveregence theorem
ˆ
Let
II th
e Green
are arbitrary fun
s identiti
ctions
es
V S
v dV n v dS
v u vu v C
nablasdot = sdot
= nablaisin
int intv v
v
ˆ ( ) ( )
ˆ ( )V S
V S
u v dV n u v dS
u v u v dV un v dS
nablasdot nabla = sdot nabla
nabla sdotnabla + nablasdotnabla = sdotnabla
int int
int int
MingsianMingsian R R BaiBai46
2 -( ) -(1)V S
vu v u v dV u dSnpart
nabla sdotnabla + nabla =partint int firs(Greens identt ity)
2
If
( ) --(2)
(1) (2) V S
u vuv u v u dV v dSn
harrpart
nabla sdotnabla + nabla =part
minus
int int
( )2 2
V S
dv duu v v u dV u v dSdn dn
⎛ ⎞nabla minus nabla = minus⎜ ⎟⎝ ⎠int int
second(Greens identity)
MingsianMingsian R R BaiBai47
( )
( )
( ) ( )
1st id ( )
Greens identities in the 1D case integration by parts
b b bb ba a
a a ab
ba
ab
ba
a
v udx v u v u dx u v uv dx uv
u v u v vu dx vu
uv vu dx uv vu
primeprime prime prime prime prime prime primeprime prime= minus rArr + =
prime prime primeprime primeharr + =
primeprime primeprime prime primerArr minus = minus
int int int
int
2
2nd id ( )
Good for any adjoint operators in addition to Good also for and integration by parts Important for theory amp reciprocity
From th
Gree
e 1s
2-
t
ns functi
identity
D 1-
tak 1
D
e
on
u
nablararr
=
int
2
V S
vv dV dSnpartnabla =part
rArr int int2Ex 0 st 0 on the boundary 0 ( 1st id u u unabla = = rArr equiv 以 証)
MingsianMingsian R R BaiBai48
1
Thm1 Stokes theorem Let be defined in Let be a piecewise-smooth orientable surface bounded by a piec
III Stokes theorem and
ewise-smooth simple clos
Greens
ed curve Then
theore
m
v C R S
C
isinv
( )
ˆwhere is a unit normal to according t right hand rulo the e
ˆS C
n S
n v dS v dRsdot nablatimes = sdotint intvv v
(not necessarily planar)S
Cn vnablatimes v vv
MingsianMingsian R R BaiBai49
Pf
( )( )
circulatioWe have shown in deriving thatthe in -direction
ˆor
ˆ
n
z
z
j j j j j
j j j j jj j
vz v dx dy
v dR n v S
v dR n v S
nablatimes
=nablatimes
sdot = sdot nablatimes Δ
sdot = sdot nablatimes Δ
intsum sumint
v
v
vv v
vv v
C
S
jS
jC
jP
ˆ jn
jSΔ
MingsianMingsian R R BaiBai50
( )
( )ext surface
After the interior boundaries cancel out only the exterior boundaries (Why)
ˆ
ˆ0 QED
Alternative forms scalar
survive
va
j
j j j j jC
jC S
v dR n v S
S v dR n v dS
φ
sdot = sdot nablatimes Δ
Δ rarr rArr sdot = sdot nablatimes
sum sumint
int int
vv v
vv v
( )
riable constant vector vector variable
1 Let
2 Let
ˆ
ˆ S C
S C
a p
v n dR dR
n p dS dR p
a
v a p
φ φφ= rArr timesnabla =
timesnabla times= rArr timestimes =
int int
int int
v v
v v
v
v v
vv vv v
MingsianMingsian R R BaiBai51
( ) ( )
( )
( )
( ) ( )
1
Thm2 Greens theorem ( planar version of Stokes theorem)ˆ ˆ Let
pf By Stokes theoremˆ
ˆ ˆ ˆ ˆ ˆ ˆ
S C
S C
S C
v P x y i Q x y j C
n v dS v dR
k Pi Qj dS Pi Qj dx i dy
Q P dS Pdx Qdyx y
⎛ ⎞part partminus = +⎜ ⎟
= + isin
sdot nablatimes = sdot
⎡ ⎤sdot nablatimes + =
part part⎝ ⎠
+ sdot +⎣ ⎦
int int
int
int
int
int
v
vv v
( )
( )
( )
ˆ
ˆˆ ˆ
ˆˆ ˆ ˆ ˆ( ) ( )
0
QEDS C
j
i j kQ PPi Qj i j k
x y z x yP Q
Q P dS Pdx Qdyx y
⎛ ⎞part part part part partnablatimes + = = + + minus⎜ ⎟part part part part part⎝ ⎠
⎛ ⎞part partrArr minus = +⎜ ⎟part part⎝ ⎠int int
L L
S
C
i
j
x
y
MingsianMingsian R R BaiBai52
Potential TheorypotentiGeorge Green coined the term (l
a )
rarr 位能 勢能
例子 重力場 電磁場 流場 聲場 彈性力學
1
2
Thm1 If is in a simply connected region then the following statements are equivalent(i) There exists a scala
Conservative Field
potential functior such that
(ii
n
) 0
v
C
C
v
v R
φφ=nabla
nablatimes
isin
isin
=r
r
r
r in (Irrotational field curl-free)
cf 0 st solenoidal field divergence-free
0
(iii) in (Conservative field)C
Rv w v w
v R Rd
nablasdot = exist nabla
sdot =
= times
int
r r r
r
r
r
MingsianMingsian R R BaiBai53
(iv) is connecting and
independent o
Ex Think
f the pa
of dynamics is the force
thB
A
v dR A B
v
sdotintrr
r
1 2
1
0
Pf
(i) (ii) ( ) 0 (zero identity cg rule) irrotational
ˆ(ii) (iii) ( ) 0 (Stokes theorem)
(iii) (iv) ( ) 0
C S
C C C
C
v
v dR n v dS
v dR v dR
v dR
φ
=
rarr nablatimes = nablatimes nabla =
rarr sdot = sdot nablatimes =
rarr sdot = minus sdot =
rArr sdot =
int int
int int int
int
r
rr
rr r
r rr r
rr r
2
independent of pathC
v dRsdotintr
A
B
1C
2C
MingsianMingsian R R BaiBai54
1 2
(iv) (i) constant ( ) ( )
--(1)
But (2)
(1) (2) QED
Note on application
Check
B
C C A
v dR v dR B A d
v dR d
d dx dy dz dRx y z
v dR dR v R
φ φ φ
φφ φ φφ φ
φ φ
rarr sdot = sdot = = minus =
rArr sdot =part part part
= + + = nabla sdot minus minus minus minus minuspart part part
sdot = nabla sdot rArr = nabla forall
int int intr rr r
rr
r
r r rr r
use simpler path (iv)if 0
find (i)v
φ⎧
nablatimes = ⎨⎩
rr
A
B
x
y
Cz
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai34
Circular Cylindrical Coordinatescossin
xyz z
ρ ϕρ ϕ
===
Ref Arfken
( )P zρ ϕ
MingsianMingsian R R BaiBai35
Polar Spherical Coordinatessin cossin sincos
x ry rz r
θ ϕθ ϕθ
===
Ref Arfken
0
0
0
sin cos sin sin coscos cos cos sin sin
sin cos
θ ϕ θ ϕ θθ ϕ θ ϕ θϕ ϕ
= + +
= + minus= minus +
r i j kθ i j kφ i j
2
2
sinsin
dS r d ddV r dr d d
θ θ ϕ
θ θ ϕ
=
=( )P r θ ϕ
MingsianMingsian R R BaiBai36
Integral TheoremsHeart of this chapter
Interchange of dimensionality Div Curl Grad in integrals High-dimensional integration
rarr
1
Thm1 Divergence theorem Let be a closed region bounded by piecewise smooth orientable surface and let (con
I Divergence theor
tinuous derivatives up to the first order
em (Gauss theorem)
)
VS v Cisin
r
ˆ If denotes the outward unit normal on
the ˆn V S
v dV n v dS
n S
nablasdot = sdotint intr r
MingsianMingsian R R BaiBai37
jS
jP
jV
S
0
ˆBy definition ( ) lim
ˆ( )
ˆ( )
j
j
j
j
Sj V
j
j j S
j j Sj j
n v dSv P
V
v P V n v dS
v P V n v dS
Δ rarr
⎧ ⎫sdot⎪ ⎪nabla sdot ⎨ ⎬Δ⎪ ⎪⎩ ⎭
nabla sdot Δ asymp sdot
nabla sdot Δ = sdot
int
intsum sumint
rr
r r
r r
Pf
MingsianMingsian R R BaiBai38
0
After interior surfaces cancel out only exterior surface (Why Opposite normal vectors)
ˆ( )
volumes exterior surfaces
ˆlim
QE
survive
j
j
j j Sj j
V SV
v p V n v dS
v dV n v dSΔ rarr
nabla sdot Δ = sdot
rArr nablasdot = sdot
sum sumint
int int
r r
r r
D
MingsianMingsian R R BaiBai39
( )
scalar variable constant vectorvector variable
Alternative formsLet be a an arbitrary and
a
ˆ
1 Le
(
ˆt
V
V S
S
V
v ap
v d
v dV vn d
V n v dS
v va
va dV v a v a
S
nablasdot = sdot
= rArr
nablasdot = nabla sdot + nabla
=
sdot
nablaint intint int
int
r
v
r r
r r
r r rQ ( )
( )
ˆ)
2 Let ˆV S
V
V S
dV n va dS
v a p
a p
p dV n p d
d p
S
V a
⎛ ⎞= sdot⎜ ⎟
⎝ ⎠
= times rArr
nablasdot times = sdot nabla
times
times
nabla = times
int int
int int
int
r
r r r
r r v
r
vQ
r
( ) ( ) ( )
( ) ( )
ˆ
ˆV S
V S
a p dV n a p dS
a p dV a n p dS
⎛ ⎞⎡ ⎤minus sdot nablatimes = sdot times⎣ ⎦⎜ ⎟⎜ ⎟⎜ ⎟rArr minus sdot nablatimes = minus sdot times⎜ ⎟⎝ ⎠
int int
int int
r rv v
r rv v
MingsianMingsian R R BaiBai40
( )
( )
From Leibnitzs rule
(1-D) ( )
( ) ( )
B t
A t
d f x t dxdt
f x t dx B ttpart prime= +part
int( )
( )( ( ) ) ( )
B t
A tf B t t A tprimeminusint ( ( ) )
(assume ( ) ( ) )
f A t t
A t B t const= =
ContinuityEx The Equation
rate of increase of mass
mass densityV
dM d dVdt dt
ρ
ρ
= int
flow velocityvv
MingsianMingsian R R BaiBai41
ˆ ˆ ( )
(
Conservation
) ( Divergence theorem)
of mass
( ) 0
V S S
V
V
dV n v dS n v dSt
v dV
v dVt
ρ ρ ρ
ρ
ρ ρ⎡ ⎤⎢ ⎥⎣ ⎦
part = minus sdot = minus sdotpart
= minus nablasdot
partrArr +nablasdot =part
int int int
int
int
v v
vQ
v
ˆOn the other hand rate influof mass through n ox i t
Extended to 3-D for time-independent fixed boundary
(3-D) ( ) ( )
Thus
S
V V
V
S V n v dS
Sd f x y z t dV f x y z t dVdt tdM dVdt t
ρ
ρ
= minus sdot
part=part
part=part
int
int int
intv
Fixed control volume (Eulerian coordinates)Otherwise Raynoldrsquos transport theoremMoving control volume Lagrangian coordinates
MingsianMingsian R R BaiBai42
But the control volime is only arbitrary
In particular for flow
( ) 0
con
(continuity equati
stant
on)
Divergence the
0 incompress
Note is t
ible
orem h
V
v
vt
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
e mother of the other integral theorems
MingsianMingsian R R BaiBai43
Ex Archimedes principle ( ) The buoyant force ( ) on a body with volume immersed in a fluid is equal to the weight of the fluid
ˆ displaced by the object ie In the followin
B
V
gVρ= minusf k
阿基米德原理
浮力
ˆg figure the ˆ with and being the fluid density gravitational
acceleration depth and outward normal unit vector to the surface
fhydraulic pressure g z
g z
S
ρ
ρ
= minusp n
n
1
the modified divergence theorem
ˆ
Hint
scalar fuctionV S
vdV v dS v Cnabla = isinint int n
MingsianMingsian R R BaiBai44
Levelsurfacefluid
V
S
airyx
z
ˆf g zρ= minusp n
ˆPf ( ) ( )
ˆ ˆ ˆ ˆ ˆ (0 0 ) v v
B fS S V
V V
g z g zdS dS dV
g dV g dV gV
ρ ρ
ρ ρ ρ
= = minus = minus nabla
= minus + + = minus = minus
int int int
int int
f p n
i j k k k
k
MingsianMingsian R R BaiBai45
Note This can also be thought of as simple balance of forces (resultant force that acts on and weight of filled with fluid)S V
2
~ a little history about George Green Recall the diveregence theorem
ˆ
Let
II th
e Green
are arbitrary fun
s identiti
ctions
es
V S
v dV n v dS
v u vu v C
nablasdot = sdot
= nablaisin
int intv v
v
ˆ ( ) ( )
ˆ ( )V S
V S
u v dV n u v dS
u v u v dV un v dS
nablasdot nabla = sdot nabla
nabla sdotnabla + nablasdotnabla = sdotnabla
int int
int int
MingsianMingsian R R BaiBai46
2 -( ) -(1)V S
vu v u v dV u dSnpart
nabla sdotnabla + nabla =partint int firs(Greens identt ity)
2
If
( ) --(2)
(1) (2) V S
u vuv u v u dV v dSn
harrpart
nabla sdotnabla + nabla =part
minus
int int
( )2 2
V S
dv duu v v u dV u v dSdn dn
⎛ ⎞nabla minus nabla = minus⎜ ⎟⎝ ⎠int int
second(Greens identity)
MingsianMingsian R R BaiBai47
( )
( )
( ) ( )
1st id ( )
Greens identities in the 1D case integration by parts
b b bb ba a
a a ab
ba
ab
ba
a
v udx v u v u dx u v uv dx uv
u v u v vu dx vu
uv vu dx uv vu
primeprime prime prime prime prime prime primeprime prime= minus rArr + =
prime prime primeprime primeharr + =
primeprime primeprime prime primerArr minus = minus
int int int
int
2
2nd id ( )
Good for any adjoint operators in addition to Good also for and integration by parts Important for theory amp reciprocity
From th
Gree
e 1s
2-
t
ns functi
identity
D 1-
tak 1
D
e
on
u
nablararr
=
int
2
V S
vv dV dSnpartnabla =part
rArr int int2Ex 0 st 0 on the boundary 0 ( 1st id u u unabla = = rArr equiv 以 証)
MingsianMingsian R R BaiBai48
1
Thm1 Stokes theorem Let be defined in Let be a piecewise-smooth orientable surface bounded by a piec
III Stokes theorem and
ewise-smooth simple clos
Greens
ed curve Then
theore
m
v C R S
C
isinv
( )
ˆwhere is a unit normal to according t right hand rulo the e
ˆS C
n S
n v dS v dRsdot nablatimes = sdotint intvv v
(not necessarily planar)S
Cn vnablatimes v vv
MingsianMingsian R R BaiBai49
Pf
( )( )
circulatioWe have shown in deriving thatthe in -direction
ˆor
ˆ
n
z
z
j j j j j
j j j j jj j
vz v dx dy
v dR n v S
v dR n v S
nablatimes
=nablatimes
sdot = sdot nablatimes Δ
sdot = sdot nablatimes Δ
intsum sumint
v
v
vv v
vv v
C
S
jS
jC
jP
ˆ jn
jSΔ
MingsianMingsian R R BaiBai50
( )
( )ext surface
After the interior boundaries cancel out only the exterior boundaries (Why)
ˆ
ˆ0 QED
Alternative forms scalar
survive
va
j
j j j j jC
jC S
v dR n v S
S v dR n v dS
φ
sdot = sdot nablatimes Δ
Δ rarr rArr sdot = sdot nablatimes
sum sumint
int int
vv v
vv v
( )
riable constant vector vector variable
1 Let
2 Let
ˆ
ˆ S C
S C
a p
v n dR dR
n p dS dR p
a
v a p
φ φφ= rArr timesnabla =
timesnabla times= rArr timestimes =
int int
int int
v v
v v
v
v v
vv vv v
MingsianMingsian R R BaiBai51
( ) ( )
( )
( )
( ) ( )
1
Thm2 Greens theorem ( planar version of Stokes theorem)ˆ ˆ Let
pf By Stokes theoremˆ
ˆ ˆ ˆ ˆ ˆ ˆ
S C
S C
S C
v P x y i Q x y j C
n v dS v dR
k Pi Qj dS Pi Qj dx i dy
Q P dS Pdx Qdyx y
⎛ ⎞part partminus = +⎜ ⎟
= + isin
sdot nablatimes = sdot
⎡ ⎤sdot nablatimes + =
part part⎝ ⎠
+ sdot +⎣ ⎦
int int
int
int
int
int
v
vv v
( )
( )
( )
ˆ
ˆˆ ˆ
ˆˆ ˆ ˆ ˆ( ) ( )
0
QEDS C
j
i j kQ PPi Qj i j k
x y z x yP Q
Q P dS Pdx Qdyx y
⎛ ⎞part part part part partnablatimes + = = + + minus⎜ ⎟part part part part part⎝ ⎠
⎛ ⎞part partrArr minus = +⎜ ⎟part part⎝ ⎠int int
L L
S
C
i
j
x
y
MingsianMingsian R R BaiBai52
Potential TheorypotentiGeorge Green coined the term (l
a )
rarr 位能 勢能
例子 重力場 電磁場 流場 聲場 彈性力學
1
2
Thm1 If is in a simply connected region then the following statements are equivalent(i) There exists a scala
Conservative Field
potential functior such that
(ii
n
) 0
v
C
C
v
v R
φφ=nabla
nablatimes
isin
isin
=r
r
r
r in (Irrotational field curl-free)
cf 0 st solenoidal field divergence-free
0
(iii) in (Conservative field)C
Rv w v w
v R Rd
nablasdot = exist nabla
sdot =
= times
int
r r r
r
r
r
MingsianMingsian R R BaiBai53
(iv) is connecting and
independent o
Ex Think
f the pa
of dynamics is the force
thB
A
v dR A B
v
sdotintrr
r
1 2
1
0
Pf
(i) (ii) ( ) 0 (zero identity cg rule) irrotational
ˆ(ii) (iii) ( ) 0 (Stokes theorem)
(iii) (iv) ( ) 0
C S
C C C
C
v
v dR n v dS
v dR v dR
v dR
φ
=
rarr nablatimes = nablatimes nabla =
rarr sdot = sdot nablatimes =
rarr sdot = minus sdot =
rArr sdot =
int int
int int int
int
r
rr
rr r
r rr r
rr r
2
independent of pathC
v dRsdotintr
A
B
1C
2C
MingsianMingsian R R BaiBai54
1 2
(iv) (i) constant ( ) ( )
--(1)
But (2)
(1) (2) QED
Note on application
Check
B
C C A
v dR v dR B A d
v dR d
d dx dy dz dRx y z
v dR dR v R
φ φ φ
φφ φ φφ φ
φ φ
rarr sdot = sdot = = minus =
rArr sdot =part part part
= + + = nabla sdot minus minus minus minus minuspart part part
sdot = nabla sdot rArr = nabla forall
int int intr rr r
rr
r
r r rr r
use simpler path (iv)if 0
find (i)v
φ⎧
nablatimes = ⎨⎩
rr
A
B
x
y
Cz
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai35
Polar Spherical Coordinatessin cossin sincos
x ry rz r
θ ϕθ ϕθ
===
Ref Arfken
0
0
0
sin cos sin sin coscos cos cos sin sin
sin cos
θ ϕ θ ϕ θθ ϕ θ ϕ θϕ ϕ
= + +
= + minus= minus +
r i j kθ i j kφ i j
2
2
sinsin
dS r d ddV r dr d d
θ θ ϕ
θ θ ϕ
=
=( )P r θ ϕ
MingsianMingsian R R BaiBai36
Integral TheoremsHeart of this chapter
Interchange of dimensionality Div Curl Grad in integrals High-dimensional integration
rarr
1
Thm1 Divergence theorem Let be a closed region bounded by piecewise smooth orientable surface and let (con
I Divergence theor
tinuous derivatives up to the first order
em (Gauss theorem)
)
VS v Cisin
r
ˆ If denotes the outward unit normal on
the ˆn V S
v dV n v dS
n S
nablasdot = sdotint intr r
MingsianMingsian R R BaiBai37
jS
jP
jV
S
0
ˆBy definition ( ) lim
ˆ( )
ˆ( )
j
j
j
j
Sj V
j
j j S
j j Sj j
n v dSv P
V
v P V n v dS
v P V n v dS
Δ rarr
⎧ ⎫sdot⎪ ⎪nabla sdot ⎨ ⎬Δ⎪ ⎪⎩ ⎭
nabla sdot Δ asymp sdot
nabla sdot Δ = sdot
int
intsum sumint
rr
r r
r r
Pf
MingsianMingsian R R BaiBai38
0
After interior surfaces cancel out only exterior surface (Why Opposite normal vectors)
ˆ( )
volumes exterior surfaces
ˆlim
QE
survive
j
j
j j Sj j
V SV
v p V n v dS
v dV n v dSΔ rarr
nabla sdot Δ = sdot
rArr nablasdot = sdot
sum sumint
int int
r r
r r
D
MingsianMingsian R R BaiBai39
( )
scalar variable constant vectorvector variable
Alternative formsLet be a an arbitrary and
a
ˆ
1 Le
(
ˆt
V
V S
S
V
v ap
v d
v dV vn d
V n v dS
v va
va dV v a v a
S
nablasdot = sdot
= rArr
nablasdot = nabla sdot + nabla
=
sdot
nablaint intint int
int
r
v
r r
r r
r r rQ ( )
( )
ˆ)
2 Let ˆV S
V
V S
dV n va dS
v a p
a p
p dV n p d
d p
S
V a
