Giai_tich_3

8/8/2019 Giai_tich_3

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TRNG AI HOC A LATKHOA TOAN - TIN HOC

TA LE LI - O NGUYEN SN

GIAI TCH 3(Giao Trnh)

--Lu hanh noi bo--a Lat 2008


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G i a i T c h 3

T a L e L i - o N g u y e n S n

M u c l u c

C h n g I . T c h p h a n p h u t h u o c t h a m s o

1 . T c h p h a n p h u t h u o c t h a m s o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 . T c h p h a n s u y r o n g p h u t h u o c t h a m s o . . . . . . . . . . . . . . . . . . . . . . . 9

3 . C a c t c h p h a n E u l e r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 4

C h n g I I . T c h p h a n h a m s o t r e n a t a p

1 . a t a p k h a v i t r o n g Rn

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 9

2 . T c h p h a n h a m s o t r e n a t a p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 4

C h n g I I I . D a n g v i p h a n

1 . D a n g

k- t u y e n t n h p h a n o i x n g . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1

2 . D a n g v i p h a n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3

3 . B o e P o i n c a r e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 7

C h n g I V . T c h p h a n d a n g v i p h a n

1 . n h h n g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1

2 . T c h p h a n d a n g v i p h a n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 4

3 . C o n g t h c S t o k e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 7

B a i t a p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3


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4

I. Tch phn ph thuc tham s

1 Tch phn ph thuc tham s

1.1 nh ngha

nh ngha 1. Xt hm f(x, t) = f(x1, . . . , xn, t1, . . . , tm) xc nh trn minX T Rn Rm. Gi s X o -c (Jordan) v vi mi gi tr ca t T cnh, hm f(x, t) kh tch theo x trn X. Khi tch phn

I(t) =X

f(x, t)dx (1)

l hm theo bin t = (t1, . . . , tm), gi l tch phn ph thuc tham s vi mtham s t1, . . . , tm.

1.2 Tnh lin tc

nh l 1. Nu f(x, t) lin tc trn X T Rn Rm, y X, T l cc tpcompact, th tch phn

I(t) = X

f(x, t)dx

lin tc trn T.

Chng minh. C nh t0 T. Ta s chng minh vi mi > 0, tn ti > 0 saocho vi mi t T, d(t, t0) < ta c | I(t) I(t0) |< .T nh ngha suy ra

| I(t) I(t0) |=X

(f(x, t) f(x, t0))dx

X

| f(x, t) f(x, t0) | dx.

Do f lin tc trn compact nn lin tc u trn , tc l tn ti > 0 sao cho

| f(x, t) f(x, t) |< v(X)

vi mi (x, t), (x, t) X T, d((x, t), (x, t)) < .T , vi d(t, t0) < ta c

| I(t) I(t0) |< v(X) v(X)

= .


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5

2

V d. 1) Ta c limt0

11

x2 + t2dx =

11

|x|dx = 1 v hm x2 + t2 lin tc trn[1, 1] [, ].2) Kho st tnh lin tc ti im (0, 0) ca hm f(x, t) =

xt2ex

2t2 nu t = 00 nu t = 0

.

Nu f(x, t) lin tc ti (0, 0), th f(x, t) lin tc trn [0, 1] [, ]. Khi , tchphn I(t) =

10

f(x, t)dx lin tc trn [, ] . Nh-ng ta c

limt0 I(t) = limt0

10 xt

2

ex2t2

= 1

2 limt0

10 e

x2t2

d(x2

t2

)

= 12

limt0

(et2 1) = 1

2= 0 = I(0).

Vy, hm f(x, t) khng lin tc ti (0, 0).

Sau y chng ta s kho st mt tng qut ha ca nh l 1 trong tr-ng hpX = [a, b].

nh l 2. Cho f(x, t) lin tc trn [a, b]T, vi T l tp compact v a(t), b(t)l hai hm lin tc trn T sao cho a(t), b(t) [a, b] vi mi t T. Khi , tch

phn

I(t) =

b(t)a(t)

f(x, t)dx

lin tc trn T.

