Dr.-
Ing.
R. M
arkl
ein
-EFT
I -S
S 05
1
Elek
trom
agne
tisch
e Fe
ldth
eorie
I (E
FT I)
/El
ectr
omag
netic
Fie
ld T
heor
y I (
EFT
I)
3rd
Lect
ure
/ 3.
Vor
lesu
ng
Univ
ersi
ty o
f Kas
sel
Dep
t. El
ectr
ical
Eng
inee
ring
/ Co
mpu
ter
Scie
nce
(FB
16)
Elec
trom
agne
tic F
ield
The
ory
(FG
TET
)W
ilhel
msh
öher
Alle
e 71
Off
ice:
Roo
m 2
113
/ 21
15D
-341
21 K
asse
l
Univ
ersi
tät K
asse
lFa
chbe
reic
h El
ektr
otec
hnik
/ In
form
atik
(F
B 16
)Fa
chge
biet
The
oret
isch
e El
ektr
otec
hnik
(F
G T
ET)
Wilh
elm
shöh
er A
llee
71Bü
ro: R
aum
211
3 /
2115
D-3
4121
Kas
selD
r.-I
ng. R
ené
Mar
klei
nm
arkl
ein@
uni-
kass
el.d
eht
tp:/
/ww
w.te
t.e-t
echn
ik.u
ni-k
asse
l.de
http
://w
ww
.uni
-kas
sel.d
e/fb
16/t
et/m
arkl
ein/
inde
x.ht
ml
Dr.-
Ing.
R. M
arkl
ein
-EFT
I -S
S 05
2
Diff
eren
t Coo
rdin
ate
Syst
ems
/ Ve
rsch
iede
ne K
oord
inat
ensy
stem
e
●Ca
rtes
ian
(Rec
tang
ular
) Coo
rdin
ate
Syst
em/
Kart
esis
ches
Koo
rdin
aten
syst
em
●Cy
lindr
ical
Coo
rdin
ate
Syst
em/
Zylin
derk
oord
inat
ensy
stem
●Sp
heric
al C
oord
inat
e Sy
stem
/ Ku
gelk
oord
inat
ensy
stem
Wha
t is
the
bene
fit o
f the
Use
of a
Pro
blem
Mat
ched
Co
ordi
nate
Sys
tem
s?
/ W
as is
t der
Nut
zen
der
Verw
endu
ng e
ines
pro
blem
ange
pass
ten
Koor
dina
tens
yste
men
?
(Eas
ier)
Sol
utio
n of
the
Prob
lem
und
er C
once
rn!/
(E
infa
cher
e) L
ösun
g de
s be
trac
htet
en P
robl
ems?
Dr.-
Ing.
R. M
arkl
ein
-EFT
I -S
S 05
3
Posi
tion
Vect
or /
Ort
svek
tor
(Pos
ition
svek
tor)
()
()
()
=
(
)(
)(
)x
yz
xy
zx
yz
xy
z
RR
R
xy
z
=+
+
++
=+
+
RRR
RR
RR
Re
Re
Re
ee
e
Cart
esia
n Co
ordi
nate
Sys
tem
/ Ka
rtes
isch
es K
oord
inat
ensy
stem
Vect
oria
l Vec
tor
Com
pone
nts
/ Ve
ktor
ielle
Vek
tork
ompo
nent
en(
)(
,,
)(
)(
,,
)
()
(,
,)
xx
xx
yy
yy
zz
zz
Rxyz
xRxyz
y
Rxyz
z
==
==
==
RR
ee
RR
ee
RR
ee
Scal
ar V
ecto
r Co
mpo
nent
s/
Skal
are
Vekt
orko
mpo
nent
en(
,,
)(
,,
)
(,
,)
x y zRxyz
xRxyz
y
Rxyz
z
= = =
Ort
hono
rmal
Uni
t Vec
tors
/
Ort
hono
rmal
e Ei
nhei
tsve
ktor
en,
,
||
||
||
1x
yz
xy
zx
yz
⊥⊥
==
=
eee
ee
ee
ee
Coor
dina
tes
/ Ko
ordi
nate
n,
,;
,
,xyz
xyz
−∞<
<∞
y
z
x
xxe
yye
zze
R
Dr.-
Ing.