⎛ ⎞= sdot⎜ ⎟
⎝ ⎠
= times rArr
nablasdot times = sdot nabla
times
times
nabla = times
int int
int int
int
r
r r r
r r v
r
vQ
r
( ) ( ) ( )
( ) ( )
ˆ
ˆV S
V S
a p dV n a p dS
a p dV a n p dS
⎛ ⎞⎡ ⎤minus sdot nablatimes = sdot times⎣ ⎦⎜ ⎟⎜ ⎟⎜ ⎟rArr minus sdot nablatimes = minus sdot times⎜ ⎟⎝ ⎠
int int
int int
r rv v
r rv v
MingsianMingsian R R BaiBai40
( )
( )
From Leibnitzs rule
(1-D) ( )
( ) ( )
B t
A t
d f x t dxdt
f x t dx B ttpart prime= +part
int( )
( )( ( ) ) ( )
B t
A tf B t t A tprimeminusint ( ( ) )
(assume ( ) ( ) )
f A t t
A t B t const= =
ContinuityEx The Equation
rate of increase of mass
mass densityV
dM d dVdt dt
ρ
ρ
= int
flow velocityvv
MingsianMingsian R R BaiBai41
ˆ ˆ ( )
(
Conservation
) ( Divergence theorem)
of mass
( ) 0
V S S
V
V
dV n v dS n v dSt
v dV
v dVt
ρ ρ ρ
ρ
ρ ρ⎡ ⎤⎢ ⎥⎣ ⎦
part = minus sdot = minus sdotpart
= minus nablasdot
partrArr +nablasdot =part
int int int
int
int
v v
vQ
v
ˆOn the other hand rate influof mass through n ox i t
Extended to 3-D for time-independent fixed boundary
(3-D) ( ) ( )
Thus
S
V V
V
S V n v dS
Sd f x y z t dV f x y z t dVdt tdM dVdt t
ρ
ρ
= minus sdot
part=part
part=part
int
int int
intv
Fixed control volume (Eulerian coordinates)Otherwise Raynoldrsquos transport theoremMoving control volume Lagrangian coordinates
MingsianMingsian R R BaiBai42
But the control volime is only arbitrary
In particular for flow
( ) 0
con
(continuity equati
stant
on)
Divergence the
0 incompress
Note is t
ible
orem h
V
v
vt
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
e mother of the other integral theorems
MingsianMingsian R R BaiBai43
Ex Archimedes principle ( ) The buoyant force ( ) on a body with volume immersed in a fluid is equal to the weight of the fluid
ˆ displaced by the object ie In the followin
B
V
gVρ= minusf k
阿基米德原理
浮力
ˆg figure the ˆ with and being the fluid density gravitational
acceleration depth and outward normal unit vector to the surface
fhydraulic pressure g z
g z
S
ρ
ρ
= minusp n
n
1
the modified divergence theorem
ˆ
Hint
scalar fuctionV S
vdV v dS v Cnabla = isinint int n
MingsianMingsian R R BaiBai44
Levelsurfacefluid
V
S
airyx
z
ˆf g zρ= minusp n
ˆPf ( ) ( )
ˆ ˆ ˆ ˆ ˆ (0 0 ) v v
B fS S V
V V
g z g zdS dS dV
g dV g dV gV
ρ ρ
ρ ρ ρ
= = minus = minus nabla
= minus + + = minus = minus
int int int
int int
f p n
i j k k k
k
MingsianMingsian R R BaiBai45
Note This can also be thought of as simple balance of forces (resultant force that acts on and weight of filled with fluid)S V
2
~ a little history about George Green Recall the diveregence theorem
ˆ
Let
II th
e Green
are arbitrary fun
s identiti
ctions
es
V S
v dV n v dS
v u vu v C
nablasdot = sdot
= nablaisin
int intv v
v
ˆ ( ) ( )
ˆ ( )V S
V S
u v dV n u v dS
u v u v dV un v dS
nablasdot nabla = sdot nabla
nabla sdotnabla + nablasdotnabla = sdotnabla
int int
int int
MingsianMingsian R R BaiBai46
2 -( ) -(1)V S
vu v u v dV u dSnpart
nabla sdotnabla + nabla =partint int firs(Greens identt ity)
2
If
( ) --(2)
(1) (2) V S
u vuv u v u dV v dSn
harrpart
nabla sdotnabla + nabla =part
minus
int int
( )2 2
V S
dv duu v v u dV u v dSdn dn
⎛ ⎞nabla minus nabla = minus⎜ ⎟⎝ ⎠int int
second(Greens identity)
MingsianMingsian R R BaiBai47
( )
( )
( ) ( )
1st id ( )
Greens identities in the 1D case integration by parts
b b bb ba a
a a ab
ba
ab
ba
a
v udx v u v u dx u v uv dx uv
u v u v vu dx vu
uv vu dx uv vu
primeprime prime prime prime prime prime primeprime prime= minus rArr + =
prime prime primeprime primeharr + =
primeprime primeprime prime primerArr minus = minus
int int int
int
2
2nd id ( )
Good for any adjoint operators in addition to Good also for and integration by parts Important for theory amp reciprocity
From th
Gree
e 1s
2-
t
ns functi
identity
D 1-
tak 1
D
e
on
u
nablararr
=
int
2
V S
vv dV dSnpartnabla =part
rArr int int2Ex 0 st 0 on the boundary 0 ( 1st id u u unabla = = rArr equiv 以 証)
MingsianMingsian R R BaiBai48
1
Thm1 Stokes theorem Let be defined in Let be a piecewise-smooth orientable surface bounded by a piec
III Stokes theorem and
ewise-smooth simple clos
Greens
ed curve Then
theore
m
v C R S
C
isinv
( )
ˆwhere is a unit normal to according t right hand rulo the e
ˆS C
n S
n v dS v dRsdot nablatimes = sdotint intvv v
(not necessarily planar)S
Cn vnablatimes v vv
MingsianMingsian R R BaiBai49
Pf
( )( )
circulatioWe have shown in deriving thatthe in -direction
ˆor
ˆ
n
z
z
j j j j j
j j j j jj j
vz v dx dy
v dR n v S
v dR n v S
nablatimes
=nablatimes
sdot = sdot nablatimes Δ
sdot = sdot nablatimes Δ
intsum sumint
v
v
vv v
vv v
C
S
jS
jC
jP
ˆ jn
jSΔ
MingsianMingsian R R BaiBai50
( )
( )ext surface
After the interior boundaries cancel out only the exterior boundaries (Why)
ˆ
ˆ0 QED
Alternative forms scalar
survive
va
j
j j j j jC
jC S
v dR n v S
S v dR n v dS
φ
sdot = sdot nablatimes Δ
Δ rarr rArr sdot = sdot nablatimes
sum sumint
int int
vv v
vv v
( )
riable constant vector vector variable
1 Let
2 Let
ˆ
ˆ S C
S C
a p
v n dR dR
n p dS dR p
a
v a p
φ φφ= rArr timesnabla =
timesnabla times= rArr timestimes =
int int
int int
v v
v v
v
v v
vv vv v
MingsianMingsian R R BaiBai51
( ) ( )
( )
( )
( ) ( )
1
Thm2 Greens theorem ( planar version of Stokes theorem)ˆ ˆ Let
pf By Stokes theoremˆ
ˆ ˆ ˆ ˆ ˆ ˆ
S C
S C
S C
v P x y i Q x y j C
n v dS v dR
k Pi Qj dS Pi Qj dx i dy
Q P dS Pdx Qdyx y
⎛ ⎞part partminus = +⎜ ⎟
= + isin
sdot nablatimes = sdot
⎡ ⎤sdot nablatimes + =
part part⎝ ⎠
+ sdot +⎣ ⎦
int int
int
int
int
int
v
vv v
( )
( )
( )
ˆ
ˆˆ ˆ
ˆˆ ˆ ˆ ˆ( ) ( )
0
QEDS C
j
i j kQ PPi Qj i j k
x y z x yP Q
Q P dS Pdx Qdyx y
⎛ ⎞part part part part partnablatimes + = = + + minus⎜ ⎟part part part part part⎝ ⎠
⎛ ⎞part partrArr minus = +⎜ ⎟part part⎝ ⎠int int
L L
S
C
i
j
x
y
MingsianMingsian R R BaiBai52
Potential TheorypotentiGeorge Green coined the term (l
a )
rarr 位能 勢能
例子 重力場 電磁場 流場 聲場 彈性力學
1
2
Thm1 If is in a simply connected region then the following statements are equivalent(i) There exists a scala
Conservative Field
potential functior such that
(ii
n
) 0
v
C
C
v
v R
φφ=nabla
nablatimes
isin
isin
=r
r
r
r in (Irrotational field curl-free)
cf 0 st solenoidal field divergence-free
0
(iii) in (Conservative field)C
Rv w v w
v R Rd
nablasdot = exist nabla
sdot =
= times
int
r r r
r
r
r
MingsianMingsian R R BaiBai53
(iv) is connecting and
independent o
Ex Think
f the pa
of dynamics is the force
thB
A
v dR A B
v
sdotintrr
r
1 2
1
0
Pf
(i) (ii) ( ) 0 (zero identity cg rule) irrotational
ˆ(ii) (iii) ( ) 0 (Stokes theorem)
(iii) (iv) ( ) 0
C S
C C C
C
v
v dR n v dS
v dR v dR
v dR
φ
=
rarr nablatimes = nablatimes nabla =
rarr sdot = sdot nablatimes =
rarr sdot = minus sdot =
rArr sdot =
int int
int int int
int
r
rr
rr r
r rr r
rr r
2
independent of pathC
v dRsdotintr
A
B
1C
2C
MingsianMingsian R R BaiBai54
1 2
(iv) (i) constant ( ) ( )
--(1)
But (2)
(1) (2) QED
Note on application
Check
B
C C A
v dR v dR B A d
v dR d
d dx dy dz dRx y z
v dR dR v R
φ φ φ
φφ φ φφ φ
φ φ
rarr sdot = sdot = = minus =
rArr sdot =part part part
= + + = nabla sdot minus minus minus minus minuspart part part
sdot = nabla sdot rArr = nabla forall
int int intr rr r
rr
r
r r rr r
use simpler path (iv)if 0
find (i)v
φ⎧
nablatimes = ⎨⎩
rr
A
B
x
y
Cz
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai36
Integral TheoremsHeart of this chapter
Interchange of dimensionality Div Curl Grad in integrals High-dimensional integration
rarr
1
Thm1 Divergence theorem Let be a closed region bounded by piecewise smooth orientable surface and let (con
I Divergence theor
tinuous derivatives up to the first order
em (Gauss theorem)
)
VS v Cisin
r
ˆ If denotes the outward unit normal on
the ˆn V S
v dV n v dS
n S
nablasdot = sdotint intr r
MingsianMingsian R R BaiBai37
jS
jP
jV
S
0
ˆBy definition ( ) lim
ˆ( )
ˆ( )
j
j
j
j
Sj V
j
j j S
j j Sj j
n v dSv P
V
v P V n v dS
v P V n v dS
Δ rarr
⎧ ⎫sdot⎪ ⎪nabla sdot ⎨ ⎬Δ⎪ ⎪⎩ ⎭
nabla sdot Δ asymp sdot
nabla sdot Δ = sdot
int
intsum sumint
rr
r r
r r
Pf
MingsianMingsian R R BaiBai38
0
After interior surfaces cancel out only exterior surface (Why Opposite normal vectors)
ˆ( )
volumes exterior surfaces
ˆlim
QE
survive
j
j
j j Sj j
V SV
v p V n v dS
v dV n v dSΔ rarr
nabla sdot Δ = sdot
rArr nablasdot = sdot
sum sumint
int int
r r
r r
D
MingsianMingsian R R BaiBai39
( )
scalar variable constant vectorvector variable
Alternative formsLet be a an arbitrary and
a
ˆ
1 Le
(
ˆt
V
V S
S
V
v ap
v d
v dV vn d
V n v dS
v va
va dV v a v a
S
nablasdot = sdot
= rArr
nablasdot = nabla sdot + nabla
=
sdot
nablaint intint int
int
r
v
r r
r r
r r rQ ( )
( )
ˆ)
2 Let ˆV S
V
V S
dV n va dS
v a p
a p
p dV n p d
d p
S
V a
⎛ ⎞= sdot⎜ ⎟
⎝ ⎠
= times rArr
nablasdot times = sdot nabla
times
times
nabla = times
int int
int int
int
r
r r r
r r v
r
vQ
r
( ) ( ) ( )
( ) ( )
ˆ
ˆV S
V S
a p dV n a p dS
a p dV a n p dS
⎛ ⎞⎡ ⎤minus sdot nablatimes = sdot times⎣ ⎦⎜ ⎟⎜ ⎟⎜ ⎟rArr minus sdot nablatimes = minus sdot times⎜ ⎟⎝ ⎠
int int
int int
r rv v
r rv v
MingsianMingsian R R BaiBai40
( )
( )
From Leibnitzs rule
(1-D) ( )
( ) ( )
B t
A t
d f x t dxdt
f x t dx B ttpart prime= +part
int( )
( )( ( ) ) ( )
B t
A tf B t t A tprimeminusint ( ( ) )
(assume ( ) ( ) )
f A t t
A t B t const= =
ContinuityEx The Equation
rate of increase of mass
mass densityV
dM d dVdt dt
ρ
ρ
= int
flow velocityvv
MingsianMingsian R R BaiBai41
ˆ ˆ ( )
(
Conservation
) ( Divergence theorem)
of mass
( ) 0
V S S
V
V
dV n v dS n v dSt
v dV
v dVt
ρ ρ ρ
ρ
ρ ρ⎡ ⎤⎢ ⎥⎣ ⎦
part = minus sdot = minus sdotpart
= minus nablasdot
partrArr +nablasdot =part
int int int
int
int
v v
vQ
v
ˆOn the other hand rate influof mass through n ox i t
Extended to 3-D for time-independent fixed boundary
(3-D) ( ) ( )
Thus
S
V V
V
S V n v dS
Sd f x y z t dV f x y z t dVdt tdM dVdt t
ρ
ρ
= minus sdot
part=part
part=part
int
int int
intv
Fixed control volume (Eulerian coordinates)Otherwise Raynoldrsquos transport theoremMoving control volume Lagrangian coordinates
MingsianMingsian R R BaiBai42
But the control volime is only arbitrary
In particular for flow
( ) 0
con
(continuity equati
stant
on)
Divergence the
0 incompress
Note is t
ible
orem h
V
v
vt
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
e mother of the other integral theorems
MingsianMingsian R R BaiBai43
Ex Archimedes principle ( ) The buoyant force ( ) on a body with volume immersed in a fluid is equal to the weight of the fluid
ˆ displaced by the object ie In the followin
B
V
gVρ= minusf k
阿基米德原理
浮力
ˆg figure the ˆ with and being the fluid density gravitational
acceleration depth and outward normal unit vector to the surface
fhydraulic pressure g z
g z
S
ρ
ρ
= minusp n
n
1
the modified divergence theorem
ˆ
Hint
scalar fuctionV S
vdV v dS v Cnabla = isinint int n
MingsianMingsian R R BaiBai44
Levelsurfacefluid
V
S
airyx
z
ˆf g zρ= minusp n
ˆPf ( ) ( )
ˆ ˆ ˆ ˆ ˆ (0 0 ) v v
B fS S V
V V
g z g zdS dS dV
g dV g dV gV
ρ ρ
ρ ρ ρ
= = minus = minus nabla
= minus + + = minus = minus
int int int
int int
f p n
i j k k k
k
MingsianMingsian R R BaiBai45
Note This can also be thought of as simple balance of forces (resultant force that acts on and weight of filled with fluid)S V
2
~ a little history about George Green Recall the diveregence theorem
ˆ
Let
II th
e Green
are arbitrary fun
s identiti
ctions
es
V S
v dV n v dS
v u vu v C
nablasdot = sdot
= nablaisin
int intv v
v
ˆ ( ) ( )
ˆ ( )V S
V S
u v dV n u v dS
u v u v dV un v dS
nablasdot nabla = sdot nabla
nabla sdotnabla + nablasdotnabla = sdotnabla
int int
int int
MingsianMingsian R R BaiBai46
2 -( ) -(1)V S
vu v u v dV u dSnpart
nabla sdotnabla + nabla =partint int firs(Greens identt ity)
2
If
( ) --(2)
(1) (2) V S
u vuv u v u dV v dSn
harrpart
nabla sdotnabla + nabla =part
minus
int int
( )2 2
V S
dv duu v v u dV u v dSdn dn
⎛ ⎞nabla minus nabla = minus⎜ ⎟⎝ ⎠int int
second(Greens identity)
MingsianMingsian R R BaiBai47
( )
( )
( ) ( )
1st id ( )
Greens identities in the 1D case integration by parts
b b bb ba a
a a ab
ba
ab
ba
a
v udx v u v u dx u v uv dx uv
u v u v vu dx vu
uv vu dx uv vu
primeprime prime prime prime prime prime primeprime prime= minus rArr + =
prime prime primeprime primeharr + =
primeprime primeprime prime primerArr minus = minus
int int int
int
2
2nd id ( )
Good for any adjoint operators in addition to Good also for and integration by parts Important for theory amp reciprocity
From th
Gree
e 1s
2-
t
ns functi
identity
D 1-
tak 1
D
e
on
u
nablararr
=
int
2
V S
vv dV dSnpartnabla =part
rArr int int2Ex 0 st 0 on the boundary 0 ( 1st id u u unabla = = rArr equiv 以 証)
MingsianMingsian R R BaiBai48
1
Thm1 Stokes theorem Let be defined in Let be a piecewise-smooth orientable surface bounded by a piec
III Stokes theorem and
ewise-smooth simple clos
Greens
ed curve Then
theore
m
v C R S
C
isinv
( )
ˆwhere is a unit normal to according t right hand rulo the e
ˆS C
n S
n v dS v dRsdot nablatimes = sdotint intvv v
(not necessarily planar)S
Cn vnablatimes v vv
MingsianMingsian R R BaiBai49
Pf
( )( )
circulatioWe have shown in deriving thatthe in -direction
ˆor
ˆ
n
z
z
j j j j j
j j j j jj j
vz v dx dy
v dR n v S
v dR n v S
nablatimes
=nablatimes
sdot = sdot nablatimes Δ
sdot = sdot nablatimes Δ
intsum sumint
v
v
vv v
vv v
C
S
jS
jC
jP
ˆ jn
jSΔ
MingsianMingsian R R BaiBai50
( )
( )ext surface
After the interior boundaries cancel out only the exterior boundaries (Why)
ˆ
ˆ0 QED
Alternative forms scalar
survive
va
j
j j j j jC
jC S
v dR n v S
S v dR n v dS
φ
sdot = sdot nablatimes Δ
Δ rarr rArr sdot = sdot nablatimes
sum sumint
int int
vv v
vv v
( )
riable constant vector vector variable
1 Let
2 Let
ˆ
ˆ S C
S C
a p
v n dR dR
n p dS dR p
a
v a p
φ φφ= rArr timesnabla =
timesnabla times= rArr timestimes =
int int
int int
v v
v v
v
v v
vv vv v
MingsianMingsian R R BaiBai51
( ) ( )
( )
( )
( ) ( )
1
Thm2 Greens theorem ( planar version of Stokes theorem)ˆ ˆ Let
pf By Stokes theoremˆ
ˆ ˆ ˆ ˆ ˆ ˆ
S C
S C
S C
v P x y i Q x y j C
n v dS v dR
k Pi Qj dS Pi Qj dx i dy
Q P dS Pdx Qdyx y
⎛ ⎞part partminus = +⎜ ⎟
= + isin
sdot nablatimes = sdot
⎡ ⎤sdot nablatimes + =
part part⎝ ⎠
+ sdot +⎣ ⎦
int int
int
int
int
int
v
vv v
( )
( )
( )
ˆ
ˆˆ ˆ
ˆˆ ˆ ˆ ˆ( ) ( )
0
QEDS C
j
i j kQ PPi Qj i j k
x y z x yP Q
Q P dS Pdx Qdyx y
⎛ ⎞part part part part partnablatimes + = = + + minus⎜ ⎟part part part part part⎝ ⎠
⎛ ⎞part partrArr minus = +⎜ ⎟part part⎝ ⎠int int
L L
S
C
i
j
x
y
MingsianMingsian R R BaiBai52
Potential TheorypotentiGeorge Green coined the term (l
a )
rarr 位能 勢能
例子 重力場 電磁場 流場 聲場 彈性力學
1
2
Thm1 If is in a simply connected region then the following statements are equivalent(i) There exists a scala
Conservative Field
potential functior such that
(ii
n
) 0
v
C
C
v
v R
φφ=nabla
nablatimes
isin
isin
=r
r
r
r in (Irrotational field curl-free)
cf 0 st solenoidal field divergence-free
0
(iii) in (Conservative field)C
Rv w v w
v R Rd
nablasdot = exist nabla
sdot =
= times
int
r r r
r
r
r
MingsianMingsian R R BaiBai53
(iv) is connecting and
independent o
Ex Think
f the pa
of dynamics is the force
thB
A
v dR A B
v
sdotintrr
r
1 2
1
0
Pf
(i) (ii) ( ) 0 (zero identity cg rule) irrotational
ˆ(ii) (iii) ( ) 0 (Stokes theorem)
(iii) (iv) ( ) 0
C S
C C C
C
v
v dR n v dS
v dR v dR
v dR
φ
=
rarr nablatimes = nablatimes nabla =
rarr sdot = sdot nablatimes =
rarr sdot = minus sdot =
rArr sdot =
int int
int int int
int
r
rr
rr r
r rr r
rr r
2
independent of pathC
v dRsdotintr
A
B
1C
2C