Chng minh. Do f lin tc trn tp compact nn gii ni, tc l tn ti M > 0sao cho | f(x, y) | M vi mi (x, t) [a, b] T. C nh t0 T ta c:

| I(t) I(t0) |= a(t0)

a(t) f(x, t)dx +b(t)

b(t0)f(x, t)dx +b(t0)

a(t0)[f(x, t) f(x, t0)]dxa(t0)a(t)

f(x, t)dx

+ b(t)b(t0)

f(x, t)dx

+b(t0)a(t0)

(f(x, t) f(x, t0))dx

M | a(t) a(t0) | +M | b(t) b(t0) | +b(t0)a(t0)

| f(x, t) f(x, t0) | dx.


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Khng nh suy ra t tnh lin tc ca a(t), b(t) v nh l 1. 2

V d. Do hm1

1 + x2 + t2lin tc trn [0, 1] [, ] v cc hm (t) = t,

(t) = cos t lin tc trn [, ], ta c

limt0

cos tt

dx

1 + x2 + t2dx =

10

dx

1 + x2=

4.

1.3 Tnh kh vi.

nh l 3. Nu f(x, t) v cc o hm ring fti

(x, t), i = 1, . . . , m, lin tc

trn X T Rn Rm, y X, T l cc tp compact, th tch phn

I(t) =

X

f(x, t)dx

kh vi trno

T v vi mi i ta c:

I

ti(t) =

X

f

ti(x, t)dx.

Chng minh. Vi mi t0 o

T c nh ta c:

I(t0 + hiei) I(t0)hi

=

X

f(x, t0 + hiei) f(x, t0)hi

dx.

trong ei l c s chnh tc ca Rm. p dng nh l gi tr trung bnh chohm 1 bin ta c:

f(x, t0 + hiei) f(x, t0) =f

ti (x, t0 + ihiei)hi, 0 < i < 1

Khi :I(t0 + hiei) I(t0)hi X

f

ti(x, t0)dx

=X

[f

ti(x, t0 + ihiei) f

ti(x, t0)]dx


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7

S dng tnh lin tc ca

f

ti (x, t) trn compact XT v l lun nh- trong chngminh nh l 1 suy ra

I

ti(t0) = lim

hi0

I(t0 + hiei) I(t0)hi

=

X

f

ti(x, t)dx.

Tnh lin tc caI

ti(t) trn T suy ra t nh l 1 2

V d. Xt I(t) =/2

01

cos xln

1 + t cos x

1

t cos x

dx, t (1, 1). Ta c cc hm

f(x, t) =

1

cos xln

1 + t cos x

1 t cos x nu x = /22t nu x = /2

f

t(x, t) =

2

1 t2 cos2 x,

lin tc trn [0, /2] [1 + , 1 ]. Vy, theo nh l trn

I(t) = 2

/20

dx

1 t2 cos2 x = 20

du

1 t2 + u2 =

1 t2 .

T , I(t) = arcsin t + C. V I(0) = 0, nn C = 0. Vy, I(t) = arcsin t.

nh l 4. Nu f(x, t) v cc o hm ringf

ti(x, t), i = 1, . . . , m, lin tc

trn [a, b] T, y T l tp compact trongRm, (t), (t) kh vi trn T v(t), (t) [a, b] vi mi t T, th tch phn

I(t) =

b(t)a(t)

f(x, t)dx

kh vi trno

T v vi mi i ta c:

I

ti(t) =

(t)(t)

f

ti(x, t)dx + f((t), t)

ti(t) f((t), t)

ti(t).


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Chng minh. Xt hm m + 2 bin

F(t,u,v) =

vu

f(x, t)dx, (t,u,v) D = T [a, b] [a, b].

Ta s ch ra rng F(t,u,v) l hm kh vi. Vi mi u, v c nh, t nh l 3,suy ra

F

ti(t,u,v) =

vu

f

ti(x, t)dx.