R. M
arkl
ein
-EFT
I -S
S 05
4
Fiel
d Ve
ctor
/ Fe
ldve
ktor
Cart
esia
n Co
ordi
nate
Sys
tem
/ Ka
rtes
isch
es K
oord
inat
ensy
stem
x
Ort
hono
rmal
Uni
t Vec
tors
/
Ort
hono
rmal
e Ei
nhei
tsve
ktor
en,
,
||
||
||
1
xy
z
xy
z
xy
z
⊥⊥
==
=
eee
ee
e
ee
e
Coor
dina
tes
/ Ko
ordi
nate
n,
,xyz
()
()
()
()
= A
(,
,)
A(
,,
)A
(,
,)
xy
z
xy
zx
yz
xyz
xyz
xyz
=+
+
++
AR
AR
AR
AR
ee
e
y
z
xxe
yye
zze
R
()
AR
xeye
zex y z
−∞<
<∞
−∞<
<∞
−∞<
<∞
Lim
its/
Gre
nzen
Arbi
trar
y Ve
ctor
Fie
ld/
Belie
bige
s Ve
ktor
feld
: Per
pend
icul
ar /
Senk
rech
t⊥
Dr.-
Ing.
R. M
arkl
ein
-EFT
I -S
S 05
5
Not
atio
n an
d Fi
eld
Qua
ntiti
es /
Not
atio
n un
d Fe
ldgr
ößen
()
()
()
()
3
12
31
12
3
3 V
ecto
r Com
pone
nts /
3 V
ekto
rkom
pone
nten
,,
,,
= E
(,
,,)
E(
,,
,)E
(,
,,)
= E
(,
,,)
= E
(,
,,)
ii
ii
xy
z
xy
zx
yz
xx
i
xx
tt
tt
xyzt
xyzt
xyzt
xxxt
xxxt
=
=+
+
++
∑
ER
ER
ER
ER
ee
e
e
e
()
()
()
()
()
()
()
()
()
()
9 D
yadi
c Com
pone
nts /
9 dy
adisc
he K
ompo
nent
en
,,
,,
,,
,
,,
,
(,
,,)
(,
,,)
(,
,,)
+(
,,
,)(
xxxy
xz
yxyy
yz
zxzy
zz
xxxy
xzxx
xy
xz
yxyy
yx
tt
tt
tt
t
tt
t
xyzt
xyzt
xyzt
xyzt
x
εε
εε
εε
ε
εε
ε
εε
ε
εε
=+
+
++
+
++
+
=+
+
+
RR
RR
RR
R
RR
R
ee
ee
ee
ee
33
12
31
j1 1
23
,,
,)(
,,
,)
+(
,,
,)(
,,
,)(
,,
,)
(
,,
,)
(
,,
,)
ij
ij
ij
ij
yzyy
yz
zxzy
zzzx
zy
zz
xx
xx
i xx
xx
yzt
xyzt
xyzt
xyzt
xyzt
xxxt
xxxt
ε
εε
ε
ε
ε==
+
++
= =
∑∑
ee
ee
ee
ee
ee
ee
ee
Vect
or /
Vek
tor:
El
ectr
ic F
ield
Str
engt
h/
Elek
tris
che
Feld
stär
keD
yad
/ D
yade
: Pe
rmitt
ivity
Dya
d/
Perm
ittiv
itäts
dyad
e
with
Ein
stei
n’s
Sum
mat
ion
Conv
entio
n /
mit
Eins
tein
sche
r Su
mm
atio
nsko
nven
tion
Eins
tein
‘s S
umm
atio
n Co
nven
tion:
If a
inde
x ap
pear
s tw
o tim
es a
t one
sid
e of
an
equa
tion
(and
not
at t
he o
ther
sid
e), t
he in
dex
is a
utom
atic
ally
sum
med
ove
r 1
to 3
. /
Eins
tein
sche
Sum
men
konv
entio
n: W
enn
ein
Inde
x au
f ein
er S
eite
ein
er G
leic
hung
zw
eim
al v
orko
mm
t (un
d au
f der
and
eren
nic
ht),
wird
dar
über
von
1 b
is 3
sum
mie
rt.
Dr.-
Ing.