MingsianMingsian R R BaiBai54
1 2
(iv) (i) constant ( ) ( )
--(1)
But (2)
(1) (2) QED
Note on application
Check
B
C C A
v dR v dR B A d
v dR d
d dx dy dz dRx y z
v dR dR v R
φ φ φ
φφ φ φφ φ
φ φ
rarr sdot = sdot = = minus =
rArr sdot =part part part
= + + = nabla sdot minus minus minus minus minuspart part part
sdot = nabla sdot rArr = nabla forall
int int intr rr r
rr
r
r r rr r
use simpler path (iv)if 0
find (i)v
φ⎧
nablatimes = ⎨⎩
rr
A
B
x
y
Cz
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai37
jS
jP
jV
S
0
ˆBy definition ( ) lim
ˆ( )
ˆ( )
j
j
j
j
Sj V
j
j j S
j j Sj j
n v dSv P
V
v P V n v dS
v P V n v dS
Δ rarr
⎧ ⎫sdot⎪ ⎪nabla sdot ⎨ ⎬Δ⎪ ⎪⎩ ⎭
nabla sdot Δ asymp sdot
nabla sdot Δ = sdot
int
intsum sumint
rr
r r
r r
Pf
MingsianMingsian R R BaiBai38
0
After interior surfaces cancel out only exterior surface (Why Opposite normal vectors)
ˆ( )
volumes exterior surfaces
ˆlim
QE
survive
j
j
j j Sj j
V SV
v p V n v dS
v dV n v dSΔ rarr
nabla sdot Δ = sdot
rArr nablasdot = sdot
sum sumint
int int
r r
r r
D
MingsianMingsian R R BaiBai39
( )
scalar variable constant vectorvector variable
Alternative formsLet be a an arbitrary and
a
ˆ
1 Le
(
ˆt
V
V S
S
V
v ap
v d
v dV vn d
V n v dS
v va
va dV v a v a
S
nablasdot = sdot
= rArr
nablasdot = nabla sdot + nabla
=
sdot
nablaint intint int
int
r
v
r r
r r
r r rQ ( )
( )
ˆ)
2 Let ˆV S
V
V S
dV n va dS
v a p
a p
p dV n p d
d p
S
V a
⎛ ⎞= sdot⎜ ⎟
⎝ ⎠
= times rArr
nablasdot times = sdot nabla
times
times
nabla = times
int int
int int
int
r
r r r
r r v
r
vQ
r
( ) ( ) ( )
( ) ( )
ˆ
ˆV S
V S
a p dV n a p dS
a p dV a n p dS
⎛ ⎞⎡ ⎤minus sdot nablatimes = sdot times⎣ ⎦⎜ ⎟⎜ ⎟⎜ ⎟rArr minus sdot nablatimes = minus sdot times⎜ ⎟⎝ ⎠
int int
int int
r rv v
r rv v
MingsianMingsian R R BaiBai40
( )
( )
From Leibnitzs rule
(1-D) ( )
( ) ( )
B t
A t
d f x t dxdt
f x t dx B ttpart prime= +part
int( )
( )( ( ) ) ( )
B t
A tf B t t A tprimeminusint ( ( ) )
(assume ( ) ( ) )
f A t t
A t B t const= =
ContinuityEx The Equation
rate of increase of mass
mass densityV
dM d dVdt dt
ρ
ρ
= int
flow velocityvv
MingsianMingsian R R BaiBai41
ˆ ˆ ( )
(
Conservation
) ( Divergence theorem)
of mass
( ) 0
V S S
V
V
dV n v dS n v dSt
v dV
v dVt
ρ ρ ρ
ρ
ρ ρ⎡ ⎤⎢ ⎥⎣ ⎦
part = minus sdot = minus sdotpart
= minus nablasdot
partrArr +nablasdot =part
int int int
int
int
v v
vQ
v
ˆOn the other hand rate influof mass through n ox i t
Extended to 3-D for time-independent fixed boundary
(3-D) ( ) ( )
Thus
S
V V
V
S V n v dS
Sd f x y z t dV f x y z t dVdt tdM dVdt t
ρ
ρ
= minus sdot
part=part
part=part
int
int int
intv
Fixed control volume (Eulerian coordinates)Otherwise Raynoldrsquos transport theoremMoving control volume Lagrangian coordinates
MingsianMingsian R R BaiBai42
But the control volime is only arbitrary
In particular for flow
( ) 0
con
(continuity equati
stant
on)
Divergence the
0 incompress
Note is t
ible
orem h
V
v
vt
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
e mother of the other integral theorems
MingsianMingsian R R BaiBai43
Ex Archimedes principle ( ) The buoyant force ( ) on a body with volume immersed in a fluid is equal to the weight of the fluid
ˆ displaced by the object ie In the followin
B
V
gVρ= minusf k
阿基米德原理
浮力
ˆg figure the ˆ with and being the fluid density gravitational
acceleration depth and outward normal unit vector to the surface
fhydraulic pressure g z
g z
S
ρ
ρ
= minusp n
n
1
the modified divergence theorem
ˆ
Hint
scalar fuctionV S
vdV v dS v Cnabla = isinint int n
MingsianMingsian R R BaiBai44
Levelsurfacefluid
V
S
airyx
z
ˆf g zρ= minusp n
ˆPf ( ) ( )
ˆ ˆ ˆ ˆ ˆ (0 0 ) v v
B fS S V
V V
g z g zdS dS dV
g dV g dV gV
ρ ρ
ρ ρ ρ
= = minus = minus nabla
= minus + + = minus = minus
int int int
int int
f p n
i j k k k
k
MingsianMingsian R R BaiBai45
Note This can also be thought of as simple balance of forces (resultant force that acts on and weight of filled with fluid)S V
2
~ a little history about George Green Recall the diveregence theorem
ˆ
Let
II th
e Green
are arbitrary fun
s identiti
ctions
es
V S
v dV n v dS
v u vu v C
nablasdot = sdot
= nablaisin
int intv v
v
ˆ ( ) ( )
ˆ ( )V S
V S
u v dV n u v dS
u v u v dV un v dS
nablasdot nabla = sdot nabla
nabla sdotnabla + nablasdotnabla = sdotnabla
int int
int int
MingsianMingsian R R BaiBai46
2 -( ) -(1)V S
vu v u v dV u dSnpart
nabla sdotnabla + nabla =partint int firs(Greens identt ity)
2
If
( ) --(2)
(1) (2) V S
u vuv u v u dV v dSn
harrpart
nabla sdotnabla + nabla =part
minus
int int
( )2 2
V S
dv duu v v u dV u v dSdn dn
⎛ ⎞nabla minus nabla = minus⎜ ⎟⎝ ⎠int int
second(Greens identity)
MingsianMingsian R R BaiBai47
( )
( )
( ) ( )
1st id ( )
Greens identities in the 1D case integration by parts
b b bb ba a
a a ab
ba
ab
ba
a
v udx v u v u dx u v uv dx uv
u v u v vu dx vu
uv vu dx uv vu
primeprime prime prime prime prime prime primeprime prime= minus rArr + =
prime prime primeprime primeharr + =
primeprime primeprime prime primerArr minus = minus
int int int
int
2
2nd id ( )
Good for any adjoint operators in addition to Good also for and integration by parts Important for theory amp reciprocity
From th
Gree
e 1s
2-
t
ns functi
identity
D 1-
tak 1
D
e
on
u
nablararr
=
int
2
V S
vv dV dSnpartnabla =part
rArr int int2Ex 0 st 0 on the boundary 0 ( 1st id u u unabla = = rArr equiv 以 証)
MingsianMingsian R R BaiBai48
1
Thm1 Stokes theorem Let be defined in Let be a piecewise-smooth orientable surface bounded by a piec
III Stokes theorem and
ewise-smooth simple clos
Greens
ed curve Then
theore
m
v C R S
C
isinv
( )
ˆwhere is a unit normal to according t right hand rulo the e
ˆS C
n S
n v dS v dRsdot nablatimes = sdotint intvv v
(not necessarily planar)S
Cn vnablatimes v vv
MingsianMingsian R R BaiBai49
Pf
( )( )
circulatioWe have shown in deriving thatthe in -direction
ˆor
ˆ
n
z
z
j j j j j
j j j j jj j
vz v dx dy
v dR n v S
v dR n v S
nablatimes
=nablatimes
sdot = sdot nablatimes Δ
sdot = sdot nablatimes Δ
intsum sumint
v
v
vv v
vv v
C
S
jS
jC
jP
ˆ jn
jSΔ
MingsianMingsian R R BaiBai50
( )
( )ext surface
After the interior boundaries cancel out only the exterior boundaries (Why)
ˆ
ˆ0 QED
Alternative forms scalar
survive
va
j
j j j j jC
jC S
v dR n v S
S v dR n v dS
φ
sdot = sdot nablatimes Δ
Δ rarr rArr sdot = sdot nablatimes
sum sumint
int int
vv v
vv v
( )
riable constant vector vector variable
1 Let
2 Let
ˆ
ˆ S C
S C
a p
v n dR dR
n p dS dR p
a
v a p
φ φφ= rArr timesnabla =
timesnabla times= rArr timestimes =
int int
int int
v v
v v
v
v v
vv vv v
MingsianMingsian R R BaiBai51
( ) ( )
( )
( )
( ) ( )
1
Thm2 Greens theorem ( planar version of Stokes theorem)ˆ ˆ Let
pf By Stokes theoremˆ
ˆ ˆ ˆ ˆ ˆ ˆ
S C
S C
S C
v P x y i Q x y j C
n v dS v dR
k Pi Qj dS Pi Qj dx i dy
Q P dS Pdx Qdyx y
⎛ ⎞part partminus = +⎜ ⎟
= + isin
sdot nablatimes = sdot
⎡ ⎤sdot nablatimes + =
part part⎝ ⎠
+ sdot +⎣ ⎦
int int
int
int
int
int
v
vv v
( )
( )
( )
ˆ
ˆˆ ˆ
ˆˆ ˆ ˆ ˆ( ) ( )
0
QEDS C
j
i j kQ PPi Qj i j k
x y z x yP Q
Q P dS Pdx Qdyx y
⎛ ⎞part part part part partnablatimes + = = + + minus⎜ ⎟part part part part part⎝ ⎠
⎛ ⎞part partrArr minus = +⎜ ⎟part part⎝ ⎠int int
L L
S
C
i
j
x
y
MingsianMingsian R R BaiBai52
Potential TheorypotentiGeorge Green coined the term (l
a )
rarr 位能 勢能
例子 重力場 電磁場 流場 聲場 彈性力學
1
2
Thm1 If is in a simply connected region then the following statements are equivalent(i) There exists a scala
Conservative Field
potential functior such that
(ii
n
) 0
v
C
C
v
v R
φφ=nabla
nablatimes
isin
isin
=r
r
r
r in (Irrotational field curl-free)
cf 0 st solenoidal field divergence-free
0
(iii) in (Conservative field)C
Rv w v w
v R Rd
nablasdot = exist nabla
sdot =
= times
int
r r r
r
r
r
MingsianMingsian R R BaiBai53
(iv) is connecting and
independent o
Ex Think
f the pa
of dynamics is the force
thB
A
v dR A B
v
sdotintrr
r
1 2
1
0
Pf
(i) (ii) ( ) 0 (zero identity cg rule) irrotational
ˆ(ii) (iii) ( ) 0 (Stokes theorem)
(iii) (iv) ( ) 0
C S
C C C
C
v
v dR n v dS
v dR v dR
v dR
φ
=
rarr nablatimes = nablatimes nabla =
rarr sdot = sdot nablatimes =
rarr sdot = minus sdot =
rArr sdot =
int int
int int int
int
r
rr
rr r
r rr r
rr r
2
independent of pathC
v dRsdotintr
A
B
1C
2C
MingsianMingsian R R BaiBai54
1 2
(iv) (i) constant ( ) ( )
--(1)
But (2)
(1) (2) QED
Note on application
Check
B
C C A
v dR v dR B A d
v dR d
d dx dy dz dRx y z
v dR dR v R
φ φ φ
φφ φ φφ φ
φ φ
rarr sdot = sdot = = minus =
rArr sdot =part part part
= + + = nabla sdot minus minus minus minus minuspart part part
sdot = nabla sdot rArr = nabla forall
int int intr rr r
rr
r
r r rr r
use simpler path (iv)if 0
find (i)v
φ⎧
nablatimes = ⎨⎩
rr
A
B
x
y
Cz
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai38
0
After interior surfaces cancel out only exterior surface (Why Opposite normal vectors)
ˆ( )
volumes exterior surfaces
ˆlim
QE
survive
j
j
j j Sj j
V SV
v p V n v dS
v dV n v dSΔ rarr
nabla sdot Δ = sdot
rArr nablasdot = sdot
sum sumint
int int
r r
r r
D
MingsianMingsian R R BaiBai39
( )
scalar variable constant vectorvector variable
Alternative formsLet be a an arbitrary and
a
ˆ
1 Le
(
ˆt
V
V S
S
V
v ap
v d
v dV vn d
V n v dS
v va
va dV v a v a
S
nablasdot = sdot
= rArr
nablasdot = nabla sdot + nabla
=
sdot
nablaint intint int
int
r
v
r r
r r
r r rQ ( )
( )
ˆ)
2 Let ˆV S
V
V S
dV n va dS
v a p
a p
p dV n p d
d p
S
V a
⎛ ⎞= sdot⎜ ⎟
⎝ ⎠
= times rArr
nablasdot times = sdot nabla
times
times
nabla = times
int int
int int
int
r
r r r
r r v
r
vQ
r
( ) ( ) ( )
( ) ( )
ˆ
ˆV S
V S
a p dV n a p dS
a p dV a n p dS
⎛ ⎞⎡ ⎤minus sdot nablatimes = sdot times⎣ ⎦⎜ ⎟⎜ ⎟⎜ ⎟rArr minus sdot nablatimes = minus sdot times⎜ ⎟⎝ ⎠
int int
int int
r rv v
r rv v
MingsianMingsian R R BaiBai40
( )
( )
From Leibnitzs rule
(1-D) ( )
( ) ( )
B t
A t
d f x t dxdt
f x t dx B ttpart prime= +part
int( )
( )( ( ) ) ( )
B t
A tf B t t A tprimeminusint ( ( ) )
(assume ( ) ( ) )
f A t t
A t B t const= =
ContinuityEx The Equation
rate of increase of mass
mass densityV
dM d dVdt dt
ρ
ρ
= int
flow velocityvv
MingsianMingsian R R BaiBai41
ˆ ˆ ( )
(
Conservation
) ( Divergence theorem)
of mass
( ) 0
V S S
V
V
dV n v dS n v dSt
v dV
v dVt
ρ ρ ρ
ρ
ρ ρ⎡ ⎤⎢ ⎥⎣ ⎦
part = minus sdot = minus sdotpart
= minus nablasdot
partrArr +nablasdot =part
int int int
int
int
v v
vQ
v
ˆOn the other hand rate influof mass through n ox i t
Extended to 3-D for time-independent fixed boundary
(3-D) ( ) ( )
Thus
S
V V
V
S V n v dS
Sd f x y z t dV f x y z t dVdt tdM dVdt t
ρ
ρ
= minus sdot
part=part
part=part
int
int int
intv
Fixed control volume (Eulerian coordinates)Otherwise Raynoldrsquos transport theoremMoving control volume Lagrangian coordinates
MingsianMingsian R R BaiBai42
But the control volime is only arbitrary
In particular for flow
( ) 0
con
(continuity equati
stant
on)
Divergence the
0 incompress
Note is t
ible
orem h
V
v
vt
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
e mother of the other integral theorems
MingsianMingsian R R BaiBai43
Ex Archimedes principle ( ) The buoyant force ( ) on a body with volume immersed in a fluid is equal to the weight of the fluid
ˆ displaced by the object ie In the followin
B
V
gVρ= minusf k
阿基米德原理
浮力
ˆg figure the ˆ with and being the fluid density gravitational
acceleration depth and outward normal unit vector to the surface
fhydraulic pressure g z
g z
S
ρ
ρ
= minusp n
n
1
the modified divergence theorem
ˆ
Hint
scalar fuctionV S
vdV v dS v Cnabla = isinint int n
MingsianMingsian R R BaiBai44
Levelsurfacefluid
V
S
airyx
z
ˆf g zρ= minusp n
ˆPf ( ) ( )
ˆ ˆ ˆ ˆ ˆ (0 0 ) v v
B fS S V
V V
g z g zdS dS dV
g dV g dV gV
ρ ρ
ρ ρ ρ
= = minus = minus nabla
= minus + + = minus = minus
int int int
int int
f p n
i j k k k
k
MingsianMingsian R R BaiBai45
Note This can also be thought of as simple balance of forces (resultant force that acts on and weight of filled with fluid)S V
2
~ a little history about George Green Recall the diveregence theorem
ˆ
Let
II th
e Green
are arbitrary fun
s identiti
ctions
es
V S
v dV n v dS
v u vu v C
nablasdot = sdot
= nablaisin
int intv v
v
ˆ ( ) ( )
ˆ ( )V S
V S
u v dV n u v dS
u v u v dV un v dS
nablasdot nabla = sdot nabla
nabla sdotnabla + nablasdotnabla = sdotnabla
int int
int int
MingsianMingsian R R BaiBai46
2 -( ) -(1)V S
vu v u v dV u dSnpart
nabla sdotnabla + nabla =partint int firs(Greens identt ity)
2
If
( ) --(2)
(1) (2) V S
u vuv u v u dV v dSn
harrpart
nabla sdotnabla + nabla =part
minus
int int
( )2 2
V S
dv duu v v u dV u v dSdn dn
⎛ ⎞nabla minus nabla = minus⎜ ⎟⎝ ⎠int int
second(Greens identity)
MingsianMingsian R R BaiBai47
( )
( )
( ) ( )
1st id ( )
Greens identities in the 1D case integration by parts
b b bb ba a
a a ab
ba
ab
ba
a
v udx v u v u dx u v uv dx uv
u v u v vu dx vu
uv vu dx uv vu
primeprime prime prime prime prime prime primeprime prime= minus rArr + =
prime prime primeprime primeharr + =
primeprime primeprime prime primerArr minus = minus
int int int
int
2
2nd id ( )
Good for any adjoint operators in addition to Good also for and integration by parts Important for theory amp reciprocity
From th
Gree
e 1s
2-
t
ns functi
identity
D 1-
tak 1
D
e
on
u
nablararr
=
int
2
V S
vv dV dSnpartnabla =part
rArr int int2Ex 0 st 0 on the boundary 0 ( 1st id u u unabla = = rArr equiv 以 証)
MingsianMingsian R R BaiBai48
1
Thm1 Stokes theorem Let be defined in Let be a piecewise-smooth orientable surface bounded by a piec
III Stokes theorem and
ewise-smooth simple clos
Greens
ed curve Then
theore
m
v C R S
C
isinv
( )
ˆwhere is a unit normal to according t right hand rulo the e
ˆS C
n S
n v dS v dRsdot nablatimes = sdotint intvv v
(not necessarily planar)S
Cn vnablatimes v vv
MingsianMingsian R R BaiBai49
Pf
( )( )
circulatioWe have shown in deriving thatthe in -direction
ˆor
ˆ
n
z
z
j j j j j
j j j j jj j
vz v dx dy
v dR n v S
v dR n v S
nablatimes
=nablatimes
sdot = sdot nablatimes Δ
sdot = sdot nablatimes Δ
intsum sumint
v
v
vv v
vv v
C
S
jS
jC
jP
ˆ jn
jSΔ
MingsianMingsian R R BaiBai50
( )
( )ext surface
After the interior boundaries cancel out only the exterior boundaries (Why)
ˆ
ˆ0 QED
Alternative forms scalar
survive
va
j
j j j j jC
jC S
v dR n v S
S v dR n v dS
φ
sdot = sdot nablatimes Δ
Δ rarr rArr sdot = sdot nablatimes
sum sumint
int int
vv v
vv v
( )
riable constant vector vector variable
1 Let
2 Let
ˆ
ˆ S C
S C
a p
v n dR dR
n p dS dR p
a
v a p
φ φφ= rArr timesnabla =
timesnabla times= rArr timestimes =
int int
int int
v v
v v
v
v v
vv vv v
MingsianMingsian R R BaiBai51
( ) ( )
( )
( )
( ) ( )
1
Thm2 Greens theorem ( planar version of Stokes theorem)ˆ ˆ Let
pf By Stokes theoremˆ
ˆ ˆ ˆ ˆ ˆ ˆ
S C
S C
S C
v P x y i Q x y j C
n v dS v dR
k Pi Qj dS Pi Qj dx i dy
Q P dS Pdx Qdyx y
⎛ ⎞part partminus = +⎜ ⎟
= + isin
sdot nablatimes = sdot
⎡ ⎤sdot nablatimes + =
part part⎝ ⎠
+ sdot +⎣ ⎦
int int
int
int
int
int
v
vv v
( )
( )
( )
ˆ
ˆˆ ˆ
ˆˆ ˆ ˆ ˆ( ) ( )
0
QEDS C
j
i j kQ PPi Qj i j k
x y z x yP Q
Q P dS Pdx Qdyx y
⎛ ⎞part part part part partnablatimes + = = + + minus⎜ ⎟part part part part part⎝ ⎠
⎛ ⎞part partrArr minus = +⎜ ⎟part part⎝ ⎠int int
L L
S
C
i
j
x
y
MingsianMingsian R R BaiBai52
Potential TheorypotentiGeorge Green coined the term (l
a )
rarr 位能 勢能
例子 重力場 電磁場 流場 聲場 彈性力學
1
2
Thm1 If is in a simply connected region then the following statements are equivalent(i) There exists a scala
Conservative Field
potential functior such that
(ii
n
) 0
v
C
C
v
v R
φφ=nabla
nablatimes
isin
isin
=r
r