V phi ca ng thc trn -c xem nh- l tich phn ph thuc cc tham st,u,v.Hm

f

ti(x, t) xem nh- l hm theo cc bin x,t,u,v lin tc trn [a, b]D. T

nh l 2, vi a(t,u,v) = u, b(t,u,v) = v, suy raF

ti(t,u,v) l hm lin tc

trn D. Ngoi ra ta cn c

F

u(t,u,v) = f(u, t) v F

v(t,u,v) = f(v, t)

u l nhng hm lin tc trn D. Vy, hm F(t,u,v) kh vi.

Hm I(t) -c xem nh- l hm hp I(t) = F(t, (t), (t)). T , hm I(t)kh vi v

I

ti(t) =

F

ti(t, (t), (t)) +

F

u(t, (t), (t))

ti(t) +

F

v(t, (t), (t))

ti(t)

=(t)(t)

f

ti(x, t)dx + f((t), t)

ti(t) f((t), t)

ti(t).

2

V d. Xt tch phn I(t) =sin t

t

etxdx. Theo nh l trn, hm I(t) kh vi v

I(t) =

sin tt

xetxdx + et sin t cos t et2.


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9

2 Tch phn suy rng ph thuc tham s2.1 Cc nh ngha

nh ngha 2. Gi s hm f(x, t) xc nh trn [a,) T, T R, sao cho vimi t T c nh , hm f(x, t) kh tch trn [a, b], vi mi b > a. Tch phn

I(t) =

a

f(x, t)dx (1),

gi l tch phn suy rng loi 1 ph thuc tham s. Tch phn (1) gi l hi t

ti t0 nuu tch phna

f(x, t0)dx hi t, tc l tn ti limb

ba

f(x, t0)dx = I(t0)

hu hn.Tch phn (1) gi l hi t trn T nuu hi t ti mi im ca T, tc l

> 0,t T,a0(, t) > a, sao cho b a0 =b

f(x, t)

< .Tch phn (1) gi l hi t u trn T nuu

> 0,a0() > a, sao cho b a0,t T = b

f(x, t) < .

nh ngha 3. Gi s hm f(x, t) xc nh trn [a, b) T, T R, sao cho vimi t T c nh , hm f(x, t) kh tch trn mi on [a, b ], > 0 . Tch

phn

J(t) =

ba

f(x, t)dx = lim0+

ba

f(x, t)dx, (2)

gi l tch phn suy rng loi 2 ph thuc tham s. Tch phn (2) gi l hi t

ti t0 nuu tch phn

ba

f(x, t0)dx hi t, tc l tn ti lim0

ba

f(x, t0)dx = J(t0)

hu hn.Tch phn (2) gi l hi t trn T nuu hi t ti mi im ca T, tc l

> 0,t T,(, t) > 0, sao cho 0 < < =

bb

f(x, t)

< .


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Tch phn (2) gi l hi t u trn T nuu

> 0,0() > 0, sao cho 0 < < ,t T =

bb

f(x, t)

< .Ch . 1) T-ng t, ta nh ngha

I(t) =b

f(x, t)dx = lima

ba

f(x, t)f(x, t),

J(t) =b

a

f(x, t)dx = lim0

+

b

a+

f(x, t)f(x, t),

v cng c khi nim hi t, hi t u t-ng ng.2) Vic kho st tch phn suy rng ph thuc tham s loi 2 -c thc hinhon ton t-ng t nh- loi 1, t nh ngha cc khi nim n cc tnh cht.Do , trong mc ny, ta ch kho st tch phn suy rng ph thuc tham s

I(t) =a

f(x, t)dx.

V d. Xt tch phn I(t) =0

textdx. Khi

a) I(t) hi t trn (0,

) v

> 0,t T,a0 = ln t ,b > a0 =b

text = ebt < .

b) I(t) khng hi t u trn (0,) v vi (0, 1), vi mi a0 > 0, nu chnb = a0 v t t bt ng thc 0 < t .

c) I(t) hi t u trn Tr = [r,), vi r > 0. Tht vy, ta c

> 0,a0 =ln

r ,b a0,t Tr =b

text = ebt < ea0r < .