R. M
arkl
ein
-EFT
I -S
S 05
6
Posi
tion
Vect
or /
Ort
svek
tor
(Pos
ition
svek
tor)
()
()
()
=
(
)(
)(
)(
)(
)
()
rz
rz
rz
rz
RR
R
rzϕ
ϕϕ
ϕϕ
ϕ
=+
+
++
=+
RRR
RR
RR
Re
Re
Re
ee
Cylin
dric
al C
oord
inat
e Sy
stem
/ Z
ylin
derk
oord
inat
ensy
stem
Vect
oria
l Vec
tor
Com
pone
nts
/ Ve
ktor
ielle
Vek
tork
ompo
nent
en(
)(
)(
)(
)(
)
()
()
rr
rr
zz
zz
Rr
r
Rz
zϕ
ϕϕ
==
= ==
RR
ee
RR
0
RR
ee
Scal
ar V
ecto
r Co
mpo
nent
s/
Skal
are
Vekt
orko
mpo
nent
en(
,,
)(
)(
,,
)0
(,
,)
rr
zz
Rr
zr
Rr
z
Rr
zz
ϕ
ϕϕ
ϕ ϕ
= = =
e e
Ort
hono
rmal
Uni
t Vec
tors
/
Ort
hono
rmal
e Ei
nhei
tsve
ktor
en(
),(
),
()
()
|(
)||
()|
||
1r
z
rz
rz
ϕ
ϕϕ
ϕϕ
ϕϕ
ϕϕ
⊥⊥
==
=
ee
e
ee
ee
ee
Coor
dina
tes
/ Ko
ordi
nate
n,
,;
0
,02
,r
zr
zϕ
ϕπ
≤<∞
≤<
−∞<
<∞
y
z
x
()
rr
ϕe
zze
R
ϕ
Dr.-
Ing.
R. M
arkl
ein
-EFT
I -S
S 05
7
Fiel
d Ve
ctor
/ F
eldv
ekto
rCy
lindr
ical
Coo
rdin
ate
Syst
em/
Zylin
derk
oord
inat
ensy
stem
()
()
()
()
()
()
= A
(,
,)
(,
,)
(,
,)
rz
rz
rz
rz
Ar
zAr
zϕ
ϕϕ
ϕϕ
ϕϕ
ϕ
=+
+ ++
AR
AR
AR
AR
ee
e
Ort
hono
rmal
Uni
t Vec
tors
/
Ort
hono
rmal
e Ei
nhei
tsve
ktor
en(
),(
),
()
()
()
()
1
rz
rz
rz
ϕ
ϕ ϕ
ϕϕ
ϕϕ
ϕϕ
⊥⊥
==
=
ee
e
ee
e
ee
e
Coor
dina
tes
/ Ko
ordi
nate
n,
,r
zϕ
y
z
x
()
rr
ϕe
zze
R
ϕ
()
AR
()
rϕ
e
ze
()
ϕϕ
e
0 02
r
zϕ
π≤
<∞
≤<
−∞<
<∞
Lim
its/
Gre
nzen
Arbi
trar
y Ve
ctor
Fie
ld/
Belie
bige
s Ve
ktor
feld
: Per
pend
icul
ar /
Senk
rech
t⊥
Dr.-
Ing.
R. M
arkl
ein
-EFT
I -S
S 05
8
Posi
tion
Vect
or /
Ort
svek
tor
(Pos
ition
svek
tor)
()
()
()
=
(
)(
,)
()
(,
)
(
)(
)
(,
)
R RR
R
RR
R R
ϑϕ
ϑϑ
ϕϕϑ
ϕϑϕ
ϕ
ϑϕ
=+
+ +
+
=
RR
RR
RR
R
Re
Re
Re
e
Sphe
rical
Coo
rdin
ate
Syst
em/
Kuge
lkoo
rdin
aten
syst
em
Vect
oria
l Vec
tor
Com
pone
nts
/ Ve
ktor
ielle
Vek
tork
ompo
nent
en(
)(
,,
)(
,)
(,
)(
)(
,,
)(
,)
()
(,
,)
()
RR
RR
RR
RRR
RR
ϑϑ
ϑ
ϕϕ
ϕ
ϑϕ
ϑϕ
ϑϕ
ϑϕ
ϑϕ
ϑϕ
ϕ
==
==
==
RR
ee
RR
e0
RR
e0
Scal
ar V
ecto
r Co
mpo
nent
s/
Skal
are
Vekt
orko
mpo
nent
en(
,,
),(
,,
),(
,,
)RRR
RR
RR
ϑϕ
ϑϕ
ϑϕ
ϑϕ
Ort
hono
rmal
Uni
t Vec
tors
/
Ort
hono
rmal
e Ei
nhei
tsve
ktor
en,
,
||
||
||
1R R
R
ϑϕ
ϑϕ
ϑϕ
⊥⊥
==
=
eee
ee
ee
ee
Coor
dina
tes
/ Ko
ordi
nate
n,
,;
0
,0;0
2R
Rϑϕ
ϑπ
ϕπ
≤<∞
≤≤
≤<
y
z
x
ϕ
Rϑ
()
,R
Rϑ
ϕe
Dr.-
Ing.