r
r in (Irrotational field curl-free)
cf 0 st solenoidal field divergence-free
0
(iii) in (Conservative field)C
Rv w v w
v R Rd
nablasdot = exist nabla
sdot =
= times
int
r r r
r
r
r
MingsianMingsian R R BaiBai53
(iv) is connecting and
independent o
Ex Think
f the pa
of dynamics is the force
thB
A
v dR A B
v
sdotintrr
r
1 2
1
0
Pf
(i) (ii) ( ) 0 (zero identity cg rule) irrotational
ˆ(ii) (iii) ( ) 0 (Stokes theorem)
(iii) (iv) ( ) 0
C S
C C C
C
v
v dR n v dS
v dR v dR
v dR
φ
=
rarr nablatimes = nablatimes nabla =
rarr sdot = sdot nablatimes =
rarr sdot = minus sdot =
rArr sdot =
int int
int int int
int
r
rr
rr r
r rr r
rr r
2
independent of pathC
v dRsdotintr
A
B
1C
2C
MingsianMingsian R R BaiBai54
1 2
(iv) (i) constant ( ) ( )
--(1)
But (2)
(1) (2) QED
Note on application
Check
B
C C A
v dR v dR B A d
v dR d
d dx dy dz dRx y z
v dR dR v R
φ φ φ
φφ φ φφ φ
φ φ
rarr sdot = sdot = = minus =
rArr sdot =part part part
= + + = nabla sdot minus minus minus minus minuspart part part
sdot = nabla sdot rArr = nabla forall
int int intr rr r
rr
r
r r rr r
use simpler path (iv)if 0
find (i)v
φ⎧
nablatimes = ⎨⎩
rr
A
B
x
y
Cz
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai39
( )
scalar variable constant vectorvector variable
Alternative formsLet be a an arbitrary and
a
ˆ
1 Le
(
ˆt
V
V S
S
V
v ap
v d
v dV vn d
V n v dS
v va
va dV v a v a
S
nablasdot = sdot
= rArr
nablasdot = nabla sdot + nabla
=
sdot
nablaint intint int
int
r
v
r r
r r
r r rQ ( )
( )
ˆ)
2 Let ˆV S
V
V S
dV n va dS
v a p
a p
p dV n p d
d p
S
V a
⎛ ⎞= sdot⎜ ⎟
⎝ ⎠
= times rArr
nablasdot times = sdot nabla
times
times
nabla = times
int int
int int
int
r
r r r
r r v
r
vQ
r
( ) ( ) ( )
( ) ( )
ˆ
ˆV S
V S
a p dV n a p dS
a p dV a n p dS
⎛ ⎞⎡ ⎤minus sdot nablatimes = sdot times⎣ ⎦⎜ ⎟⎜ ⎟⎜ ⎟rArr minus sdot nablatimes = minus sdot times⎜ ⎟⎝ ⎠
int int
int int
r rv v
r rv v
MingsianMingsian R R BaiBai40
( )
( )
From Leibnitzs rule
(1-D) ( )
( ) ( )
B t
A t
d f x t dxdt
f x t dx B ttpart prime= +part
int( )
( )( ( ) ) ( )
B t
A tf B t t A tprimeminusint ( ( ) )
(assume ( ) ( ) )
f A t t
A t B t const= =
ContinuityEx The Equation
rate of increase of mass
mass densityV
dM d dVdt dt
ρ
ρ
= int
flow velocityvv
MingsianMingsian R R BaiBai41
ˆ ˆ ( )
(
Conservation
) ( Divergence theorem)
of mass
( ) 0
V S S
V
V
dV n v dS n v dSt
v dV
v dVt
ρ ρ ρ
ρ
ρ ρ⎡ ⎤⎢ ⎥⎣ ⎦
part = minus sdot = minus sdotpart
= minus nablasdot
partrArr +nablasdot =part
int int int
int
int
v v
vQ
v
ˆOn the other hand rate influof mass through n ox i t
Extended to 3-D for time-independent fixed boundary
(3-D) ( ) ( )
Thus
S
V V
V
S V n v dS
Sd f x y z t dV f x y z t dVdt tdM dVdt t
ρ
ρ
= minus sdot
part=part
part=part
int
int int
intv
Fixed control volume (Eulerian coordinates)Otherwise Raynoldrsquos transport theoremMoving control volume Lagrangian coordinates
MingsianMingsian R R BaiBai42
But the control volime is only arbitrary
In particular for flow
( ) 0
con
(continuity equati
stant
on)
Divergence the
0 incompress
Note is t
ible
orem h
V
v
vt
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
e mother of the other integral theorems
MingsianMingsian R R BaiBai43
Ex Archimedes principle ( ) The buoyant force ( ) on a body with volume immersed in a fluid is equal to the weight of the fluid
ˆ displaced by the object ie In the followin
B
V
gVρ= minusf k
阿基米德原理
浮力
ˆg figure the ˆ with and being the fluid density gravitational
acceleration depth and outward normal unit vector to the surface
fhydraulic pressure g z
g z
S
ρ
ρ
= minusp n
n
1
the modified divergence theorem
ˆ
Hint
scalar fuctionV S
vdV v dS v Cnabla = isinint int n
MingsianMingsian R R BaiBai44
Levelsurfacefluid
V
S
airyx
z
ˆf g zρ= minusp n
ˆPf ( ) ( )
ˆ ˆ ˆ ˆ ˆ (0 0 ) v v
B fS S V
V V
g z g zdS dS dV
g dV g dV gV
ρ ρ
ρ ρ ρ
= = minus = minus nabla
= minus + + = minus = minus
int int int
int int
f p n
i j k k k
k
MingsianMingsian R R BaiBai45
Note This can also be thought of as simple balance of forces (resultant force that acts on and weight of filled with fluid)S V
2
~ a little history about George Green Recall the diveregence theorem
ˆ
Let
II th
e Green
are arbitrary fun
s identiti
ctions
es
V S
v dV n v dS
v u vu v C
nablasdot = sdot
= nablaisin
int intv v
v
ˆ ( ) ( )
ˆ ( )V S
V S
u v dV n u v dS
u v u v dV un v dS
nablasdot nabla = sdot nabla
nabla sdotnabla + nablasdotnabla = sdotnabla
int int
int int
MingsianMingsian R R BaiBai46
2 -( ) -(1)V S
vu v u v dV u dSnpart
nabla sdotnabla + nabla =partint int firs(Greens identt ity)
2
If
( ) --(2)
(1) (2) V S
u vuv u v u dV v dSn
harrpart
nabla sdotnabla + nabla =part
minus
int int
( )2 2
V S
dv duu v v u dV u v dSdn dn
⎛ ⎞nabla minus nabla = minus⎜ ⎟⎝ ⎠int int
second(Greens identity)
MingsianMingsian R R BaiBai47
( )
( )
( ) ( )
1st id ( )
Greens identities in the 1D case integration by parts
b b bb ba a
a a ab
ba
ab
ba
a
v udx v u v u dx u v uv dx uv
u v u v vu dx vu
uv vu dx uv vu
primeprime prime prime prime prime prime primeprime prime= minus rArr + =
prime prime primeprime primeharr + =
primeprime primeprime prime primerArr minus = minus
int int int
int
2
2nd id ( )
Good for any adjoint operators in addition to Good also for and integration by parts Important for theory amp reciprocity
From th
Gree
e 1s
2-
t
ns functi
identity
D 1-
tak 1
D
e
on
u
nablararr
=
int
2
V S
vv dV dSnpartnabla =part
rArr int int2Ex 0 st 0 on the boundary 0 ( 1st id u u unabla = = rArr equiv 以 証)
MingsianMingsian R R BaiBai48
1
Thm1 Stokes theorem Let be defined in Let be a piecewise-smooth orientable surface bounded by a piec
III Stokes theorem and
ewise-smooth simple clos
Greens
ed curve Then
theore
m
v C R S
C
isinv
( )
ˆwhere is a unit normal to according t right hand rulo the e
ˆS C
n S
n v dS v dRsdot nablatimes = sdotint intvv v
(not necessarily planar)S
Cn vnablatimes v vv
MingsianMingsian R R BaiBai49
Pf
( )( )
circulatioWe have shown in deriving thatthe in -direction
ˆor
ˆ
n
z
z
j j j j j
j j j j jj j
vz v dx dy
v dR n v S
v dR n v S
nablatimes
=nablatimes
sdot = sdot nablatimes Δ
sdot = sdot nablatimes Δ
intsum sumint
v
v
vv v
vv v
C
S
jS
jC
jP
ˆ jn
jSΔ
MingsianMingsian R R BaiBai50
( )
( )ext surface
After the interior boundaries cancel out only the exterior boundaries (Why)
ˆ
ˆ0 QED
Alternative forms scalar
survive
va
j
j j j j jC
jC S
v dR n v S
S v dR n v dS
φ
sdot = sdot nablatimes Δ
Δ rarr rArr sdot = sdot nablatimes
sum sumint
int int
vv v
vv v
( )
riable constant vector vector variable
1 Let
2 Let
ˆ
ˆ S C
S C
a p
v n dR dR
n p dS dR p
a
v a p
φ φφ= rArr timesnabla =
timesnabla times= rArr timestimes =
int int
int int
v v
v v
v
v v
vv vv v
MingsianMingsian R R BaiBai51
( ) ( )
( )
( )
( ) ( )
1
Thm2 Greens theorem ( planar version of Stokes theorem)ˆ ˆ Let
pf By Stokes theoremˆ
ˆ ˆ ˆ ˆ ˆ ˆ
S C
S C
S C
v P x y i Q x y j C
n v dS v dR
k Pi Qj dS Pi Qj dx i dy
Q P dS Pdx Qdyx y
⎛ ⎞part partminus = +⎜ ⎟
= + isin
sdot nablatimes = sdot
⎡ ⎤sdot nablatimes + =
part part⎝ ⎠
+ sdot +⎣ ⎦
int int
int
int
int
int
v
vv v
( )
( )
( )
ˆ
ˆˆ ˆ
ˆˆ ˆ ˆ ˆ( ) ( )
0
QEDS C
j
i j kQ PPi Qj i j k
x y z x yP Q
Q P dS Pdx Qdyx y
⎛ ⎞part part part part partnablatimes + = = + + minus⎜ ⎟part part part part part⎝ ⎠
⎛ ⎞part partrArr minus = +⎜ ⎟part part⎝ ⎠int int
L L
S
C
i
j
x
y
MingsianMingsian R R BaiBai52
Potential TheorypotentiGeorge Green coined the term (l
a )
rarr 位能 勢能
例子 重力場 電磁場 流場 聲場 彈性力學
1
2
Thm1 If is in a simply connected region then the following statements are equivalent(i) There exists a scala
Conservative Field
potential functior such that
(ii
n
) 0
v
C
C
v
v R
φφ=nabla
nablatimes
isin
isin
=r
r
r
r in (Irrotational field curl-free)
cf 0 st solenoidal field divergence-free
0
(iii) in (Conservative field)C
Rv w v w
v R Rd
nablasdot = exist nabla
sdot =
= times
int
r r r
r
r
r
MingsianMingsian R R BaiBai53
(iv) is connecting and
independent o
Ex Think
f the pa
of dynamics is the force
thB
A
v dR A B
v
sdotintrr
r
1 2
1
0
Pf
(i) (ii) ( ) 0 (zero identity cg rule) irrotational
ˆ(ii) (iii) ( ) 0 (Stokes theorem)
(iii) (iv) ( ) 0
C S
C C C
C
v
v dR n v dS
v dR v dR
v dR
φ
=
rarr nablatimes = nablatimes nabla =
rarr sdot = sdot nablatimes =
rarr sdot = minus sdot =
rArr sdot =
int int
int int int
int
r
rr
rr r
r rr r
rr r
2
independent of pathC
v dRsdotintr
A
B
1C
2C
MingsianMingsian R R BaiBai54
1 2
(iv) (i) constant ( ) ( )
--(1)
But (2)
(1) (2) QED
Note on application
Check
B
C C A
v dR v dR B A d
v dR d
d dx dy dz dRx y z
v dR dR v R
φ φ φ
φφ φ φφ φ
φ φ
rarr sdot = sdot = = minus =
rArr sdot =part part part
= + + = nabla sdot minus minus minus minus minuspart part part
sdot = nabla sdot rArr = nabla forall
int int intr rr r
rr
r
r r rr r
use simpler path (iv)if 0
find (i)v
φ⎧
nablatimes = ⎨⎩
rr
A
B
x
y
Cz
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai40
( )
( )
From Leibnitzs rule
(1-D) ( )
( ) ( )
B t
A t
d f x t dxdt
f x t dx B ttpart prime= +part
int( )
( )( ( ) ) ( )
B t
A tf B t t A tprimeminusint ( ( ) )
(assume ( ) ( ) )
f A t t
A t B t const= =
ContinuityEx The Equation
rate of increase of mass
mass densityV
dM d dVdt dt
ρ
ρ
= int
flow velocityvv
MingsianMingsian R R BaiBai41
ˆ ˆ ( )
(
Conservation
) ( Divergence theorem)
of mass
( ) 0
V S S
V
V
dV n v dS n v dSt
v dV
v dVt
ρ ρ ρ
ρ
ρ ρ⎡ ⎤⎢ ⎥⎣ ⎦
part = minus sdot = minus sdotpart
= minus nablasdot
partrArr +nablasdot =part
int int int
int
int
v v
vQ
v
ˆOn the other hand rate influof mass through n ox i t
Extended to 3-D for time-independent fixed boundary
(3-D) ( ) ( )
Thus
S
V V
V
S V n v dS
Sd f x y z t dV f x y z t dVdt tdM dVdt t
ρ
ρ
= minus sdot
part=part
part=part
int
int int
intv
Fixed control volume (Eulerian coordinates)Otherwise Raynoldrsquos transport theoremMoving control volume Lagrangian coordinates
MingsianMingsian R R BaiBai42
But the control volime is only arbitrary
In particular for flow
( ) 0
con
(continuity equati
stant
on)
Divergence the
0 incompress
Note is t
ible
orem h
V
v
vt
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
e mother of the other integral theorems
MingsianMingsian R R BaiBai43
Ex Archimedes principle ( ) The buoyant force ( ) on a body with volume immersed in a fluid is equal to the weight of the fluid
ˆ displaced by the object ie In the followin
B
V
gVρ= minusf k
阿基米德原理
浮力
ˆg figure the ˆ with and being the fluid density gravitational
acceleration depth and outward normal unit vector to the surface
fhydraulic pressure g z
g z
S
ρ
ρ
= minusp n
n
1
the modified divergence theorem
ˆ
Hint
scalar fuctionV S
vdV v dS v Cnabla = isinint int n
MingsianMingsian R R BaiBai44
Levelsurfacefluid
V
S
airyx
z
ˆf g zρ= minusp n
ˆPf ( ) ( )
ˆ ˆ ˆ ˆ ˆ (0 0 ) v v
B fS S V
V V
g z g zdS dS dV
g dV g dV gV
ρ ρ
ρ ρ ρ
= = minus = minus nabla
= minus + + = minus = minus
int int int
int int
f p n
i j k k k
k
MingsianMingsian R R BaiBai45
Note This can also be thought of as simple balance of forces (resultant force that acts on and weight of filled with fluid)S V
2
~ a little history about George Green Recall the diveregence theorem
ˆ
Let
II th
e Green
are arbitrary fun
s identiti
ctions
es
V S
v dV n v dS
v u vu v C
nablasdot = sdot
= nablaisin
int intv v
v
ˆ ( ) ( )
ˆ ( )V S
V S
u v dV n u v dS
u v u v dV un v dS
nablasdot nabla = sdot nabla
nabla sdotnabla + nablasdotnabla = sdotnabla
int int
int int
MingsianMingsian R R BaiBai46
2 -( ) -(1)V S
vu v u v dV u dSnpart
nabla sdotnabla + nabla =partint int firs(Greens identt ity)
2
If
( ) --(2)
(1) (2) V S
u vuv u v u dV v dSn
harrpart
nabla sdotnabla + nabla =part
minus
int int
( )2 2
V S
dv duu v v u dV u v dSdn dn
⎛ ⎞nabla minus nabla = minus⎜ ⎟⎝ ⎠int int
second(Greens identity)
MingsianMingsian R R BaiBai47
( )
( )
( ) ( )
1st id ( )
Greens identities in the 1D case integration by parts
b b bb ba a
a a ab
ba
ab
ba
a
v udx v u v u dx u v uv dx uv
u v u v vu dx vu
uv vu dx uv vu
primeprime prime prime prime prime prime primeprime prime= minus rArr + =
prime prime primeprime primeharr + =
primeprime primeprime prime primerArr minus = minus
int int int
int
2
2nd id ( )
Good for any adjoint operators in addition to Good also for and integration by parts Important for theory amp reciprocity
From th
Gree
e 1s
2-
t
ns functi
identity
D 1-
tak 1
D
e
on
u
nablararr
=
int
2
V S
vv dV dSnpartnabla =part
rArr int int2Ex 0 st 0 on the boundary 0 ( 1st id u u unabla = = rArr equiv 以 証)
MingsianMingsian R R BaiBai48
1
Thm1 Stokes theorem Let be defined in Let be a piecewise-smooth orientable surface bounded by a piec
III Stokes theorem and
ewise-smooth simple clos
Greens
ed curve Then
theore
m
v C R S
C
isinv
( )
ˆwhere is a unit normal to according t right hand rulo the e
ˆS C
n S
n v dS v dRsdot nablatimes = sdotint intvv v
(not necessarily planar)S
Cn vnablatimes v vv
MingsianMingsian R R BaiBai49
Pf
( )( )
circulatioWe have shown in deriving thatthe in -direction
ˆor
ˆ
n
z
z
j j j j j
j j j j jj j
vz v dx dy
v dR n v S
v dR n v S
nablatimes
=nablatimes
sdot = sdot nablatimes Δ
sdot = sdot nablatimes Δ
intsum sumint
v
v
vv v
vv v
C
S
jS
jC
jP
ˆ jn
jSΔ
MingsianMingsian R R BaiBai50
( )
( )ext surface
After the interior boundaries cancel out only the exterior boundaries (Why)
ˆ
ˆ0 QED
Alternative forms scalar
survive
va
j
j j j j jC
jC S
v dR n v S
S v dR n v dS
φ
sdot = sdot nablatimes Δ
Δ rarr rArr sdot = sdot nablatimes
sum sumint
int int
vv v
vv v
( )
riable constant vector vector variable
1 Let
2 Let
ˆ
ˆ S C
S C
a p
v n dR dR
n p dS dR p
a
v a p
φ φφ= rArr timesnabla =
timesnabla times= rArr timestimes =
int int
int int
v v
v v
v
v v
vv vv v
MingsianMingsian R R BaiBai51
( ) ( )
( )
( )
( ) ( )
1
Thm2 Greens theorem ( planar version of Stokes theorem)ˆ ˆ Let
pf By Stokes theoremˆ
ˆ ˆ ˆ ˆ ˆ ˆ
S C
S C
S C
v P x y i Q x y j C
n v dS v dR
k Pi Qj dS Pi Qj dx i dy
Q P dS Pdx Qdyx y
⎛ ⎞part partminus = +⎜ ⎟
= + isin
sdot nablatimes = sdot
⎡ ⎤sdot nablatimes + =
part part⎝ ⎠
+ sdot +⎣ ⎦
int int
int
int
int
int
v
vv v
( )
( )
( )
ˆ
ˆˆ ˆ
ˆˆ ˆ ˆ ˆ( ) ( )
0
QEDS C
j
i j kQ PPi Qj i j k
x y z x yP Q
Q P dS Pdx Qdyx y
⎛ ⎞part part part part partnablatimes + = = + + minus⎜ ⎟part part part part part⎝ ⎠
⎛ ⎞part partrArr minus = +⎜ ⎟part part⎝ ⎠int int
L L
S
C
i
j
x
y
MingsianMingsian R R BaiBai52
Potential TheorypotentiGeorge Green coined the term (l
a )
rarr 位能 勢能
例子 重力場 電磁場 流場 聲場 彈性力學
1
2
Thm1 If is in a simply connected region then the following statements are equivalent(i) There exists a scala
Conservative Field
potential functior such that
(ii
n
) 0
v
C
C
v
v R
φφ=nabla
nablatimes
isin
isin
=r
r
r
r in (Irrotational field curl-free)
cf 0 st solenoidal field divergence-free
0
(iii) in (Conservative field)C
Rv w v w
v R Rd
nablasdot = exist nabla
sdot =
= times
int
r r r
r
r
r
MingsianMingsian R R BaiBai53
(iv) is connecting and
independent o
Ex Think
f the pa
of dynamics is the force
thB
A
v dR A B
v
sdotintrr
r
1 2
1
0
Pf
(i) (ii) ( ) 0 (zero identity cg rule) irrotational
ˆ(ii) (iii) ( ) 0 (Stokes theorem)
(iii) (iv) ( ) 0
C S
C C C
C
v
v dR n v dS
v dR v dR
v dR
φ
=
rarr nablatimes = nablatimes nabla =
rarr sdot = sdot nablatimes =
rarr sdot = minus sdot =
rArr sdot =
int int
int int int
int
r
rr
rr r
r rr r
rr r
2
independent of pathC
v dRsdotintr
A
B
1C
2C
MingsianMingsian R R BaiBai54
1 2
(iv) (i) constant ( ) ( )
--(1)
But (2)
(1) (2) QED
Note on application
Check
B
C C A
v dR v dR B A d
v dR d
d dx dy dz dRx y z
v dR dR v R