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2.2 Mt s tiu chun hi t u

nh l 5. (Tiu chun Cauchy) Tch phn I(t) =a

f(x, t)dx hi t u trn

T khi v ch khi

> 0,a0() > a, sao cho b1, b2 a0,t T =

b2b1

f(x, t)

< . ()

Chng minh. Gi s I(t) =

af(x, t)dx hi t u trn T. Khi , iu kin ()

suy ra t bt ng thcb2

b1

f(x, t)

b1

f(x, t)

+

b2

f(x, t)

Ng-c li, vi t c nh, iu kin () suy ra I(t) hi t. Trong (), cho b2 0,suy ra I(t hi t u theo nh ngha. 2

nh l 6. (Tiu chun Weierstrass) Gi s(1) tn ti hm (x) sao cho |f(x, t)| (x), x a, t T,(2) tch phn

a

(x)dx hi t.

Khi , tch phn I(t) =a

f(x, t)dx hi t u trn T.

Chng minh. Theo tiu chun Cauchy i vi tch phn suy rng hi t, vi mi > 0, tn ti a0 sao cho

b2b1

(x)

< , b1, b2 a0.Suy ra,

b2b1

f(x, t)

b2

b1

|f(x, t)|

b2

b1

(x) < .

Theo nh l 5, tch phn I(t) hi t u. 2

kho st tnh cht ca tch phn suy rng ph thuc tham s hi t u, chngta thit lp mi quan h gia n v dy hm hi t u.


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Mnh 1. Gi s tch phn I(t) =

a f(x, t)dx hi t u trn T v (an), vi

an > a. l dy s sao cho limn

an = . Khi , dy hm

In(t) =

ana

f(x, t)dx

hi t u ti hm s I(t) trn T.

Chng minh. Do I(t) =

a f(x, t)dx hi t trn T nn dy hm (In(t)) hi t tiI(t) trn T. V I(t) hi t u nn vi mi > 0, tn ti a0 sao cho

b

f(x, t)

< , b > a0,t T.V lim

nan = nn tn ti N > 0 sao cho vi mi n N, ta c an b. Vy,

ta c

|In(t) I(t)| =

ana

f(x, t)a

f(x, t)

=

an

f(x, t)

< ,

vi mi n N, vi mi t T. T , In(t) hi t u ti I(t) trn T. 2

2.2.1 Tnh lin tc

nh l 7. Nu hm f(x, t) lin tc trn [a,) [c, d] v tch phn I(t) =a

f(x, t)dx hi t trn trn [c, d], th I(t) lin tc trn [c, d].

Chng minh. Gi (an), vi an > a. l dy s sao cho limn

an = v xt dyhm

In(t) =

ana

f(x, t)dx, t [c, d].

Vi mi n c nh, theo nh l 1, hm In(t) lin tc trn [c, d]. Theo mnh 1, dy hm (In(t)) hi t u ti I(t). Theo nh l v tnh lin tc ca dy hmhi t u, I(t) lin tc trn [c, d]. 2


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2.2.2 Tnh kh vi

nh l 8. Gi s

(a) Hm f(x, t) lin tc v c o hm ringf

t(x, t) lin tc trn [a,) [c, d].

(b) Tch phn I(t) =a

f(x, t)dx hi t trn [c, d].

(c) Tch phna

f

t(x, t)dx hi t u trn [c, d].

Khi , hm I(t) kh vi trn [c, d] v ta c cng thc I(t) =a

f

t(x, t)dx.

Chng minh. Xt dy hm

In(t) =

a+na


Vi mi n, theo nh l 3, hm In(t) kh vi trn [c, d] v

In(t) =

a+na

f

t(x, t)dx, t [c, d].

Ta c lim In(t) = I(t) v lim In(t) =

a

f

t

(x, t)dx. Theo mnh 1, dy hm

In(t) hi t u trn [c, d]. Theo nh l v tnh kh vi ca dy hm hi t u,I(t) kh vi trn [c, d] v

I(t) =

limn

In(t)

= limn

In(t) =

a

f

t(x, t)dx.