R. M
arkl
ein
-EFT
I -S
S 05
9
Fiel
d Ve
ctor
/ Fe
ldve
ktor
Sphe
rical
Coo
rdin
ate
Syst
em/
Kuge
lkoo
rdin
aten
syst
em
()
()
()
()
()
()
()
,
= A
(,
,)
,(
,,
),
(,
,)
R RR
t
RAR
AR
ϑϕ
ϑϕ
ϑϕ
ϑϕ
ϑϕ
ϑϕ
ϑϕ
ϑϕ
ϕ
=+
+
++
AR
AR
AR
AR
ee
e
Ort
hono
rmal
Uni
t Vec
tors
/
Ort
hono
rmal
e Ei
nhei
tsve
ktor
en(
)(
)(
)(
)(
)(
)(
)(
)(
)
,,
,,
,,
|,
||
,|
||
1
R R R
ϑϕ
ϑϕ
ϑϕ
ϑϕ
ϑϕ
ϕ
ϑϕ
ϑϕ
ϕ
ϑϕ
ϑϕ
ϕ
⊥⊥
==
=
ee
e
ee
e
ee
e
Coor
dina
tes
/Ko
ordi
nate
n,
,Rϑϕ
y
z
x
ϕ
Rϑ
()
,R
Rϑ
ϕe
()
AR ()
ϕϕ
e
()
,R
ϑϕ
e
()
,ϑ
ϑϕ
e
Arbi
trar
y Ve
ctor
Fie
ld/
Belie
bige
s Ve
ktor
feld
Lim
its/
Gre
nzen
: Per
pend
icul
ar /
Senk
rech
t⊥
0 0 02
R ϑπ
ϕπ
≤<∞
≤≤
≤<
Dr.-
Ing.
R. M
arkl
ein
-EFT
I -S
S 05
10
Sphe
rical
Coo
rdin
ates
/Ku
gelk
oord
inat
enCy
lindr
ical
Coor
dina
tes
/Zy
linde
rkoo
rdin
aten
Cart
esia
nCo
ordi
nate
s/
Kart
esis
che
Koor
dina
ten
x y z
cos
sin
r rz
ϕ ϕ
sin
cos
sin
sin
cos
R R R
ϑϕ
ϑϕ
ϑ
22
arct
an
xy y x
z+r zϕ
sin
cos
R R
ϑϕ
ϑ
22
2
22
arct
an
arct
an
xy
z
xy
zy x
++
+
22
arct
an
rz r z
ϕ
+R ϑ ϕ
Tran
sfor
mat
ion
Tabl
e /
Umre
chnu
ngst
abel
le
z
y
x
ϕ
Rϑ
Coor
dina
tes
of D
iffer
ent
Coor
dina
te S
yste
ms
/Ko
ordi
nate
n ve
rsch
iede
nen
Koor
dina
tens
yste
men
Dr.-
Ing.
R. M
arkl
ein
-EFT
I -S
S 05
11
cos
sin
cos
xr
Rϕ
ϑϕ
==
1.Fo
rmul
ate
xas
a fu
nctio
n of
the
cylin
der
and
sphe
rical
coo
rdin
ates
./
Form
ulie
re x
als
Funk
tion
der
Zylin
der-
und
Kuge
lkoo
rdin
aten
.
2.Fo
rmul
ate
ras
a fu
nctio
n of
the
Cart
esia
n an
d sp
heric
al c
oord
inat
es.
/ Fo
rmul
iere
ral
s Fu
nktio
n de
r Ka
rtes
isch
en u
nd K
ugel
koor
dina
ten.
3.Fo
rmul
ate
as
a fu
nctio
n of
the
cylin
der
coor
dina
tes.
/ Fo
rmul
iere
als
Fun
ktio
n de
r Zy
linde
rkoo
rdin
aten
.