φ φ φ
φφ φ φφ φ
φ φ
rarr sdot = sdot = = minus =
rArr sdot =part part part
= + + = nabla sdot minus minus minus minus minuspart part part
sdot = nabla sdot rArr = nabla forall
int int intr rr r
rr
r
r r rr r
use simpler path (iv)if 0
find (i)v
φ⎧
nablatimes = ⎨⎩
rr
A
B
x
y
Cz
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai41
ˆ ˆ ( )
(
Conservation
) ( Divergence theorem)
of mass
( ) 0
V S S
V
V
dV n v dS n v dSt
v dV
v dVt
ρ ρ ρ
ρ
ρ ρ⎡ ⎤⎢ ⎥⎣ ⎦
part = minus sdot = minus sdotpart
= minus nablasdot
partrArr +nablasdot =part
int int int
int
int
v v
vQ
v
ˆOn the other hand rate influof mass through n ox i t
Extended to 3-D for time-independent fixed boundary
(3-D) ( ) ( )
Thus
S
V V
V
S V n v dS
Sd f x y z t dV f x y z t dVdt tdM dVdt t
ρ
ρ
= minus sdot
part=part
part=part
int
int int
intv
Fixed control volume (Eulerian coordinates)Otherwise Raynoldrsquos transport theoremMoving control volume Lagrangian coordinates
MingsianMingsian R R BaiBai42
But the control volime is only arbitrary
In particular for flow
( ) 0
con
(continuity equati
stant
on)
Divergence the
0 incompress
Note is t
ible
orem h
V
v
vt
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
e mother of the other integral theorems
MingsianMingsian R R BaiBai43
Ex Archimedes principle ( ) The buoyant force ( ) on a body with volume immersed in a fluid is equal to the weight of the fluid
ˆ displaced by the object ie In the followin
B
V
gVρ= minusf k
阿基米德原理
浮力
ˆg figure the ˆ with and being the fluid density gravitational
acceleration depth and outward normal unit vector to the surface
fhydraulic pressure g z
g z
S
ρ
ρ
= minusp n
n
1
the modified divergence theorem
ˆ
Hint
scalar fuctionV S
vdV v dS v Cnabla = isinint int n
MingsianMingsian R R BaiBai44
Levelsurfacefluid
V
S
airyx
z
ˆf g zρ= minusp n
ˆPf ( ) ( )
ˆ ˆ ˆ ˆ ˆ (0 0 ) v v
B fS S V
V V
g z g zdS dS dV
g dV g dV gV
ρ ρ
ρ ρ ρ
= = minus = minus nabla
= minus + + = minus = minus
int int int
int int
f p n
i j k k k
k
MingsianMingsian R R BaiBai45
Note This can also be thought of as simple balance of forces (resultant force that acts on and weight of filled with fluid)S V
2
~ a little history about George Green Recall the diveregence theorem
ˆ
Let
II th
e Green
are arbitrary fun
s identiti
ctions
es
V S
v dV n v dS
v u vu v C
nablasdot = sdot
= nablaisin
int intv v
v
ˆ ( ) ( )
ˆ ( )V S
V S
u v dV n u v dS
u v u v dV un v dS
nablasdot nabla = sdot nabla
nabla sdotnabla + nablasdotnabla = sdotnabla
int int
int int
MingsianMingsian R R BaiBai46
2 -( ) -(1)V S
vu v u v dV u dSnpart
nabla sdotnabla + nabla =partint int firs(Greens identt ity)
2
If
( ) --(2)
(1) (2) V S
u vuv u v u dV v dSn
harrpart
nabla sdotnabla + nabla =part
minus
int int
( )2 2
V S
dv duu v v u dV u v dSdn dn
⎛ ⎞nabla minus nabla = minus⎜ ⎟⎝ ⎠int int
second(Greens identity)
MingsianMingsian R R BaiBai47
( )
( )
( ) ( )
1st id ( )
Greens identities in the 1D case integration by parts
b b bb ba a
a a ab
ba
ab
ba
a
v udx v u v u dx u v uv dx uv
u v u v vu dx vu
uv vu dx uv vu
primeprime prime prime prime prime prime primeprime prime= minus rArr + =
prime prime primeprime primeharr + =
primeprime primeprime prime primerArr minus = minus
int int int
int
2
2nd id ( )
Good for any adjoint operators in addition to Good also for and integration by parts Important for theory amp reciprocity
From th
Gree
e 1s
2-
t
ns functi
identity
D 1-
tak 1
D
e
on
u
nablararr
=
int
2
V S
vv dV dSnpartnabla =part
rArr int int2Ex 0 st 0 on the boundary 0 ( 1st id u u unabla = = rArr equiv 以 証)
MingsianMingsian R R BaiBai48
1
Thm1 Stokes theorem Let be defined in Let be a piecewise-smooth orientable surface bounded by a piec
III Stokes theorem and
ewise-smooth simple clos
Greens
ed curve Then
theore
m
v C R S
C
isinv
( )
ˆwhere is a unit normal to according t right hand rulo the e
ˆS C
n S
n v dS v dRsdot nablatimes = sdotint intvv v
(not necessarily planar)S
Cn vnablatimes v vv
MingsianMingsian R R BaiBai49
Pf
( )( )
circulatioWe have shown in deriving thatthe in -direction
ˆor
ˆ
n
z
z
j j j j j
j j j j jj j
vz v dx dy
v dR n v S
v dR n v S
nablatimes
=nablatimes
sdot = sdot nablatimes Δ
sdot = sdot nablatimes Δ
intsum sumint
v
v
vv v
vv v
C
S
jS
jC
jP
ˆ jn
jSΔ
MingsianMingsian R R BaiBai50
( )
( )ext surface
After the interior boundaries cancel out only the exterior boundaries (Why)
ˆ
ˆ0 QED
Alternative forms scalar
survive
va
j
j j j j jC
jC S
v dR n v S
S v dR n v dS
φ
sdot = sdot nablatimes Δ
Δ rarr rArr sdot = sdot nablatimes
sum sumint
int int
vv v
vv v
( )
riable constant vector vector variable
1 Let
2 Let
ˆ
ˆ S C
S C
a p
v n dR dR
n p dS dR p
a
v a p
φ φφ= rArr timesnabla =
timesnabla times= rArr timestimes =
int int
int int
v v
v v
v
v v
vv vv v
MingsianMingsian R R BaiBai51
( ) ( )
( )
( )
( ) ( )
1
Thm2 Greens theorem ( planar version of Stokes theorem)ˆ ˆ Let
pf By Stokes theoremˆ
ˆ ˆ ˆ ˆ ˆ ˆ
S C
S C
S C
v P x y i Q x y j C
n v dS v dR
k Pi Qj dS Pi Qj dx i dy
Q P dS Pdx Qdyx y
⎛ ⎞part partminus = +⎜ ⎟
= + isin
sdot nablatimes = sdot
⎡ ⎤sdot nablatimes + =
part part⎝ ⎠
+ sdot +⎣ ⎦
int int
int
int
int
int
v
vv v
( )
( )
( )
ˆ
ˆˆ ˆ
ˆˆ ˆ ˆ ˆ( ) ( )
0
QEDS C
j
i j kQ PPi Qj i j k
x y z x yP Q
Q P dS Pdx Qdyx y
⎛ ⎞part part part part partnablatimes + = = + + minus⎜ ⎟part part part part part⎝ ⎠
⎛ ⎞part partrArr minus = +⎜ ⎟part part⎝ ⎠int int
L L
S
C
i
j
x
y
MingsianMingsian R R BaiBai52
Potential TheorypotentiGeorge Green coined the term (l
a )
rarr 位能 勢能
例子 重力場 電磁場 流場 聲場 彈性力學
1
2
Thm1 If is in a simply connected region then the following statements are equivalent(i) There exists a scala
Conservative Field
potential functior such that
(ii
n
) 0
v
C
C
v
v R
φφ=nabla
nablatimes
isin
isin
=r
r
r
r in (Irrotational field curl-free)
cf 0 st solenoidal field divergence-free
0
(iii) in (Conservative field)C
Rv w v w
v R Rd
nablasdot = exist nabla
sdot =
= times
int
r r r
r
r
r
MingsianMingsian R R BaiBai53
(iv) is connecting and
independent o
Ex Think
f the pa
of dynamics is the force
thB
A
v dR A B
v
sdotintrr
r
1 2
1
0
Pf
(i) (ii) ( ) 0 (zero identity cg rule) irrotational
ˆ(ii) (iii) ( ) 0 (Stokes theorem)
(iii) (iv) ( ) 0
C S
C C C
C
v
v dR n v dS
v dR v dR
v dR
φ
=
rarr nablatimes = nablatimes nabla =
rarr sdot = sdot nablatimes =
rarr sdot = minus sdot =
rArr sdot =
int int
int int int
int
r
rr
rr r
r rr r
rr r
2
independent of pathC
v dRsdotintr
A
B
1C
2C
MingsianMingsian R R BaiBai54
1 2
(iv) (i) constant ( ) ( )
--(1)
But (2)
(1) (2) QED
Note on application
Check
B
C C A
v dR v dR B A d
v dR d
d dx dy dz dRx y z
v dR dR v R
φ φ φ
φφ φ φφ φ
φ φ
rarr sdot = sdot = = minus =
rArr sdot =part part part
= + + = nabla sdot minus minus minus minus minuspart part part
sdot = nabla sdot rArr = nabla forall
int int intr rr r
rr
r
r r rr r
use simpler path (iv)if 0
find (i)v
φ⎧
nablatimes = ⎨⎩
rr
A
B
x
y
Cz
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai42
But the control volime is only arbitrary
In particular for flow
( ) 0
con
(continuity equati
stant
on)
Divergence the
0 incompress
Note is t
ible
orem h
V
v
vt
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
e mother of the other integral theorems
MingsianMingsian R R BaiBai43
Ex Archimedes principle ( ) The buoyant force ( ) on a body with volume immersed in a fluid is equal to the weight of the fluid
ˆ displaced by the object ie In the followin
B
V
gVρ= minusf k
阿基米德原理
浮力
ˆg figure the ˆ with and being the fluid density gravitational
acceleration depth and outward normal unit vector to the surface
fhydraulic pressure g z
g z
S
ρ
ρ
= minusp n
n
1
the modified divergence theorem
ˆ
Hint
scalar fuctionV S
vdV v dS v Cnabla = isinint int n
MingsianMingsian R R BaiBai44
Levelsurfacefluid
V
S
airyx
z
ˆf g zρ= minusp n
ˆPf ( ) ( )
ˆ ˆ ˆ ˆ ˆ (0 0 ) v v
B fS S V
V V
g z g zdS dS dV
g dV g dV gV
ρ ρ
ρ ρ ρ
= = minus = minus nabla
= minus + + = minus = minus
int int int
int int
f p n
i j k k k
k
MingsianMingsian R R BaiBai45
Note This can also be thought of as simple balance of forces (resultant force that acts on and weight of filled with fluid)S V
2
~ a little history about George Green Recall the diveregence theorem
ˆ
Let
II th
e Green
are arbitrary fun
s identiti
ctions
es
V S
v dV n v dS
v u vu v C
nablasdot = sdot
= nablaisin
int intv v
v
ˆ ( ) ( )
ˆ ( )V S
V S
u v dV n u v dS
u v u v dV un v dS
nablasdot nabla = sdot nabla
nabla sdotnabla + nablasdotnabla = sdotnabla
int int
int int
MingsianMingsian R R BaiBai46
2 -( ) -(1)V S
vu v u v dV u dSnpart
nabla sdotnabla + nabla =partint int firs(Greens identt ity)
2
If
( ) --(2)
(1) (2) V S
u vuv u v u dV v dSn
harrpart
nabla sdotnabla + nabla =part
minus
int int
( )2 2
V S
dv duu v v u dV u v dSdn dn
⎛ ⎞nabla minus nabla = minus⎜ ⎟⎝ ⎠int int
second(Greens identity)
MingsianMingsian R R BaiBai47
( )
( )
( ) ( )
1st id ( )
Greens identities in the 1D case integration by parts
b b bb ba a
a a ab
ba
ab
ba
a
v udx v u v u dx u v uv dx uv
u v u v vu dx vu
uv vu dx uv vu
primeprime prime prime prime prime prime primeprime prime= minus rArr + =
prime prime primeprime primeharr + =
primeprime primeprime prime primerArr minus = minus
int int int
int
2
2nd id ( )
Good for any adjoint operators in addition to Good also for and integration by parts Important for theory amp reciprocity
From th
Gree
e 1s
2-
t
ns functi
identity
D 1-
tak 1
D
e
on
u
nablararr
=
int
2
V S
vv dV dSnpartnabla =part
rArr int int2Ex 0 st 0 on the boundary 0 ( 1st id u u unabla = = rArr equiv 以 証)
MingsianMingsian R R BaiBai48
1
Thm1 Stokes theorem Let be defined in Let be a piecewise-smooth orientable surface bounded by a piec
III Stokes theorem and
ewise-smooth simple clos
Greens
ed curve Then
theore
m
v C R S
C
isinv
( )
ˆwhere is a unit normal to according t right hand rulo the e
ˆS C
n S
n v dS v dRsdot nablatimes = sdotint intvv v
(not necessarily planar)S
Cn vnablatimes v vv
MingsianMingsian R R BaiBai49
Pf
( )( )
circulatioWe have shown in deriving thatthe in -direction
ˆor
ˆ
n
z
z
j j j j j
j j j j jj j
vz v dx dy
v dR n v S
v dR n v S
nablatimes
=nablatimes
sdot = sdot nablatimes Δ
sdot = sdot nablatimes Δ
intsum sumint
v
v
vv v
vv v
C
S
jS
jC
jP
ˆ jn
jSΔ
MingsianMingsian R R BaiBai50
( )
( )ext surface
After the interior boundaries cancel out only the exterior boundaries (Why)
ˆ
ˆ0 QED
Alternative forms scalar
survive
va
j
j j j j jC
jC S
v dR n v S
S v dR n v dS
φ
sdot = sdot nablatimes Δ
Δ rarr rArr sdot = sdot nablatimes
sum sumint
int int
vv v
vv v
( )
riable constant vector vector variable
1 Let
2 Let
ˆ
ˆ S C
S C
a p
v n dR dR
n p dS dR p
a
v a p
φ φφ= rArr timesnabla =
timesnabla times= rArr timestimes =
int int
int int
v v
v v
v
v v
vv vv v
MingsianMingsian R R BaiBai51
( ) ( )
( )
( )
( ) ( )
1
Thm2 Greens theorem ( planar version of Stokes theorem)ˆ ˆ Let
pf By Stokes theoremˆ
ˆ ˆ ˆ ˆ ˆ ˆ
S C
S C
S C
v P x y i Q x y j C
n v dS v dR
k Pi Qj dS Pi Qj dx i dy
Q P dS Pdx Qdyx y
⎛ ⎞part partminus = +⎜ ⎟
= + isin
sdot nablatimes = sdot
⎡ ⎤sdot nablatimes + =
part part⎝ ⎠
+ sdot +⎣ ⎦
int int
int
int
int
int
v
vv v
( )
( )
( )
ˆ
ˆˆ ˆ
ˆˆ ˆ ˆ ˆ( ) ( )
0
QEDS C
j
i j kQ PPi Qj i j k
x y z x yP Q
Q P dS Pdx Qdyx y
⎛ ⎞part part part part partnablatimes + = = + + minus⎜ ⎟part part part part part⎝ ⎠
⎛ ⎞part partrArr minus = +⎜ ⎟part part⎝ ⎠int int
L L
S
C
i
j
x
y
MingsianMingsian R R BaiBai52
Potential TheorypotentiGeorge Green coined the term (l
a )
rarr 位能 勢能
例子 重力場 電磁場 流場 聲場 彈性力學
1
2
Thm1 If is in a simply connected region then the following statements are equivalent(i) There exists a scala
Conservative Field
potential functior such that
(ii
n
) 0
v
C
C
v
v R
φφ=nabla
nablatimes
isin
isin
=r
r
r
r in (Irrotational field curl-free)
cf 0 st solenoidal field divergence-free
0
(iii) in (Conservative field)C
Rv w v w
v R Rd
nablasdot = exist nabla
sdot =
= times
int
r r r
r
r
r
MingsianMingsian R R BaiBai53
(iv) is connecting and
independent o
Ex Think
f the pa
of dynamics is the force
thB
A
v dR A B
v
sdotintrr
r
1 2
1
0
Pf
(i) (ii) ( ) 0 (zero identity cg rule) irrotational
ˆ(ii) (iii) ( ) 0 (Stokes theorem)
(iii) (iv) ( ) 0
C S
C C C
C
v
v dR n v dS
v dR v dR
v dR
φ
=
rarr nablatimes = nablatimes nabla =
rarr sdot = sdot nablatimes =
rarr sdot = minus sdot =
rArr sdot =
int int
int int int
int
r
rr
rr r
r rr r
rr r
2
independent of pathC
v dRsdotintr
A
B
1C
2C
MingsianMingsian R R BaiBai54
1 2
(iv) (i) constant ( ) ( )
--(1)
But (2)
(1) (2) QED
Note on application
Check
B
C C A
v dR v dR B A d
v dR d
d dx dy dz dRx y z
v dR dR v R
φ φ φ
φφ φ φφ φ
φ φ
rarr sdot = sdot = = minus =
rArr sdot =part part part
= + + = nabla sdot minus minus minus minus minuspart part part
sdot = nabla sdot rArr = nabla forall
int int intr rr r
rr
r
r r rr r
use simpler path (iv)if 0
find (i)v
φ⎧
nablatimes = ⎨⎩
rr
A
B
x
y
Cz
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai43
Ex Archimedes principle ( ) The buoyant force ( ) on a body with volume immersed in a fluid is equal to the weight of the fluid
ˆ displaced by the object ie In the followin
B
V
gVρ= minusf k
阿基米德原理
浮力
ˆg figure the ˆ with and being the fluid density gravitational
acceleration depth and outward normal unit vector to the surface
fhydraulic pressure g z
g z
S
ρ
ρ
= minusp n
n
1
the modified divergence theorem
ˆ
Hint
scalar fuctionV S
vdV v dS v Cnabla = isinint int n
MingsianMingsian R R BaiBai44
Levelsurfacefluid
V
S
airyx
z
ˆf g zρ= minusp n
ˆPf ( ) ( )
ˆ ˆ ˆ ˆ ˆ (0 0 ) v v
B fS S V
V V
g z g zdS dS dV
g dV g dV gV
ρ ρ
ρ ρ ρ
= = minus = minus nabla
= minus + + = minus = minus
int int int
int int
f p n
i j k k k
k
MingsianMingsian R R BaiBai45
Note This can also be thought of as simple balance of forces (resultant force that acts on and weight of filled with fluid)S V
2
~ a little history about George Green Recall the diveregence theorem
ˆ
Let
II th
e Green
are arbitrary fun
s identiti
ctions
es
V S
v dV n v dS
v u vu v C
nablasdot = sdot
= nablaisin
int intv v
v
ˆ ( ) ( )
ˆ ( )V S
V S
u v dV n u v dS
u v u v dV un v dS
nablasdot nabla = sdot nabla
nabla sdotnabla + nablasdotnabla = sdotnabla
int int
int int
MingsianMingsian R R BaiBai46
2 -( ) -(1)V S
vu v u v dV u dSnpart
nabla sdotnabla + nabla =partint int firs(Greens identt ity)
2
If
( ) --(2)
(1) (2) V S
u vuv u v u dV v dSn
harrpart
nabla sdotnabla + nabla =part
minus
int int
( )2 2
V S
dv duu v v u dV u v dSdn dn
⎛ ⎞nabla minus nabla = minus⎜ ⎟⎝ ⎠int int
second(Greens identity)
MingsianMingsian R R BaiBai47
( )
( )
( ) ( )
1st id ( )
Greens identities in the 1D case integration by parts
b b bb ba a
a a ab
ba
ab
ba
a
v udx v u v u dx u v uv dx uv
u v u v vu dx vu
uv vu dx uv vu
primeprime prime prime prime prime prime primeprime prime= minus rArr + =
prime prime primeprime primeharr + =
primeprime primeprime prime primerArr minus = minus
int int int
int
2
2nd id ( )
Good for any adjoint operators in addition to Good also for and integration by parts Important for theory amp reciprocity
From th
Gree
e 1s
2-
t
ns functi
identity
D 1-
tak 1
D
e
on
u
nablararr
=
int
2
V S
vv dV dSnpartnabla =part
rArr int int2Ex 0 st 0 on the boundary 0 ( 1st id u u unabla = = rArr equiv 以 証)
MingsianMingsian R R BaiBai48
1
Thm1 Stokes theorem Let be defined in Let be a piecewise-smooth orientable surface bounded by a piec
III Stokes theorem and
ewise-smooth simple clos
Greens
ed curve Then
theore
m
v C R S
C
isinv
( )