2

2.2.3 Tnh kh tch

nh l 9. Gi s hmf(x, t)

lin tc trn[a,) [c, d]

v tch phnI(t) =

a

f(x, t)dx hi t u trn [c, d]. Khi , hm I(t) kh tch trn [c, d] v ta c

cng thc

dc

I(t)dt =

dc

a

f(x, t)dx

dt =

a

dc

f(x, t)dt

dx


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Chng minh. Theo nh l 7, I(t) l hm lin tc trn [c, d], do kh tch. Xtdy hm

In(t) =

a+na


Vi mi n c nh, theo nh l 1, hm In(t) lin tc trn [c, d]. Theo mnh 1, dy hm (In(t)) hi t u ti I(t) trn [c, d]. Theo nh l v tnh kh tchca dy hm hi t u, ta c

d

c I(t)dt =d

c limn In(t)dt = limnd

c In(t)dt= lim

n

dc

a+na

f(x, t)dx

dt

= limn

a+na

dc

f(x, t)dx

dt =

a

dc

f(x, t)dt

.

2

3 Cc tch phn Euler

3.1 Tch phn Euler loi 13.1.1 nh ngha

Tch phn Euler loi 1 hay hm Beta l tch phn ph thuc 2 tham s dng

B(p,q) =

10

xp1(1 x)q1dx, p > 0, q > 0.

3.1.2 Cc tnh cht cu hm Beta

1) S hi t. Ta phn tch B(p,q) thnh hai tch phn

B(p,q) =

1/20

xp1(1 x)q1dx +1

1/2

xp1(1 x)q1dx = B1(p,q) + B2(p,q).


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15

Tch phn B1

hi t nu p > 0 v phn k nu p

0. iu ny suy ra t

xp1(1 x)q1 Mqxp1, Mq = max0x1/2

(1 x)q1xp1(1 x)q1 mqxp1, mq = min

0x1/2(1 x)q1.

T-ng t, tch phn B2 hi t nu q > 0 v phn k nu q 0. Nh- vy hmB(p,q) xc nh vi mi p > 0, q > 0.

2) S hi t u. Tch phn B(p,q) hi t u trn ch nht [p0, p1] [q0, q1],trong , 0 < p0 < p1, 0 < q0 < q1. iu ny suy ra t nh gi

xp1(1 x)q1 xp01(1 x)q01, x (0, 1), p p0, q q0,v sau s dng tiu chun Weierstrass.

3) Tnh lin tc. Hm B(p,q) lin tc trn min xc nh ca n. Tht vy, vimi (p,q), p > 0, q > 0, tch phn B(p,q) hi u trn [p, p + ] [q, q + ],do lin tc trn min ny.

4) Tnh i xng. Bng cch i bin x = 1 t, ta -c B(p,q) = B(q, p).

5) Cng thc truy hi. Bng cch ly tch phn tng phn t tch phn B(p,q) ta-c

B(p + 1, q + 1) =q

p + q + 1B(p + 1, q) =

q

p + q + 1B(p,q + 1).

c bit, nu m, n l cc s t nhin, th p dng lin tip cng thc trn, ta c

B(1, 1) = 1

B(p + 1, 1) =1

p + 1

B(p + 1, n) =n!

(p + n)(p + n 1) (p + 1)B(m, n) =

(n 1)!(m 1)!(m + n 1)! .


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3.2 Tch phn Euler loi 2

3.2.1 nh ngha

Tch phn Euler loi 2 hay hm Gamma l tch phn ph thuc tham s dng

(p) =

0

xp1exdx, p > 0.

3.2.2 Cc tnh cht cu hm Gamma

1) S hi t. Ta phn tch B(p,q) thnh hai tch phn

(p) =

10

xp1exdx +

1

xp1exdx = 1(p) + 2(p).

Tch phn 1(p) hi t khi p > 0. iu ny suy ra t

xp1ex xp1, x (0, 1].