22
sin
rx
yR
ϑ=
+=
22
22
22
1
(co
s)
(si
n)
cos
sin
xy
rr
rr
ϕϕ
ϕϕ
=
+=
+=
+=
22
xy
+ 22
xy
+
Exam
ples
/ B
eisp
iele
Dr.-
Ing.
R. M
arkl
ein
-EFT
I -S
S 05
12
Sphe
rical
Coo
rdin
ates
/Ku
gelk
oord
inat
enCy
lindr
ical
Coo
rdin
ates
/Zy
linde
rkoo
rdin
aten
Cart
esia
n Co
ordi
nate
s/
Kart
esis
che
Koor
dina
ten
xy
zx
yz
AA
A=
++
Ae
ee
rz
rz
AA
Aϕ
ϕ=
++
Ae
ee
RRA
AA
ϑϕ
ϑϕ
++
A=
ee
e
x y zA A A
cos
sin
sin
cos
r r
z
AA
AA A
ϕ ϕ
ϕϕ
ϕϕ
− +
sin
cos
cos
cos
sin
sin
sin
cos
sin
cos
cos
sin
R R
R
AA
AA
AA
AA
ϑϕ
ϑϕ
ϑ
ϑϕ
ϑϕ
ϕ
ϑϕ
ϑϕ
ϕ
ϑϑ
+−
++
−
cos
sin
sin
cos
xy
xy
z
AA
AA
A
ϕϕ
ϕϕ
+
−+
r zA A Aϕ
sin
cos
cos
sin
R RAA
AA
Aϑ
ϕ
ϑ
ϑϑ
ϑϑ
+ −
sin
cos
sin
sin
cos
cos
cos
cos
sin
sin
sin
cos
xy
z
xy
z
xy
AA
AA
AA
AA
ϑϕ
ϑϕ
ϑ
ϑϕ
ϑϕ
ϑ
ϕϕ
++
+−
−+
sin
cos
cos
sin
rz
rz
AA
AA
A ϕ
ϑϑ
ϑϑ
+ −RA A Aϑ ϕ
Tran
sfor
mat
ion
Tabl
e /
Umre
chnu
ngst
abel
le
Scal
ar V
ecto
r Co
mpo
nent
s in
Diff
eren
t Coo
rdin
ate
Syst
ems
/Sk
alar
e Ve
ktor
kom
pone
nten
in v
ersc
hied
enen
Koo
rdin
aten
syst
emen
Dr.-
Ing.
R. M
arkl
ein
-EFT
I -S
S 05
13
Exam
ple:
Coo
rdin
ate
Tran
sfor
mat
ion
of th
e Po
sitio
n Ve
ctor
/ Be
ispi
el: K
oord
inat
entr
ansf
orm
atio
n de
s O
rtsv
ekto
r
()
()
()
,,
,,
,,
zx
y
xy
zRxyz
Rxyz
Rxyz
xy
z=
++
Re
ee
Posi
tion
Vect
or in
the
Cart
esia
n Co
ordi
nate
Sys
tem
/
Ort
svek
tor
im K
arte
sisc
hen
Koor
dina
tens
yste
m
(,
,,
,,
)
cos
sin
(,
,,
,,
)si
nco
s(
,,
,,
,)
rx
yz
xy
xy
zx
y
zx
yz
z
Rr
zRRR
RR
Rr
zRRR
RR
Rr
zRRR
Rϕ
ϕϕ
ϕϕ
ϕϕ
ϕ
=+
=−
+=
(,
,)
(,
,)
cos
(,
,)
(,
,)
sin(
,,
)(
,,
)
x y zRr
zxr
zr
Rr
zyr
zr
Rr
zzr
zz
ϕϕ
ϕϕ
ϕϕ
ϕϕ
==
==
==
()
()
()
,,
,,
,,
cos
sinz
xy
xy
zRr
zRr
zRr
z
rr
z ϕϕ
ϕ
ϕϕ
=+
+R
ee
e
Tran
sfor
mat
ion
of th
e Co
ordi
nate
s /
Tran
sfor
mat
ion
der
Koor
dina
ten
Posi
tion
Vect
or in
the
Cart
esia
n Co
ordi
nate
Sys
tem
as
a Fu
nctio
n of
Cyl
inde
r Co
ordi
nate
s /
Ort
svek
tor
im K
arte
sisc
hen
Koor
dina
tens
yste
m a
ls F
unkt
ion
der
Zylin
derk
oord
inat
en
Tran
sfor
mat
ion
of th
e Sc
alar
Vec
tor
Com
pone
nts
/ Tr
ansf
orm
atio
n de
r ska
lare
n Ve
ktor
kom
pone
nten
22
1
co
sco
ssi
nsi
n
(c
ossi
n)
cos
sin
sin
cos
0
r zz
Rr
rr
r
Rr
r
RR
ϕ
ϕϕ
ϕϕ
ϕϕ
ϕϕ
ϕϕ
=
=+
=+
=
=−
+= =
()
rz
rz
RR
rz
ϕ=
+R
ee
Posi
tion
Vect
or in
the
Cylin
der
Coor
dina
te S
yste
m /
O
rtsv
ekto
r in
dem
Zyl
inde
rkoo
rdin
aten
syst
em
()
()
()
(,
,,
,,
)
,,
()
,,
()
,,
rz
rz
rz
ryRRR
Rr
yR
ry
Rr
yϕ
ϕϕ
ϕ
ϕϕ
ϕϕ
ϕ=
++
R
ee
e?Po
sitio
n Ve
ctor
in th
e Cy
linde
r Co
ordi
nate
Sys
tem
/
Ort
svek
tor
im Z
ylin
derk
oord
inat
ensy
stem
Dr.-
Ing.