ˆwhere is a unit normal to according t right hand rulo the e
ˆS C
n S
n v dS v dRsdot nablatimes = sdotint intvv v
(not necessarily planar)S
Cn vnablatimes v vv
MingsianMingsian R R BaiBai49
Pf
( )( )
circulatioWe have shown in deriving thatthe in -direction
ˆor
ˆ
n
z
z
j j j j j
j j j j jj j
vz v dx dy
v dR n v S
v dR n v S
nablatimes
=nablatimes
sdot = sdot nablatimes Δ
sdot = sdot nablatimes Δ
intsum sumint
v
v
vv v
vv v
C
S
jS
jC
jP
ˆ jn
jSΔ
MingsianMingsian R R BaiBai50
( )
( )ext surface
After the interior boundaries cancel out only the exterior boundaries (Why)
ˆ
ˆ0 QED
Alternative forms scalar
survive
va
j
j j j j jC
jC S
v dR n v S
S v dR n v dS
φ
sdot = sdot nablatimes Δ
Δ rarr rArr sdot = sdot nablatimes
sum sumint
int int
vv v
vv v
( )
riable constant vector vector variable
1 Let
2 Let
ˆ
ˆ S C
S C
a p
v n dR dR
n p dS dR p
a
v a p
φ φφ= rArr timesnabla =
timesnabla times= rArr timestimes =
int int
int int
v v
v v
v
v v
vv vv v
MingsianMingsian R R BaiBai51
( ) ( )
( )
( )
( ) ( )
1
Thm2 Greens theorem ( planar version of Stokes theorem)ˆ ˆ Let
pf By Stokes theoremˆ
ˆ ˆ ˆ ˆ ˆ ˆ
S C
S C
S C
v P x y i Q x y j C
n v dS v dR
k Pi Qj dS Pi Qj dx i dy
Q P dS Pdx Qdyx y
⎛ ⎞part partminus = +⎜ ⎟
= + isin
sdot nablatimes = sdot
⎡ ⎤sdot nablatimes + =
part part⎝ ⎠
+ sdot +⎣ ⎦
int int
int
int
int
int
v
vv v
( )
( )
( )
ˆ
ˆˆ ˆ
ˆˆ ˆ ˆ ˆ( ) ( )
0
QEDS C
j
i j kQ PPi Qj i j k
x y z x yP Q
Q P dS Pdx Qdyx y
⎛ ⎞part part part part partnablatimes + = = + + minus⎜ ⎟part part part part part⎝ ⎠
⎛ ⎞part partrArr minus = +⎜ ⎟part part⎝ ⎠int int
L L
S
C
i
j
x
y
MingsianMingsian R R BaiBai52
Potential TheorypotentiGeorge Green coined the term (l
a )
rarr 位能 勢能
例子 重力場 電磁場 流場 聲場 彈性力學
1
2
Thm1 If is in a simply connected region then the following statements are equivalent(i) There exists a scala
Conservative Field
potential functior such that
(ii
n
) 0
v
C
C
v
v R
φφ=nabla
nablatimes
isin
isin
=r
r
r
r in (Irrotational field curl-free)
cf 0 st solenoidal field divergence-free
0
(iii) in (Conservative field)C
Rv w v w
v R Rd
nablasdot = exist nabla
sdot =
= times
int
r r r
r
r
r
MingsianMingsian R R BaiBai53
(iv) is connecting and
independent o
Ex Think
f the pa
of dynamics is the force
thB
A
v dR A B
v
sdotintrr
r
1 2
1
0
Pf
(i) (ii) ( ) 0 (zero identity cg rule) irrotational
ˆ(ii) (iii) ( ) 0 (Stokes theorem)
(iii) (iv) ( ) 0
C S
C C C
C
v
v dR n v dS
v dR v dR
v dR
φ
=
rarr nablatimes = nablatimes nabla =
rarr sdot = sdot nablatimes =
rarr sdot = minus sdot =
rArr sdot =
int int
int int int
int
r
rr
rr r
r rr r
rr r
2
independent of pathC
v dRsdotintr
A
B
1C
2C
MingsianMingsian R R BaiBai54
1 2
(iv) (i) constant ( ) ( )
--(1)
But (2)
(1) (2) QED
Note on application
Check
B
C C A
v dR v dR B A d
v dR d
d dx dy dz dRx y z
v dR dR v R
φ φ φ
φφ φ φφ φ
φ φ
rarr sdot = sdot = = minus =
rArr sdot =part part part
= + + = nabla sdot minus minus minus minus minuspart part part
sdot = nabla sdot rArr = nabla forall
int int intr rr r
rr
r
r r rr r
use simpler path (iv)if 0
find (i)v
φ⎧
nablatimes = ⎨⎩
rr
A
B
x
y
Cz
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai44
Levelsurfacefluid
V
S
airyx
z
ˆf g zρ= minusp n
ˆPf ( ) ( )
ˆ ˆ ˆ ˆ ˆ (0 0 ) v v
B fS S V
V V
g z g zdS dS dV
g dV g dV gV
ρ ρ
ρ ρ ρ
= = minus = minus nabla
= minus + + = minus = minus
int int int
int int
f p n
i j k k k
k
MingsianMingsian R R BaiBai45
Note This can also be thought of as simple balance of forces (resultant force that acts on and weight of filled with fluid)S V
2
~ a little history about George Green Recall the diveregence theorem
ˆ
Let
II th
e Green
are arbitrary fun
s identiti
ctions
es
V S
v dV n v dS
v u vu v C
nablasdot = sdot
= nablaisin
int intv v
v
ˆ ( ) ( )
ˆ ( )V S
V S
u v dV n u v dS
u v u v dV un v dS
nablasdot nabla = sdot nabla
nabla sdotnabla + nablasdotnabla = sdotnabla
int int
int int
MingsianMingsian R R BaiBai46
2 -( ) -(1)V S
vu v u v dV u dSnpart
nabla sdotnabla + nabla =partint int firs(Greens identt ity)
2
If
( ) --(2)
(1) (2) V S
u vuv u v u dV v dSn
harrpart
nabla sdotnabla + nabla =part
minus
int int
( )2 2
V S
dv duu v v u dV u v dSdn dn
⎛ ⎞nabla minus nabla = minus⎜ ⎟⎝ ⎠int int
second(Greens identity)
MingsianMingsian R R BaiBai47
( )
( )
( ) ( )
1st id ( )
Greens identities in the 1D case integration by parts
b b bb ba a
a a ab
ba
ab
ba
a
v udx v u v u dx u v uv dx uv
u v u v vu dx vu
uv vu dx uv vu
primeprime prime prime prime prime prime primeprime prime= minus rArr + =
prime prime primeprime primeharr + =
primeprime primeprime prime primerArr minus = minus
int int int
int
2
2nd id ( )
Good for any adjoint operators in addition to Good also for and integration by parts Important for theory amp reciprocity
From th
Gree
e 1s
2-
t
ns functi
identity
D 1-
tak 1
D
e
on
u
nablararr
=
int
2
V S
vv dV dSnpartnabla =part
rArr int int2Ex 0 st 0 on the boundary 0 ( 1st id u u unabla = = rArr equiv 以 証)
MingsianMingsian R R BaiBai48
1
Thm1 Stokes theorem Let be defined in Let be a piecewise-smooth orientable surface bounded by a piec
III Stokes theorem and
ewise-smooth simple clos
Greens
ed curve Then
theore
m
v C R S
C
isinv
( )
ˆwhere is a unit normal to according t right hand rulo the e
ˆS C
n S
n v dS v dRsdot nablatimes = sdotint intvv v
(not necessarily planar)S
Cn vnablatimes v vv
MingsianMingsian R R BaiBai49
Pf
( )( )
circulatioWe have shown in deriving thatthe in -direction
ˆor
ˆ
n
z
z
j j j j j
j j j j jj j
vz v dx dy
v dR n v S
v dR n v S
nablatimes
=nablatimes
sdot = sdot nablatimes Δ
sdot = sdot nablatimes Δ
intsum sumint
v
v
vv v
vv v
C
S
jS
jC
jP
ˆ jn
jSΔ
MingsianMingsian R R BaiBai50
( )
( )ext surface
After the interior boundaries cancel out only the exterior boundaries (Why)
ˆ
ˆ0 QED
Alternative forms scalar
survive
va
j
j j j j jC
jC S
v dR n v S
S v dR n v dS
φ
sdot = sdot nablatimes Δ
Δ rarr rArr sdot = sdot nablatimes
sum sumint
int int
vv v
vv v
( )
riable constant vector vector variable
1 Let
2 Let
ˆ
ˆ S C
S C
a p
v n dR dR
n p dS dR p
a
v a p
φ φφ= rArr timesnabla =
timesnabla times= rArr timestimes =
int int
int int
v v
v v
v
v v
vv vv v
MingsianMingsian R R BaiBai51
( ) ( )
( )
( )
( ) ( )
1
Thm2 Greens theorem ( planar version of Stokes theorem)ˆ ˆ Let
pf By Stokes theoremˆ
ˆ ˆ ˆ ˆ ˆ ˆ
S C
S C
S C
v P x y i Q x y j C
n v dS v dR
k Pi Qj dS Pi Qj dx i dy
Q P dS Pdx Qdyx y
⎛ ⎞part partminus = +⎜ ⎟
= + isin
sdot nablatimes = sdot
⎡ ⎤sdot nablatimes + =
part part⎝ ⎠
+ sdot +⎣ ⎦
int int
int
int
int
int
v
vv v
( )
( )
( )
ˆ
ˆˆ ˆ
ˆˆ ˆ ˆ ˆ( ) ( )
0
QEDS C
j
i j kQ PPi Qj i j k
x y z x yP Q
Q P dS Pdx Qdyx y
⎛ ⎞part part part part partnablatimes + = = + + minus⎜ ⎟part part part part part⎝ ⎠
⎛ ⎞part partrArr minus = +⎜ ⎟part part⎝ ⎠int int
L L
S
C
i
j
x
y
MingsianMingsian R R BaiBai52
Potential TheorypotentiGeorge Green coined the term (l
a )
rarr 位能 勢能
例子 重力場 電磁場 流場 聲場 彈性力學
1
2
Thm1 If is in a simply connected region then the following statements are equivalent(i) There exists a scala
Conservative Field
potential functior such that
(ii
n
) 0
v
C
C
v
v R
φφ=nabla
nablatimes
isin
isin
=r
r
r
r in (Irrotational field curl-free)
cf 0 st solenoidal field divergence-free
0
(iii) in (Conservative field)C
Rv w v w
v R Rd
nablasdot = exist nabla
sdot =
= times
int
r r r
r
r
r
MingsianMingsian R R BaiBai53
(iv) is connecting and
independent o
Ex Think
f the pa
of dynamics is the force
thB
A
v dR A B
v
sdotintrr
r
1 2
1
0
Pf
(i) (ii) ( ) 0 (zero identity cg rule) irrotational
ˆ(ii) (iii) ( ) 0 (Stokes theorem)
(iii) (iv) ( ) 0
C S
C C C
C
v
v dR n v dS
v dR v dR
v dR
φ
=
rarr nablatimes = nablatimes nabla =
rarr sdot = sdot nablatimes =
rarr sdot = minus sdot =
rArr sdot =
int int
int int int
int
r
rr
rr r
r rr r
rr r
2
independent of pathC
v dRsdotintr
A
B
1C
2C
MingsianMingsian R R BaiBai54
1 2
(iv) (i) constant ( ) ( )
--(1)
But (2)
(1) (2) QED
Note on application
Check
B
C C A
v dR v dR B A d
v dR d
d dx dy dz dRx y z
v dR dR v R
φ φ φ
φφ φ φφ φ
φ φ
rarr sdot = sdot = = minus =
rArr sdot =part part part
= + + = nabla sdot minus minus minus minus minuspart part part
sdot = nabla sdot rArr = nabla forall
int int intr rr r
rr
r
r r rr r
use simpler path (iv)if 0
find (i)v
φ⎧
nablatimes = ⎨⎩
rr
A
B
x
y
Cz
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai45
Note This can also be thought of as simple balance of forces (resultant force that acts on and weight of filled with fluid)S V
2
~ a little history about George Green Recall the diveregence theorem
ˆ
Let
II th
e Green
are arbitrary fun
s identiti
ctions
es
V S
v dV n v dS
v u vu v C
nablasdot = sdot
= nablaisin
int intv v
v
ˆ ( ) ( )
ˆ ( )V S
V S
u v dV n u v dS
u v u v dV un v dS
nablasdot nabla = sdot nabla
nabla sdotnabla + nablasdotnabla = sdotnabla
int int
int int
MingsianMingsian R R BaiBai46
2 -( ) -(1)V S
vu v u v dV u dSnpart
nabla sdotnabla + nabla =partint int firs(Greens identt ity)
2
If
( ) --(2)
(1) (2) V S
u vuv u v u dV v dSn
harrpart
nabla sdotnabla + nabla =part
minus
int int
( )2 2
V S
dv duu v v u dV u v dSdn dn
⎛ ⎞nabla minus nabla = minus⎜ ⎟⎝ ⎠int int
second(Greens identity)
MingsianMingsian R R BaiBai47
( )
( )
( ) ( )
1st id ( )
Greens identities in the 1D case integration by parts
b b bb ba a
a a ab
ba
ab
ba
a
v udx v u v u dx u v uv dx uv
u v u v vu dx vu
uv vu dx uv vu
primeprime prime prime prime prime prime primeprime prime= minus rArr + =
prime prime primeprime primeharr + =
primeprime primeprime prime primerArr minus = minus
int int int
int
2
2nd id ( )
Good for any adjoint operators in addition to Good also for and integration by parts Important for theory amp reciprocity
From th
Gree
e 1s
2-
t
ns functi
identity
D 1-
tak 1
D
e
on
u
nablararr
=
int
2
V S
vv dV dSnpartnabla =part
rArr int int2Ex 0 st 0 on the boundary 0 ( 1st id u u unabla = = rArr equiv 以 証)
MingsianMingsian R R BaiBai48
1
Thm1 Stokes theorem Let be defined in Let be a piecewise-smooth orientable surface bounded by a piec
III Stokes theorem and
ewise-smooth simple clos
Greens
ed curve Then
theore
m
v C R S
C
isinv
( )
ˆwhere is a unit normal to according t right hand rulo the e
ˆS C
n S
n v dS v dRsdot nablatimes = sdotint intvv v
(not necessarily planar)S
Cn vnablatimes v vv
MingsianMingsian R R BaiBai49
Pf
( )( )
circulatioWe have shown in deriving thatthe in -direction
ˆor
ˆ
n
z
z
j j j j j
j j j j jj j
vz v dx dy
v dR n v S
v dR n v S
nablatimes
=nablatimes
sdot = sdot nablatimes Δ
sdot = sdot nablatimes Δ
intsum sumint
v
v
vv v
vv v
C
S
jS
jC
jP
ˆ jn
jSΔ
MingsianMingsian R R BaiBai50
( )
( )ext surface
After the interior boundaries cancel out only the exterior boundaries (Why)
ˆ
ˆ0 QED
Alternative forms scalar
survive
va
j
j j j j jC
jC S
v dR n v S
S v dR n v dS
φ
sdot = sdot nablatimes Δ
Δ rarr rArr sdot = sdot nablatimes
sum sumint
int int
vv v
vv v
( )
riable constant vector vector variable
1 Let
2 Let
ˆ
ˆ S C
S C
a p
v n dR dR
n p dS dR p
a
v a p
φ φφ= rArr timesnabla =
timesnabla times= rArr timestimes =
int int
int int
v v
v v
v
v v
vv vv v
MingsianMingsian R R BaiBai51
( ) ( )
( )
( )
( ) ( )
1
Thm2 Greens theorem ( planar version of Stokes theorem)ˆ ˆ Let
pf By Stokes theoremˆ
ˆ ˆ ˆ ˆ ˆ ˆ
S C
S C
S C
v P x y i Q x y j C
n v dS v dR
k Pi Qj dS Pi Qj dx i dy
Q P dS Pdx Qdyx y
⎛ ⎞part partminus = +⎜ ⎟
= + isin
sdot nablatimes = sdot
⎡ ⎤sdot nablatimes + =
part part⎝ ⎠
+ sdot +⎣ ⎦
int int
int
int
int
int
v
vv v
( )
( )
( )
ˆ
ˆˆ ˆ
ˆˆ ˆ ˆ ˆ( ) ( )
0
QEDS C
j
i j kQ PPi Qj i j k
x y z x yP Q
Q P dS Pdx Qdyx y
⎛ ⎞part part part part partnablatimes + = = + + minus⎜ ⎟part part part part part⎝ ⎠
⎛ ⎞part partrArr minus = +⎜ ⎟part part⎝ ⎠int int
L L
S
C
i
j
x
y
MingsianMingsian R R BaiBai52
Potential TheorypotentiGeorge Green coined the term (l
a )
rarr 位能 勢能
例子 重力場 電磁場 流場 聲場 彈性力學
1
2
Thm1 If is in a simply connected region then the following statements are equivalent(i) There exists a scala
Conservative Field
potential functior such that
(ii
n
) 0
v
C
C
v
v R
φφ=nabla
nablatimes
isin
isin
=r
r
r
r in (Irrotational field curl-free)
cf 0 st solenoidal field divergence-free
0
(iii) in (Conservative field)C
Rv w v w
v R Rd
nablasdot = exist nabla
sdot =
= times
int
r r r
r
r
r
MingsianMingsian R R BaiBai53
(iv) is connecting and
independent o
Ex Think
f the pa
of dynamics is the force
thB
A
v dR A B
v
sdotintrr
r
1 2
1
0
Pf
(i) (ii) ( ) 0 (zero identity cg rule) irrotational
ˆ(ii) (iii) ( ) 0 (Stokes theorem)
(iii) (iv) ( ) 0
C S
C C C
C
v
v dR n v dS
v dR v dR
v dR
φ
=
rarr nablatimes = nablatimes nabla =
rarr sdot = sdot nablatimes =
rarr sdot = minus sdot =
rArr sdot =
int int
int int int
int
r
rr
rr r
r rr r
rr r
2
independent of pathC
v dRsdotintr
A
B
1C
2C
MingsianMingsian R R BaiBai54
1 2
(iv) (i) constant ( ) ( )
--(1)
But (2)
(1) (2) QED
Note on application
Check
B
C C A
v dR v dR B A d
v dR d
d dx dy dz dRx y z
v dR dR v R
φ φ φ
φφ φ φφ φ
φ φ
rarr sdot = sdot = = minus =
rArr sdot =part part part
= + + = nabla sdot minus minus minus minus minuspart part part
sdot = nabla sdot rArr = nabla forall
int int intr rr r
rr
r
r r rr r
use simpler path (iv)if 0
find (i)v
φ⎧
nablatimes = ⎨⎩
rr
A
B
x
y
Cz
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai46
2 -( ) -(1)V S
vu v u v dV u dSnpart
nabla sdotnabla + nabla =partint int firs(Greens identt ity)
2
If
( ) --(2)
(1) (2) V S
u vuv u v u dV v dSn
harrpart
nabla sdotnabla + nabla =part
minus
int int
( )2 2
V S
dv duu v v u dV u v dSdn dn
⎛ ⎞nabla minus nabla = minus⎜ ⎟⎝ ⎠int int
second(Greens identity)
MingsianMingsian R R BaiBai47
( )
( )
( ) ( )
1st id ( )
Greens identities in the 1D case integration by parts
b b bb ba a
a a ab
ba
ab
ba
a
v udx v u v u dx u v uv dx uv
u v u v vu dx vu
uv vu dx uv vu
primeprime prime prime prime prime prime primeprime prime= minus rArr + =
prime prime primeprime primeharr + =
primeprime primeprime prime primerArr minus = minus
int int int
int
2
2nd id ( )
Good for any adjoint operators in addition to Good also for and integration by parts Important for theory amp reciprocity
From th
Gree
e 1s
2-
t
ns functi
identity
D 1-
tak 1
D
e
on
u
nablararr
=
int
2
V S
vv dV dSnpartnabla =part
rArr int int2Ex 0 st 0 on the boundary 0 ( 1st id u u unabla = = rArr equiv 以 証)
MingsianMingsian R R BaiBai48
1
Thm1 Stokes theorem Let be defined in Let be a piecewise-smooth orientable surface bounded by a piec
III Stokes theorem and
ewise-smooth simple clos
Greens
ed curve Then
theore
m
v C R S
C
isinv
( )
ˆwhere is a unit normal to according t right hand rulo the e
ˆS C
n S
n v dS v dRsdot nablatimes = sdotint intvv v
(not necessarily planar)S
Cn vnablatimes v vv
MingsianMingsian R R BaiBai49
Pf
( )( )
circulatioWe have shown in deriving thatthe in -direction
ˆor
ˆ
n
z
z
j j j j j
j j j j jj j
vz v dx dy
v dR n v S
v dR n v S
nablatimes
=nablatimes
sdot = sdot nablatimes Δ
sdot = sdot nablatimes Δ
intsum sumint
v
v
vv v
vv v
C
S
jS
jC
jP
ˆ jn
jSΔ
MingsianMingsian R R BaiBai50
( )
( )ext surface
After the interior boundaries cancel out only the exterior boundaries (Why)
ˆ
ˆ0 QED
Alternative forms scalar
survive
va
j
j j j j jC
jC S
v dR n v S
S v dR n v dS
φ
sdot = sdot nablatimes Δ
Δ rarr rArr sdot = sdot nablatimes
sum sumint
int int
vv v
vv v
( )
riable constant vector vector variable
1 Let
2 Let
ˆ
ˆ S C
S C
a p
v n dR dR
n p dS dR p
a
v a p
φ φφ= rArr timesnabla =
timesnabla times= rArr timestimes =
int int
int int
v v
v v
v
v v
vv vv v