R. M
arkl
ein
-EFT
I -S
S 05
14
Fara
day‘
s In
duct
ion
Law
in In
tegr
al F
orm
/Fa
rada
ysch
es In
dukt
ions
gese
tz in
Inte
gral
form
(1)
()
()
()
St
Ct
St
=∂
m(
)(
)(
)(
)d
(,
)(
,)
(,
)d
Ct
St
St
St
tt
tt
=∂
=−
−∫
∫∫∫∫
ER
dRBR
dSJ
RdS
ii
i
Fara
day‘
s In
duct
ion
Law
/ Fa
rada
ysch
es In
dukt
ions
gese
tz
Tim
e D
epen
dent
Sur
face
/Ze
itabh
ängi
ge F
läch
eTi
me
Dep
ende
nt C
onto
ur /
Zeita
bhän
gige
Kon
tur
Dr.-
Ing.
R. M
arkl
ein
-EFT
I -S
S 05
15
Fara
day‘
s In
duct
ion
Law
in In
tegr
al F
orm
/Fa
rada
ysch
es In
dukt
ions
gese
tz in
Inte
gral
form
(2)
Fara
day‘
s In
duct
ion
Law
/ Fa
rada
ysch
es In
dukt
ions
gese
tz
[](
)(
)Ct
St
=∂∫
dRi
(,)t
ER dR
(,)t
ER
dRiSc
alar
Pro
duct
of E
and
dR
= ta
ngen
tial p
roje
ctio
n of
E o
nto
dR /
Sk
alar
prod
ukt v
on E
auf
dR
= T
ange
ntia
lpro
jekt
ion
von
E au
f dR
[V]
Vect
oria
l Diff
eren
tial L
ine
Elem
ent /
Vek
torie
lles
diff
eren
tielle
s Li
nien
elem
ent
[m]
Elec
tric
Fie
ld S
tren
gth
/ El
ektr
isch
e Fe
ldst
ärke
[V/m
]
Clos
ed C
onto
ur In
tegr
al /
Ges
chlo
ssen
es K
urve
nint
egra
l[m
]
dR=
dRs
Vect
oria
l Diff
eren
tial L
ine
Elem
ent /
Ve
ktor
ielle
sdi
ffer
entie
lles
Lini
enel
emen
t
Tang
entia
l Uni
t Vec
tor
/ Ta
ngen
tiale
r Ei
nhei
tsve
ktor
Scal
ar D
iffer
entia
l Lin
e El
emen
t/ S
kala
res
diff
eren
tielle
s Li
nien
elem
ent
m(
)(
)(
)(
)d
(,
)(
,)
(,
)d
Ct
St
St
St
tt
tt
=∂=−
−∫
∫∫∫∫
ER
dRBR
dSJ
RdS
ii
i
Dr.-
Ing.
R. M
arkl
ein
-EFT
I -S
S 05
16
Diff
eren
t Pro
duct
s /
Vers
chie
dene
Pro
dukt
e
C=ABi
Scal
ar P
rodu
ct /
Ska
larp
rodu
kt
=C
AB
=C
A×B
Vect
or P
rodu
ct /
Vek
torp
rodu
kt
Dya
dic
Prod
uct /
Dya
disc
hes
Prod
ukt
Dr.-
Ing.