MingsianMingsian R R BaiBai51
( ) ( )
( )
( )
( ) ( )
1
Thm2 Greens theorem ( planar version of Stokes theorem)ˆ ˆ Let
pf By Stokes theoremˆ
ˆ ˆ ˆ ˆ ˆ ˆ
S C
S C
S C
v P x y i Q x y j C
n v dS v dR
k Pi Qj dS Pi Qj dx i dy
Q P dS Pdx Qdyx y
⎛ ⎞part partminus = +⎜ ⎟
= + isin
sdot nablatimes = sdot
⎡ ⎤sdot nablatimes + =
part part⎝ ⎠
+ sdot +⎣ ⎦
int int
int
int
int
int
v
vv v
( )
( )
( )
ˆ
ˆˆ ˆ
ˆˆ ˆ ˆ ˆ( ) ( )
0
QEDS C
j
i j kQ PPi Qj i j k
x y z x yP Q
Q P dS Pdx Qdyx y
⎛ ⎞part part part part partnablatimes + = = + + minus⎜ ⎟part part part part part⎝ ⎠
⎛ ⎞part partrArr minus = +⎜ ⎟part part⎝ ⎠int int
L L
S
C
i
j
x
y
MingsianMingsian R R BaiBai52
Potential TheorypotentiGeorge Green coined the term (l
a )
rarr 位能 勢能
例子 重力場 電磁場 流場 聲場 彈性力學
1
2
Thm1 If is in a simply connected region then the following statements are equivalent(i) There exists a scala
Conservative Field
potential functior such that
(ii
n
) 0
v
C
C
v
v R
φφ=nabla
nablatimes
isin
isin
=r
r
r
r in (Irrotational field curl-free)
cf 0 st solenoidal field divergence-free
0
(iii) in (Conservative field)C
Rv w v w
v R Rd
nablasdot = exist nabla
sdot =
= times
int
r r r
r
r
r
MingsianMingsian R R BaiBai53
(iv) is connecting and
independent o
Ex Think
f the pa
of dynamics is the force
thB
A
v dR A B
v
sdotintrr
r
1 2
1
0
Pf
(i) (ii) ( ) 0 (zero identity cg rule) irrotational
ˆ(ii) (iii) ( ) 0 (Stokes theorem)
(iii) (iv) ( ) 0
C S
C C C
C
v
v dR n v dS
v dR v dR
v dR
φ
=
rarr nablatimes = nablatimes nabla =
rarr sdot = sdot nablatimes =
rarr sdot = minus sdot =
rArr sdot =
int int
int int int
int
r
rr
rr r
r rr r
rr r
2
independent of pathC
v dRsdotintr
A
B
1C
2C
MingsianMingsian R R BaiBai54
1 2
(iv) (i) constant ( ) ( )
--(1)
But (2)
(1) (2) QED
Note on application
Check
B
C C A
v dR v dR B A d
v dR d
d dx dy dz dRx y z
v dR dR v R
φ φ φ
φφ φ φφ φ
φ φ
rarr sdot = sdot = = minus =
rArr sdot =part part part
= + + = nabla sdot minus minus minus minus minuspart part part
sdot = nabla sdot rArr = nabla forall
int int intr rr r
rr
r
r r rr r
use simpler path (iv)if 0
find (i)v
φ⎧
nablatimes = ⎨⎩
rr
A
B
x
y
Cz
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai47
( )
( )
( ) ( )
1st id ( )
Greens identities in the 1D case integration by parts
b b bb ba a
a a ab
ba
ab
ba
a
v udx v u v u dx u v uv dx uv
u v u v vu dx vu
uv vu dx uv vu
primeprime prime prime prime prime prime primeprime prime= minus rArr + =
prime prime primeprime primeharr + =
primeprime primeprime prime primerArr minus = minus
int int int
int
2
2nd id ( )
Good for any adjoint operators in addition to Good also for and integration by parts Important for theory amp reciprocity
From th
Gree
e 1s
2-
t
ns functi
identity
D 1-
tak 1
D
e
on
u
nablararr
=
int
2
V S
vv dV dSnpartnabla =part
rArr int int2Ex 0 st 0 on the boundary 0 ( 1st id u u unabla = = rArr equiv 以 証)
MingsianMingsian R R BaiBai48
1
Thm1 Stokes theorem Let be defined in Let be a piecewise-smooth orientable surface bounded by a piec
III Stokes theorem and
ewise-smooth simple clos
Greens
ed curve Then
theore
m
v C R S
C
isinv
( )
ˆwhere is a unit normal to according t right hand rulo the e
ˆS C
n S
n v dS v dRsdot nablatimes = sdotint intvv v
(not necessarily planar)S
Cn vnablatimes v vv
MingsianMingsian R R BaiBai49
Pf
( )( )
circulatioWe have shown in deriving thatthe in -direction
ˆor
ˆ
n
z
z
j j j j j
j j j j jj j
vz v dx dy
v dR n v S
v dR n v S
nablatimes
=nablatimes
sdot = sdot nablatimes Δ
sdot = sdot nablatimes Δ
intsum sumint
v
v
vv v
vv v
C
S
jS
jC
jP
ˆ jn
jSΔ
MingsianMingsian R R BaiBai50
( )
( )ext surface
After the interior boundaries cancel out only the exterior boundaries (Why)
ˆ
ˆ0 QED
Alternative forms scalar
survive
va
j
j j j j jC
jC S
v dR n v S
S v dR n v dS
φ
sdot = sdot nablatimes Δ
Δ rarr rArr sdot = sdot nablatimes
sum sumint
int int
vv v
vv v
( )
riable constant vector vector variable
1 Let
2 Let
ˆ
ˆ S C
S C
a p
v n dR dR
n p dS dR p
a
v a p
φ φφ= rArr timesnabla =
timesnabla times= rArr timestimes =
int int
int int
v v
v v
v
v v
vv vv v
MingsianMingsian R R BaiBai51
( ) ( )
( )
( )
( ) ( )
1
Thm2 Greens theorem ( planar version of Stokes theorem)ˆ ˆ Let
pf By Stokes theoremˆ
ˆ ˆ ˆ ˆ ˆ ˆ
S C
S C
S C
v P x y i Q x y j C
n v dS v dR
k Pi Qj dS Pi Qj dx i dy
Q P dS Pdx Qdyx y
⎛ ⎞part partminus = +⎜ ⎟
= + isin
sdot nablatimes = sdot
⎡ ⎤sdot nablatimes + =
part part⎝ ⎠
+ sdot +⎣ ⎦
int int
int
int
int
int
v
vv v
( )
( )
( )
ˆ
ˆˆ ˆ
ˆˆ ˆ ˆ ˆ( ) ( )
0
QEDS C
j
i j kQ PPi Qj i j k
x y z x yP Q
Q P dS Pdx Qdyx y
⎛ ⎞part part part part partnablatimes + = = + + minus⎜ ⎟part part part part part⎝ ⎠
⎛ ⎞part partrArr minus = +⎜ ⎟part part⎝ ⎠int int
L L
S
C
i
j
x
y
MingsianMingsian R R BaiBai52
Potential TheorypotentiGeorge Green coined the term (l
a )
rarr 位能 勢能
例子 重力場 電磁場 流場 聲場 彈性力學
1
2
Thm1 If is in a simply connected region then the following statements are equivalent(i) There exists a scala
Conservative Field
potential functior such that
(ii
n
) 0
v
C
C
v
v R
φφ=nabla
nablatimes
isin
isin
=r
r
r
r in (Irrotational field curl-free)
cf 0 st solenoidal field divergence-free
0
(iii) in (Conservative field)C
Rv w v w
v R Rd
nablasdot = exist nabla
sdot =
= times
int
r r r
r
r
r
MingsianMingsian R R BaiBai53
(iv) is connecting and
independent o
Ex Think
f the pa
of dynamics is the force
thB
A
v dR A B
v
sdotintrr
r
1 2
1
0
Pf
(i) (ii) ( ) 0 (zero identity cg rule) irrotational
ˆ(ii) (iii) ( ) 0 (Stokes theorem)
(iii) (iv) ( ) 0
C S
C C C
C
v
v dR n v dS
v dR v dR
v dR
φ
=
rarr nablatimes = nablatimes nabla =
rarr sdot = sdot nablatimes =
rarr sdot = minus sdot =
rArr sdot =
int int
int int int
int
r
rr
rr r
r rr r
rr r
2
independent of pathC
v dRsdotintr
A
B
1C
2C
MingsianMingsian R R BaiBai54
1 2
(iv) (i) constant ( ) ( )
--(1)
But (2)
(1) (2) QED
Note on application
Check
B
C C A
v dR v dR B A d
v dR d
d dx dy dz dRx y z
v dR dR v R
φ φ φ
φφ φ φφ φ
φ φ
rarr sdot = sdot = = minus =
rArr sdot =part part part
= + + = nabla sdot minus minus minus minus minuspart part part
sdot = nabla sdot rArr = nabla forall
int int intr rr r
rr
r
r r rr r
use simpler path (iv)if 0
find (i)v
φ⎧
nablatimes = ⎨⎩
rr
A
B
x
y
Cz
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai48
1
Thm1 Stokes theorem Let be defined in Let be a piecewise-smooth orientable surface bounded by a piec
III Stokes theorem and
ewise-smooth simple clos
Greens
ed curve Then
theore
m
v C R S
C
isinv
( )
ˆwhere is a unit normal to according t right hand rulo the e
ˆS C
n S
n v dS v dRsdot nablatimes = sdotint intvv v
(not necessarily planar)S
Cn vnablatimes v vv
MingsianMingsian R R BaiBai49
Pf
( )( )
circulatioWe have shown in deriving thatthe in -direction
ˆor
ˆ
n
z
z
j j j j j
j j j j jj j
vz v dx dy
v dR n v S
v dR n v S
nablatimes
=nablatimes
sdot = sdot nablatimes Δ
sdot = sdot nablatimes Δ
intsum sumint
v
v
vv v
vv v
C
S
jS
jC
jP
ˆ jn
jSΔ
MingsianMingsian R R BaiBai50
( )
( )ext surface
After the interior boundaries cancel out only the exterior boundaries (Why)
ˆ
ˆ0 QED
Alternative forms scalar
survive
va
j
j j j j jC
jC S
v dR n v S
S v dR n v dS
φ
sdot = sdot nablatimes Δ
Δ rarr rArr sdot = sdot nablatimes
sum sumint
int int
vv v
vv v
( )
riable constant vector vector variable
1 Let
2 Let
ˆ
ˆ S C
S C
a p
v n dR dR
n p dS dR p
a
v a p
φ φφ= rArr timesnabla =
timesnabla times= rArr timestimes =
int int
int int
v v
v v
v
v v
vv vv v
MingsianMingsian R R BaiBai51
( ) ( )
( )
( )
( ) ( )
1
Thm2 Greens theorem ( planar version of Stokes theorem)ˆ ˆ Let
pf By Stokes theoremˆ
ˆ ˆ ˆ ˆ ˆ ˆ
S C
S C
S C
v P x y i Q x y j C
n v dS v dR
k Pi Qj dS Pi Qj dx i dy
Q P dS Pdx Qdyx y
⎛ ⎞part partminus = +⎜ ⎟
= + isin
sdot nablatimes = sdot
⎡ ⎤sdot nablatimes + =
part part⎝ ⎠
+ sdot +⎣ ⎦
int int
int
int
int
int
v
vv v
( )
( )
( )
ˆ
ˆˆ ˆ
ˆˆ ˆ ˆ ˆ( ) ( )
0
QEDS C
j
i j kQ PPi Qj i j k
x y z x yP Q
Q P dS Pdx Qdyx y
⎛ ⎞part part part part partnablatimes + = = + + minus⎜ ⎟part part part part part⎝ ⎠
⎛ ⎞part partrArr minus = +⎜ ⎟part part⎝ ⎠int int
L L
S
C
i
j
x
y
MingsianMingsian R R BaiBai52
Potential TheorypotentiGeorge Green coined the term (l
a )
rarr 位能 勢能
例子 重力場 電磁場 流場 聲場 彈性力學
1
2
Thm1 If is in a simply connected region then the following statements are equivalent(i) There exists a scala
Conservative Field
potential functior such that
(ii
n
) 0
v
C
C
v
v R
φφ=nabla
nablatimes
isin
isin
=r
r
r
r in (Irrotational field curl-free)
cf 0 st solenoidal field divergence-free
0
(iii) in (Conservative field)C
Rv w v w
v R Rd
nablasdot = exist nabla
sdot =
= times
int
r r r
r
r
r
MingsianMingsian R R BaiBai53
(iv) is connecting and
independent o
Ex Think
f the pa
of dynamics is the force
thB
A
v dR A B
v
sdotintrr
r
1 2
1
0
Pf
(i) (ii) ( ) 0 (zero identity cg rule) irrotational
ˆ(ii) (iii) ( ) 0 (Stokes theorem)
(iii) (iv) ( ) 0
C S
C C C
C
v
v dR n v dS
v dR v dR
v dR
φ
=
rarr nablatimes = nablatimes nabla =
rarr sdot = sdot nablatimes =
rarr sdot = minus sdot =
rArr sdot =
int int
int int int
int
r
rr
rr r
r rr r
rr r
2
independent of pathC
v dRsdotintr
A
B
1C
2C
MingsianMingsian R R BaiBai54
1 2
(iv) (i) constant ( ) ( )
--(1)
But (2)
(1) (2) QED
Note on application
Check
B
C C A
v dR v dR B A d
v dR d
d dx dy dz dRx y z
v dR dR v R
φ φ φ
φφ φ φφ φ
φ φ
rarr sdot = sdot = = minus =
rArr sdot =part part part
= + + = nabla sdot minus minus minus minus minuspart part part
sdot = nabla sdot rArr = nabla forall
int int intr rr r
rr
r
r r rr r
use simpler path (iv)if 0
find (i)v
φ⎧
nablatimes = ⎨⎩
rr
A
B
x
y
Cz
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai49
Pf
( )( )
circulatioWe have shown in deriving thatthe in -direction
ˆor
ˆ
n
z
z
j j j j j
j j j j jj j
vz v dx dy
v dR n v S
v dR n v S
nablatimes
=nablatimes
sdot = sdot nablatimes Δ
sdot = sdot nablatimes Δ
intsum sumint
v
v
vv v
vv v
C
S
jS
jC
jP
ˆ jn
jSΔ
MingsianMingsian R R BaiBai50
( )
( )ext surface
After the interior boundaries cancel out only the exterior boundaries (Why)
ˆ
ˆ0 QED
Alternative forms scalar
survive
va
j
j j j j jC
jC S
v dR n v S
S v dR n v dS
φ
sdot = sdot nablatimes Δ
Δ rarr rArr sdot = sdot nablatimes
sum sumint
int int
vv v
vv v
( )
riable constant vector vector variable
1 Let
2 Let
ˆ
ˆ S C
S C
a p
v n dR dR
n p dS dR p
a
v a p
φ φφ= rArr timesnabla =
timesnabla times= rArr timestimes =
int int
int int
v v
v v
v
v v
vv vv v
MingsianMingsian R R BaiBai51
( ) ( )
( )
( )
( ) ( )
1
Thm2 Greens theorem ( planar version of Stokes theorem)ˆ ˆ Let
pf By Stokes theoremˆ
ˆ ˆ ˆ ˆ ˆ ˆ
S C
S C
S C
v P x y i Q x y j C
n v dS v dR
k Pi Qj dS Pi Qj dx i dy
Q P dS Pdx Qdyx y
⎛ ⎞part partminus = +⎜ ⎟
= + isin
sdot nablatimes = sdot
⎡ ⎤sdot nablatimes + =
part part⎝ ⎠
+ sdot +⎣ ⎦
int int
int
int
int
int
v
vv v
( )
( )
( )
ˆ
ˆˆ ˆ
ˆˆ ˆ ˆ ˆ( ) ( )
0
QEDS C
j
i j kQ PPi Qj i j k
x y z x yP Q
Q P dS Pdx Qdyx y
⎛ ⎞part part part part partnablatimes + = = + + minus⎜ ⎟part part part part part⎝ ⎠
⎛ ⎞part partrArr minus = +⎜ ⎟part part⎝ ⎠int int
L L
S
C
i
j
x
y
MingsianMingsian R R BaiBai52
Potential TheorypotentiGeorge Green coined the term (l
a )
rarr 位能 勢能
例子 重力場 電磁場 流場 聲場 彈性力學
1
2
Thm1 If is in a simply connected region then the following statements are equivalent(i) There exists a scala
Conservative Field
potential functior such that
(ii
n
) 0
v
C
C
v
v R
φφ=nabla
nablatimes
isin
isin
=r
r
r
r in (Irrotational field curl-free)
cf 0 st solenoidal field divergence-free
0
(iii) in (Conservative field)C
Rv w v w
v R Rd
nablasdot = exist nabla
sdot =
= times
int
r r r
r
r
r
MingsianMingsian R R BaiBai53
(iv) is connecting and
independent o
Ex Think
f the pa
of dynamics is the force
thB
A
v dR A B
v
sdotintrr
r
1 2
1
0
Pf
(i) (ii) ( ) 0 (zero identity cg rule) irrotational
ˆ(ii) (iii) ( ) 0 (Stokes theorem)
(iii) (iv) ( ) 0
C S
C C C
C
v
v dR n v dS
v dR v dR
v dR
φ
=
rarr nablatimes = nablatimes nabla =
rarr sdot = sdot nablatimes =
rarr sdot = minus sdot =
rArr sdot =
int int
int int int
int
r
rr
rr r
r rr r
rr r
2
independent of pathC
v dRsdotintr
A
B
1C
2C
MingsianMingsian R R BaiBai54
1 2
(iv) (i) constant ( ) ( )
--(1)
But (2)
(1) (2) QED
Note on application
Check
B
C C A
v dR v dR B A d
v dR d
d dx dy dz dRx y z
v dR dR v R
φ φ φ
φφ φ φφ φ
φ φ
rarr sdot = sdot = = minus =
rArr sdot =part part part
= + + = nabla sdot minus minus minus minus minuspart part part
sdot = nabla sdot rArr = nabla forall
int int intr rr r
rr
r
r r rr r
use simpler path (iv)if 0
find (i)v
φ⎧
nablatimes = ⎨⎩
rr
A
B
x
y
Cz
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai50
( )
( )ext surface
After the interior boundaries cancel out only the exterior boundaries (Why)
ˆ
ˆ0 QED
Alternative forms scalar
survive
va
j
j j j j jC
jC S
v dR n v S
S v dR n v dS
φ
sdot = sdot nablatimes Δ
Δ rarr rArr sdot = sdot nablatimes
sum sumint
int int
vv v
vv v
( )
riable constant vector vector variable
1 Let
2 Let
ˆ
ˆ S C
S C
a p
v n dR dR
n p dS dR p
a
v a p
φ φφ= rArr timesnabla =
timesnabla times= rArr timestimes =
int int
int int
v v
v v
v
v v
vv vv v
MingsianMingsian R R BaiBai51
( ) ( )
( )
( )
( ) ( )
1
Thm2 Greens theorem ( planar version of Stokes theorem)ˆ ˆ Let
pf By Stokes theoremˆ
ˆ ˆ ˆ ˆ ˆ ˆ
S C
S C
S C
v P x y i Q x y j C
n v dS v dR
k Pi Qj dS Pi Qj dx i dy
Q P dS Pdx Qdyx y
⎛ ⎞part partminus = +⎜ ⎟
= + isin
sdot nablatimes = sdot
⎡ ⎤sdot nablatimes + =
part part⎝ ⎠
+ sdot +⎣ ⎦
int int
int
int
int
int
v
vv v
( )
( )
( )
ˆ
ˆˆ ˆ
ˆˆ ˆ ˆ ˆ( ) ( )
0
QEDS C
j
i j kQ PPi Qj i j k
x y z x yP Q
Q P dS Pdx Qdyx y
⎛ ⎞part part part part partnablatimes + = = + + minus⎜ ⎟part part part part part⎝ ⎠
⎛ ⎞part partrArr minus = +⎜ ⎟part part⎝ ⎠int int
L L
S
C
i
j
x
y
MingsianMingsian R R BaiBai52
Potential TheorypotentiGeorge Green coined the term (l
a )
rarr 位能 勢能
例子 重力場 電磁場 流場 聲場 彈性力學
1
2
Thm1 If is in a simply connected region then the following statements are equivalent(i) There exists a scala
Conservative Field
potential functior such that
(ii
n
) 0
v
C
C
v
v R
φφ=nabla
nablatimes
isin
isin
=r
r
r
r in (Irrotational field curl-free)
cf 0 st solenoidal field divergence-free
0
(iii) in (Conservative field)C
Rv w v w
v R Rd
nablasdot = exist nabla
sdot =
= times
int
r r r
r
r
r
MingsianMingsian R R BaiBai53
(iv) is connecting and
independent o
Ex Think
f the pa
of dynamics is the force
thB
A
v dR A B
v
sdotintrr
r
1 2
1
0
Pf
(i) (ii) ( ) 0 (zero identity cg rule) irrotational
ˆ(ii) (iii) ( ) 0 (Stokes theorem)
(iii) (iv) ( ) 0
C S
C C C
C
v
v dR n v dS
v dR v dR
v dR
φ
=
rarr nablatimes = nablatimes nabla =
rarr sdot = sdot nablatimes =
rarr sdot = minus sdot =
rArr sdot =
int int
int int int
int
r
rr
rr r
r rr r
rr r
2
independent of pathC
v dRsdotintr
A
B
1C
2C
MingsianMingsian R R BaiBai54
1 2
(iv) (i) constant ( ) ( )
--(1)
But (2)
(1) (2) QED
Note on application
Check
B
C C A
v dR v dR B A d
v dR d
d dx dy dz dRx y z
v dR dR v R
φ φ φ
φφ φ φφ φ
φ φ
rarr sdot = sdot = = minus =
rArr sdot =part part part
= + + = nabla sdot minus minus minus minus minuspart part part
sdot = nabla sdot rArr = nabla forall
int int intr rr r
rr
r
r r rr r
use simpler path (iv)if 0
find (i)v
φ⎧
nablatimes = ⎨⎩
rr
A
B
x
y
Cz
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai51
( ) ( )
( )
( )
( ) ( )
1
Thm2 Greens theorem ( planar version of Stokes theorem)ˆ ˆ Let
pf By Stokes theoremˆ
ˆ ˆ ˆ ˆ ˆ ˆ
S C
S C
S C
v P x y i Q x y j C
n v dS v dR
k Pi Qj dS Pi Qj dx i dy
Q P dS Pdx Qdyx y
⎛ ⎞part partminus = +⎜ ⎟
= + isin
sdot nablatimes = sdot
⎡ ⎤sdot nablatimes + =
part part⎝ ⎠
+ sdot +⎣ ⎦
int int
int
int
int
int
v
vv v
( )
( )
( )
ˆ
ˆˆ ˆ
ˆˆ ˆ ˆ ˆ( ) ( )
0
QEDS C
j
i j kQ PPi Qj i j k
x y z x yP Q
Q P dS Pdx Qdyx y
⎛ ⎞part part part part partnablatimes + = = + + minus⎜ ⎟part part part part part⎝ ⎠
⎛ ⎞part partrArr minus = +⎜ ⎟part part⎝ ⎠int int
L L
S
C
i
j
x
y
MingsianMingsian R R BaiBai52
Potential TheorypotentiGeorge Green coined the term (l
a )
rarr 位能 勢能
例子 重力場 電磁場 流場 聲場 彈性力學
1
2
Thm1 If is in a simply connected region then the following statements are equivalent(i) There exists a scala
Conservative Field
potential functior such that
(ii
n
) 0
v
C
C
v
v R
φφ=nabla
nablatimes
isin
isin
=r
r
r
r in (Irrotational field curl-free)
cf 0 st solenoidal field divergence-free
0
(iii) in (Conservative field)C
Rv w v w
v R Rd
nablasdot = exist nabla
sdot =
= times
int
r r r
r
r
r
MingsianMingsian R R BaiBai53
(iv) is connecting and
independent o
Ex Think
f the pa
of dynamics is the force
thB
A
v dR A B
v
sdotintrr
r
1 2
1
0
Pf
(i) (ii) ( ) 0 (zero identity cg rule) irrotational
ˆ(ii) (iii) ( ) 0 (Stokes theorem)
(iii) (iv) ( ) 0
C S
C C C
C
v
v dR n v dS
v dR v dR
v dR
φ
=
rarr nablatimes = nablatimes nabla =
rarr sdot = sdot nablatimes =
rarr sdot = minus sdot =
rArr sdot =
int int
int int int
int
r
rr
rr r
r rr r
rr r
2
independent of pathC
v dRsdotintr
A
B
1C
2C
MingsianMingsian R R BaiBai54
1 2
(iv) (i) constant ( ) ( )
--(1)
But (2)
(1) (2) QED
Note on application
Check
B
C C A
v dR v dR B A d
v dR d
d dx dy dz dRx y z
v dR dR v R
φ φ φ
φφ φ φφ φ
φ φ
rarr sdot = sdot = = minus =
rArr sdot =part part part
= + + = nabla sdot minus minus minus minus minuspart part part
sdot = nabla sdot rArr = nabla forall
int int intr rr r
rr
r
r r rr r
use simpler path (iv)if 0
find (i)v
φ⎧
nablatimes = ⎨⎩
rr
A
B
x
y
Cz
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai52
Potential TheorypotentiGeorge Green coined the term (l
a )
rarr 位能 勢能
例子 重力場 電磁場 流場 聲場 彈性力學
1
2
Thm1 If is in a simply connected region then the following statements are equivalent(i) There exists a scala
Conservative Field
potential functior such that
(ii
n
) 0
v
C
C
v
v R
φφ=nabla
nablatimes
isin
isin
=r
r
r
r in (Irrotational field curl-free)
cf 0 st solenoidal field divergence-free
0
(iii) in (Conservative field)C
Rv w v w
v R Rd
nablasdot = exist nabla
sdot =
= times
int
r r r
r
r
r
MingsianMingsian R R BaiBai53
(iv) is connecting and
independent o
Ex Think
f the pa
of dynamics is the force
thB
A
v dR A B
v
sdotintrr
r
1 2
1
0
Pf
(i) (ii) ( ) 0 (zero identity cg rule) irrotational
ˆ(ii) (iii) ( ) 0 (Stokes theorem)
(iii) (iv) ( ) 0
C S
C C C
C
v
v dR n v dS
v dR v dR
v dR
φ
=
rarr nablatimes = nablatimes nabla =
rarr sdot = sdot nablatimes =
rarr sdot = minus sdot =
rArr sdot =
int int
int int int
int
r
rr
rr r
r rr r
rr r
2
independent of pathC
v dRsdotintr
A
B
1C
2C
MingsianMingsian R R BaiBai54
1 2
(iv) (i) constant ( ) ( )
--(1)
But (2)
(1) (2) QED
Note on application
Check
B
C C A
v dR v dR B A d
v dR d
d dx dy dz dRx y z
v dR dR v R
φ φ φ
φφ φ φφ φ
φ φ
rarr sdot = sdot = = minus =
rArr sdot =part part part
= + + = nabla sdot minus minus minus minus minuspart part part
sdot = nabla sdot rArr = nabla forall
int int intr rr r
rr
r
r r rr r
use simpler path (iv)if 0
find (i)v
φ⎧
nablatimes = ⎨⎩
rr
A
B
x
y
Cz
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai53
(iv) is connecting and
independent o
Ex Think
f the pa
of dynamics is the force
thB
A
v dR A B
v
sdotintrr
r
1 2
1
0
Pf
(i) (ii) ( ) 0 (zero identity cg rule) irrotational
ˆ(ii) (iii) ( ) 0 (Stokes theorem)
(iii) (iv) ( ) 0
C S
C C C
C
v
v dR n v dS
v dR v dR
v dR
φ
=
rarr nablatimes = nablatimes nabla =
rarr sdot = sdot nablatimes =
rarr sdot = minus sdot =
rArr sdot =
int int
int int int
int
r
rr
rr r
r rr r
rr r
2
independent of pathC
v dRsdotintr
A
B
1C
2C
MingsianMingsian R R BaiBai54
1 2
(iv) (i) constant ( ) ( )
--(1)
But (2)
(1) (2) QED
Note on application
Check
B
C C A
v dR v dR B A d
v dR d
d dx dy dz dRx y z
v dR dR v R
φ φ φ
φφ φ φφ φ
φ φ
rarr sdot = sdot = = minus =
rArr sdot =part part part
= + + = nabla sdot minus minus minus minus minuspart part part
sdot = nabla sdot rArr = nabla forall
int int intr rr r
rr
r
r r rr r
use simpler path (iv)if 0
find (i)v
φ⎧
nablatimes = ⎨⎩
rr
A
B
x
y
Cz
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai54
1 2
(iv) (i) constant ( ) ( )
--(1)
But (2)
(1) (2) QED
Note on application
Check
B
C C A
v dR v dR B A d
v dR d
d dx dy dz dRx y z
v dR dR v R
φ φ φ
φφ φ φφ φ
φ φ
rarr sdot = sdot = = minus =
rArr sdot =part part part
= + + = nabla sdot minus minus minus minus minuspart part part
sdot = nabla sdot rArr = nabla forall
int int intr rr r
rr
r
r r rr r
use simpler path (iv)if 0
find (i)v
φ⎧
nablatimes = ⎨⎩
rr
A
B
x
y
Cz
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai55
Ex Irrotational incompressible flowLet be the flow velocity For irrotational flow 0 ( 0) velocity potentialIf in addition the fluid i
inviscid
incos
mpres b
si le
vv
v
v
μφ φ
nablatimes = =rArr =nabla
nablasdot
v
vv
v
v
2
0Thus 0 (Laplace equation) potential flow
v φ φ
=
nablasdot = nablasdotnabla = nabla =v
x
y
n
ˆv Uiasympv
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai56
incomp
Recall
ressib
(
le
For flow
) 0 (continuity e
constant 0
quatio
n)v
v
t
ρ
ρ ρpart +nablasdot =part
=nablasdot =
v
v
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai57
2 2
2 2
2 2
ˆBC as
ˆ ˆ ˆ
0 as
(with a constant) as --(1) on the wall
ˆ ˆ 0 --(2
v Ui x y
v i j Uix y
U x yx y
Ux x y
v n v nn
φ φφ
φ φ
φ
φφperp
asymp + rarrinfinpart part
= nabla = + asymppart part
part partrArr asymp asymp + rarrinfin
part part
rArr = + rarrinfin
part= sdot = sdotnabla = =
part
v
v
v )
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai58
2 22
2 2
2 2
PDE 0work on scalar instead of vector
BC as pricecomplicated BCs 0 on the wall
In summary
x yUx x y
n
φ φφ
φφ
⎫part partnabla = + = ⎪part part ⎪⎪
⎬+ rarrinfin rarr⎪part ⎪=⎪part ⎭
( )
2
Ex Evaluate if 11 0 2 4
ˆˆ ˆ and where sin cos 2 2 2
B
A
d A B
ui vj wku z y x v x z w xz z y
π π⎛ ⎞= =⎜ ⎟⎝ ⎠
= + +
= minus = minus = + minus
intF R
F
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai59
( ) ( ) ( )
2
2 21
sin cos 2 2 2Sol ˆˆ ˆ
ˆˆ ˆ2 2 2 2 sin sin
conservative fieldThus there exists a scalar potential function such that
sin cos (
u z y x v x z w xz z y
i j k
i z z j x x kx y z
u v w
z y x xz y x cx
φ φφ φ
= minus = minus = + minus
part part partnablatimes = = minus + + minus + minus + =part part part
rArr=nabla
part = minus rArr = + +part
F 0
F
)
cos
y z
xyφpart =part
2 cosz xminus = 1 1 2
22
( ) ( ) 2 ( )
cos 2 ( )
2
c y z c y z yz c zy
xz y x yz c z
xzz
φφ
part+ rArr = minus +part
= + minus +part =part
2z y+ minus 2xz= 2yminus2
2 2 3( ) ( )2zc z c z cprime+ rArr = +
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai60
( )
( )
32
2
Since the choice of is immaterial let 0
cos 22
( ) ( ) 02 114
2 1 15 7 2 4 2 22 4 2 4 2 2
Alternatively knowing that
da
is
tum
d
in
B
A
B
A
czxz y x yz
d B A
d
φ
πφ φ φ π φ
π ππ
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
rArr = + minus +
= minus = minus
= + minus + minus + = + minus
int
int
F R
F R
( ) ( ) ( )
ependent of the
path connecting and choose an easy integration path
11 11 01 024
A Bπ π π π⎛ ⎞⎜ ⎟⎝ ⎠
rarr rarr rarr
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai61
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
2
02 11 01 02
11 0111 114 4
02
111 1
4
411 11 01 0
sin cos
2
2 2 2
sin | cos 2 | 2 2 |
B
A
x xyzz
u z y x v x z w xz z y
d udx vdy wdz udx vdy wdz
z y x dx x z dy xz z y
π π π π
π π π π
π
ππ
π π π π
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
====
= minus = minus = +
rarr rarr
minus
rarr
= + + = + + + +
= minus + minus + + minus
int int int int int
int int
F R
( ) ( ) ( )
( )
2
01
0 2
1 14
22
2
4 1
1 sin 1 2 2
15 7 2cos 32 4 2 2
ydz
x dx dy z z dz
zx x z
π
π
π
ππ
π
ππ
==
⎛ ⎞⎜ ⎟⎝ ⎠
= minus + minus minus + +
= + + + + = + minus
int
int int int
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai62
( ) ( ) ( ) ( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
000 000
00 0
000 00 0
sin cos 2
Yet another approach choose an integration path
0 0 0 0
0 0 x y z x y z
x x y x y z
x x y
u z y x v x z w
x y z d udx vdy wdz
u
x x y x y z
dx vdy wdz
φ
= minus = minus
= = + +
= + + +
rarr
+
rarr rarr
int int
int int int
F R
( )
( )
0 0 02
2
2 2
0 cos 2 2
cos 2215 7 2 0 2 11
4 4 2 2
yx z
B
A
xz z y
dx x dy xz z y dz
zxz y x yz
d π πφ π φ
= + minus
⎛ ⎞⎜ ⎟⎝ ⎠
= + + + minus
= + minus +
= minus = + minus
int int int
int F R
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai63
HW (Wylie)
155 3(b) 4 25(d) 57156 21 23 41 47157 1(b) 2(c) 9 14 18 41 46
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai64
Ex a general framework to unify amp
(vectors) electric field in
electric flux density
tensity magnetic field intensi
Maxwells Electromagnetic Wave Equat
t
ion
s
yD E
EH
ε= =
rarr
=
=
Defv
v
v
v電通量密度
電 磁
2 current density ( ) conductivity ( ) (scalars)
permi
(electric displmagnetic flux
ttivity
ac
( ) permeability (
densitement)
y
)
J AB H
m Eμ
σ σ
εμ
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
⎧ ⎫⎨ ⎬⎩ ⎭
= =
==
= =
Def
v v
v v磁通量密度
導電
介電
導磁
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai65
8 12 70 0
0 0
1 3 10 ms 8854 10 Fm 4 10 Hm
This prompts Maxwell to think magnetic field and electric field are related and light is an electromagnetic wave
Note
c ε μ πε μ
minus minus= = times = times = times
rarr
charge density
total electric charge
( total magnetic flux)
( total electric flux)
(total current)
V
M MS
E E ES S
S
Q
q QdV
N B dS
N D dS N E dS
i N J dS
φ φ
ε φ φ φ
=
=
sdot =
sdot = rArr sdot =
= sdot
int
int
int int
int
v v
v v v v
v v
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai66
Faradays law (1)
Amperes law (2)
Gauss law for electric fields
(3)
Electromagnetic
Gauss law for magnetic fiel
laws
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
εφ
part⎧ sdot = minus⎪ part⎪nablatimes⎨ part⎪ sdot = = +⎪ part⎩
sdot = =
nablasdot
int
int
int
v v
v v
v v
ds
0 (4)MS
N B dS φ
⎧⎪⎪⎪⎨⎪⎪ sdot = =⎪⎩
intv v
電生磁
磁生電
EMF (voltage)
(故磁極不可能為單極)
iMφ
i
C
Eφ
S
q
Nv
Dv
out fluxS
Nv
Bv
out flux
H
C
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai67
Apply theorem to Faradays law in
0 ( is the surface bounded by )
-------(5)
(D
Stokes Eq (
i
1)
M
C S S S
S
BE dR N EdS N BdS N
BEt
dSt t t
BN E dS S Ct
S
φ
⎛ ⎞⎜ ⎟⎝ ⎠
part part partsdot = sdotnablatimes = minus = minus sdot
partnablatimes = minus
= minus sdotpart part part
partrArr sdot nablatimes +
part
=part
rArr forall
int int int int
int
vv v v v v v
v
v
v v
v
v
Eqff
eren
(2)tial form of Faradays law)
Apply theorem to Amperes law in
Stok
(6)
es
C S S
H dR N H dS i N J dS
H J S
sdot = sdotnablatimes = = sdot
rArrnablatimes = forall
int int intv v v v v v
v v
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai68
The current density consists of two parts ie a density due to the flow of electric charg displaceme
c
n
on
t
duct
curr
ion curr
e and a density due
e
ntto time variat
ent
i
c
J
J Eσ=
v
v v
on of the electric field eg a capacitor電容器
Two ways of setting up a magnetic field
Thus
d
c d
EJt
EJ J J Et
ε
σ ε
part=
part
part= + = +
part
vv
vv v v v
It can be shown that
+ -
Eq(2)
i
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai69
( )
Giancoli p788For a capacitor with area and gap
( ) (displacement curre
X
nt)
( )
Ed
Ed d
S S S
d
A dAq CV Ed AEd
q AEit t t
Ei N J dS N E dS N dSt t t
EJ MMFt
ε ε
φε ε
φ εε ε
ε
⎛ ⎞= = =⎜ ⎟⎝ ⎠
partpart part= = =part part part
part part part= sdot = = sdot = sdot
part part part
partrArr = rarr
part
int int intv
v v v v v
vv
cf MEMFtφpart
= minuspart
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai70
Hence Eq (6) becomes
--(7) (Differential form of Amperes law)
Next apply the divergence theorem to the Gauss EF law in Eq (3)
----- -(8S V V
N D dS D dV q Q dV
V
EH Et
D Q
σ ε partnablatimes = +part
nabla
sdot = nablasdot = =
rArr forallsdot =
int int int
v
v
v
v v
v
v)
(Differential form of Gauss EF law)
Similarly apply the divergence theorem to Gauss MF law in Eq (4)
0 ( is a closed surface)
(9)(Differential from of
0
MS V
B
N B dS B dV S
V
φ = sdot = nablasdot =
rArr forall minusminusminusnablasdot = minusminus
int intr
rv v
Gauss MF law)
電生磁
(Helmholtz Decomposition)v φ ψ= nabla +nablatimes vv
irrotational source-free solenoidal
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai71
2
Eq(7)
Take the curl of Eq (5) D-form of Faradays law
( )
( ) ( ) by vector identity
( ) ( ) --(10)
Consider a free space where there are n
BEt
E E Bt
EH Et t t
μ μ σ ε
partnablatimes nablatimes = minusnablatimespart
partrArrnabla nablasdot minusnabla = minus nablatimespart
part part part= minus nablatimes = minus +part part part
Q
vr
r r r
rr r
o charges nor conduction current ie 0 0 0Recall Eq (8) D-form of Gauss EF law ( ) 0 --(11)
cQ J E
D E E Q
σ σ
ε ε
= = = =
nablasdot =nablasdot = nablasdot = =
vv v
r r r
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai72
8
22 2
2
1 3 10
( Maxwells wave equation electric field
Eqs (10) and (11) lead to --(12)
Similarly take the curl of Eq (7) D-form of Amperes law
( ) (
for the )
E
Ec Et
c
H
m sμε
σ
partnabla =part
nablatimes nablatimes
=
nabla
times
times
=
=r
rr
r
光速
2
) ( ) ( free space)
( ) ( ) by vector identity
From Eq (9) D-form of Gauss MF law 0
Also from Eq (5) D-form of Faradays law
E Et t
H H Et
B HBEt
ε ε
ε
μ
part part+ = nablatimespart part
partrArrnabla nablasdot minusnabla = nablatimespart
nablasdot = nablasdot =
partnablatimes = minuspart
rr
Q
r r r
r r
rr
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai73
22 2
2
22
2
( )
--(1
(Maxw for t
3)
ells wave equation magnetic fiehe )l
ds
B H
Bc
BH Et
t
t
B
μ
ε ε
=
part part
partnabla =part
nabla = minus nablatimes =part part
there4
v vQ
v
r
v
r r
Ev
Bv
Change of magnetic field generates electric field and vice versa
Travels with speed of light c
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai74
Good dielectric ( 0)σ =
Good conductor ( )σ rarrinfin
22
2
22
2
(wave equation)
EEtHHt
με
με
partnabla =partpartnabla =part
rr
rr
2
2
(heat equation)
EEtHHt
μσ
μσ
partnabla =partpartnabla =part
rr
rr
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai75
By divergence thm
( ) 0
For arbitrary 0
For steady current
Ex Continuity eq
S V V
V
S V
QJ NdS JdV dVt
QJ dVt
QV Jt
dq dI J NdS QdVdt dt
partsdot = nabla sdot = minus
part
partnabla sdot + =
part
part+nabla sdot =
partpart
= sdot = minus = minus
int int int
int
int int
r r r
r
r
r r
電磁學補充
0 0
0 (KCL)iiV S
Q Jt
JdV J NdS I
= rarrnablasdot =partnabla sdot = sdot = =sumint int
r
r r r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai76
Faradays law (1)
Amperes law
Maxwells equati
(2)
Gauss law for electric fields
(3
ons (integral form)
)
Gauss law for
M
C
EC
C
ES
E dRt
H dR i it
N D dS q
φ
φ
φ
⎧⎪⎪⎨⎪⎪⎩
partsdot = minuspart
nablatimespartsdot = = +part
sdot = =nablasdot
int
int
int
v v
v v
v v
magnetic fields
0 (4)S
N B dS
⎧⎪⎪⎪⎨⎪⎪⎪⎩
sdot =intv v
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
MingsianMingsian R R BaiBai77
Faradays law
Amperes law
Gauss law for electric fields Gauss law for magnet
Maxwells equations (differential form
ic fields 0
)
BEt
EH EtD Q
B
σ ε
partnablatimes = minuspart
partnablatimes = +part
nablasdot =
nablasdot =
vv
vv v
v
r
X