R. M
arkl
ein
-EFT
I -S
S 05
17
Scal
ar P
rodu
ct (D
ot o
r In
ner
Prod
uct)
/ Sk
alar
prod
ukt (
Punk
tpro
dukt
ode
r in
nere
s Pr
oduk
t) (1
)
cos
(,
)
cos
AB
ABAB
φ
φ
=∠
=
AB
AB
AB
i
cosAB
Bφ
=
ABφ
A
B
cosAB
Aφ
=
ABφEn
clos
ed A
ngle
/
Eing
esch
loss
ener
Win
kel
cos
cosBA AB
BA ABφ φ
= = =
AB
BA
ii
()
()
cos
cos
ABAB
φφ
=−
cos
arcc
os
AB ABφ φ
=
=
AB
AB
AB
AB
i
i
Dr.-
Ing.
R. M
arkl
ein
-EFT
I -S
S 05
18
Scal
ar P
rodu
ct (D
ot o
r In
ner
Prod
uct)
/ Sk
alar
prod
ukt (
Punk
tpro
dukt
ode
r in
nere
s Pr
oduk
t) (2
)
10
0
01
0 10
0
()
()
+
+
xy
zx
yz
xy
zx
yz
xx
xy
xz
xx
xy
xz
yx
yy
yz
yx
yy
yz
zx
zy
zz
zx
zy
zz
AA
AB
BB
AB
AB
AB
AB
AB
AB
AB
AB
AB
A
==
=
==
= ==
=
=+
++
+
=+
+
++
++
=
AB
ee
ee
ee
ee
ee
ee
ee
ee
ee
ee
ee
ee
ii
ii
i
ii
i
ii
i
xx
yy
zz
BAB
AB
++
12
31
23
12
31
23
11
22
33
3 1
()
()
()
()
i
i
xy
zx
yz
xy
zx
yz
xx
yy
zz
xx
xx
xx
xx
xx
xx
xx
xx
xx
xx
iAA
AB
BB
AB
AB
AB
AA
AB
BB
AB
AB
AB
AB
=
=+
++
+
=+
+
=+
++
+
=+
+
=∑
AB
ee
ee
ee
ee
ee
ee
ii
i
xy
z⊥
⊥e
ee
Ort
hono
rmal
Uni
t Vec
tors
/
Ort
hono
rmal
e Ei
nhei
tsve
ktor
en
1 0 0
xx
xy
xz
= = =
ee
ee
eei i i
0 1 0
yx
yy
yz
= = =
ee
ee
eei i i
0 0 1
zx
zy
zz
= = =
ee
ee
eei i i
1 2 3
xx
yx
zx
= = =
Cart
esia
n Co
ordi
nate
s /
Kart
esis
che
Koor
dina
ten
Dr.-
Ing.
R. M
arkl
ein
-EFT
I -S
S 05
19
Scal
ar P
rodu
ct (D
ot o
r In
ner
Prod
uct)
/ Sk
alar
prod
ukt (
Punk
tpro
dukt
ode
r in
nere
s Pr
oduk
t) (3
)
33
11
33
11
33
11
()
()
or/o
der
ij
ij
ij
ij
ij
ij
ij
ij
ij
ij
ij x i
xy
zx
yz
xy
zx
yz
xx
xx
ij
xx
xx
ij
xx
xx
ij
xx
xx
xx
ij
B
AA
AB
BB
AB
AB
AB
AB
AB
δ
δ
δ
==
==
==
=
=
=
=+
++
+
= = = = =
∑∑
∑∑
∑∑
AB
ee
ee
ee
ee
ee
ee
ee
ii
i
i
i
i
ij
x j xx
jj
ii
xij
x
A AB
xx
AB
AB
δ= =
=
1 0ij
ij
ij
δ=
=
≠
Kron
ecke
r D
elta
/
Kron
ecke
r-D
elta
with
Ein
stei
n’s
Sum
mat
ion
Conv
entio
n /
mit
Eins
tein
sche
r Su
mm
atio
nsko
nven
tion
Eins
tein
‘s S
umm
atio
n Co
nven
tion:
If a
inde
x ap
pear
s tw
o tim
es a
t one
sid
e of
an
equa
tion
(and
not
at t
he o
ther
sid
e),
the
inde
x is
aut
omat
ical
ly s
umm
ed o
ver
1 to
3. /
Ei
nste
insc
he S
umm
enko
nven
tion:
Wen
n ei
n In
dex
auf e
iner
Se
ite e
iner
Gle
ichu
ng z
wei
mal
vor
kom
mt (
und
auf d
er
ande
ren
nich
t), w
ird d
arüb
er v
on 1
bis
3 s
umm
iert
.
Dr.-
Ing.
R. M
arkl
ein
-EFT
I -S
S 05
20
Mag
nitu
de o
f a V
ecto
r /
Betr
ag e
ines
Vek
tors
10
0
01
0 10
0
(AA
A)
(AA
A)
A
AA
AA
A
+
AA
AA
AA
+ A
AA
AA
A
xy
zx
yz
xy
zx
yz
xx
xy
xz
xx
xy
xz
yx
yy
yz
yx
yy
yz
zx
zy
zz
zx
zy
zz
==
=
==
= ==
=
= =+
++
+
=
++
+
+
+
+
AAA e
ee
ee
e
ee
ee
ee
ee
ee
ee
ee
ee
ee
i
i
ii
i
ii
i
ii
i
1 2
22
2
A
AA
AA
A
AA
A
A
xx
yy
zz
xy
z
=+
+
=+
+
=
33
11
2
ij
ij
ij
ij
ij
ij
ij
i
xx
xx
ij
xx
xx
xx
xx
x
AB
AA
AA
A
δ
==
=
= = = = =
∑∑
AAA
ee
ee
ee
i
i
i
i
Dr.-
Ing.
R. M
arkl
ein
-EFT
I -S
S 05
21
Exam
ple:
Pos
ition
Vec
tor
and
Elec
tric
Fie
ld S
tren
gth
Vect
or /
Beis
piel
: Ort
svek
tor
und
elek
tris
cher
Fel
dstä
rkev
ekto
r
(,
,)
R(
,,
)R
(,
,)
R(
,,
)
xy
zx
yz
xy
z
xyz
xyz
xyz
xyz
xy
z=
++
=+
+R
ee
ee
eeCa
rtes
ian
Coor
dina
te S
yste
m/
Kart
esis
ches
Koo
rdin
aten
syst
em
(,
)(
,,
,)
E(
,,
,)
E(
,,
,)
E(
,,
,)
xy
zx
yz
txyzt
xyzt
xyzt
xyzt
= =+
+ER
Ee
ee
Elec
tric
Fie
ld S
tren
gth
Vect
or /
El
ektr
isch
e Fe
ldst
ärke
vekt
or
22
2
(,
,)
ˆ (,
,)
(,
,)
x
yz
xyz
xyz
xyz
xy
z
xy
z
=
++
=+
+
RR
R ee
e
()(
)2
22
(,
,)
(,
,)
(,
,)
x
yz
xy
z
xyz
xyz
xyz
xy
zx
yz
xy
z
= =+
++
+
=+
+
RR
R
ee
ee
ee
i
i
22
2
(,
,)
ˆ (,
,)
(,
,)
EE
E
EE
E
xy
zx
yz
xy
z
xyz
xyz
xyz
=
++
=+
+
EE
Ee
ee
()(
)2
22
(,
,)
(,
,)
(,
,)
EE
EE
EE
E
EE
xy
zx
yz
xy
zx
yz
xy
z
xyz
xyz
xyz
= =+
++
+
=+
+
EE
E
ee
ee
ee
i
i
Posi
tion
Vect
or /
O
rtsv
ekto
r
Mag
nitu
de o
f the
Pos
ition
Vec
tor
(Dis
tanc
e) /
Be
trag
des
Ort
svek
tor
(Abs
tand
)
Mag
nitu
de o
f the
Ele
ctric
Fie
ld S
tren
gth
Vect
or
(Str
engt
h) /
Bet
rag
des
elek
tris
che
Feld
stär
keve
ktor
s (S
tärk
e)
Posi
tion
Unit
Vect
or (D
irect
ion)
/
Ort
sein
heits
vekt
or(R
icht
ung)
Elec
tric
Fie
ld S
tren
gth
Unit
Vect
or (D
irect
ion)
/
Elek
tris
che
Feld
stär
keei
nhei
tsve
ktor
(Ric
htun
g)