Tch phn 2(p) hi t khi p > 0. iu ny suy ra t

limx

xp1ex

1

xp+1

= limx

=x2p

ex= 0, v

1

1

xp+1< .

Suy ra, tch phn (p) =0

xp1exdx hi t khi p > 0.

2) S hi t u. Tch phn 1(p) hi t u trn mi on [p0.p1], vi p1 > p0 > 0.iu ny suy ra t

xp1

ex

xp01

(0 < x 1)10 x

p01

< ,xp1ex xp11ex, (1 x < ),

1

xp01ex < .

3) Tnh lin tc. T tnh hi t u suy ra hm (p) lin tc trn min xc nhca n.


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4) Cng thc truy hi. Bng cch tch phn tng phn, ta c

(p + 1) =

0

xpexdx = limb

xpex

b

0

+ p

b0

xp1exdx

= p(p).

Nu n l s t nhin, th p dng lin tip cng thc trn, ta c

(p + n) = (n + p 1)(n + p 2) p(p).

Ni ring, (1) = 1, (n + 1) = n!, (1/2) =0

exx

dx = 20

ex2

dx =

.

5) Lin h vi hm Beta. Bng php i binx = ty

,t > 0

, ta c

(p)

tp=

0

yp1etydy.

Thay p bi p + q v t bi t + 1 ta -c

(p + q)

(1 + t)p+q=

0

yp+q1e(1+t)ydy.

Nhn hai v ca ng thc trn vi tp1 ri ly tch phn theo t t 0 n ta-c

(p + q)

0

tp1

(1 + t)p+qdy =

0

0

tp1etyyp+q1eydy

dt.

i bin x =t

1 + t, ta -c B(p,q) =

0

tp1

(1 + t)p+q. Mt khc, c th i th t

tch phn v phi (hy kim chng iu ny nh- bi tp). T

(p + q)B(p,q) =0

0

tp1etyyp+q1etydt

dy

=

0

yp+q1

ey (p)

yp

dy

= (a)0

yq1eydy = (p)(q).

Vy. ta c cng thc

B(p,q) =(p)(q)

(p + q).


18/64


19/64

I I . T c h p h a n h a m s o t r e n a t a p k h a v i

1 . A T A P K H A V I T R O N G Rn

1 . 1 n g c o n g . T a p c o n C Rn c g o i l a n g c o n g t r n l p

Cp(p 1) n e u u m o i x C, t o n t a i l a n c a n m V Rn c u a x , k h o a n g m I R , v a : I Rn

t h u o c l p Cp , (t) = (x1(t), , xn(t)) , s a o c h o : ( 1 ) : I C V l a 1 - 1 . ( 2 ) (t) = (x1(t), , x

n(t)) = 0, v i m o i t I.

K h i o (, I) c g o i l a m o t t h a m s o h o a c u a C t a i x.

st E sx

"!#

V e c t o r (t) g o i l a v e c t o r t i e p x u c c u a C t a i x. T a c o p h n g t r n h t h a m s o c u a n g t h a n g t i e p x u c v i C t a i (t0):

x = (t0) + s(t0), s R

V d u . T r o n g R2 .

a ) n g t r o n c o t h e c h o b i t h a m s o h o a : x = a cos t, y = a sin t, t [0, 2).

b ) T h a m s o h o a : x = a cos t, y = a sin t, z = bt, t (0, H), m o t a n g x o a n .

B a i t a p : V i e t c u t h e p h n g t r n h t i e p t u y e n k h i n = 2 h a y n = 3 . N h a n x e t . i e u k i e n (t) = 0 b a o a m c h o n g c o n g k h o n g c o g o c h a y i e m

l u i . C h a n g h a n , n e u (t) = (t3, t2) t h n g c o n g c o i e m l u i t a i (0, 0), c o n (t) =(t3, |t|3) , t h n g c o n g c o i e m g o c t a i (0, 0) .

1 . 2 M a t c o n g . T a p c o n S Rn c g o i l a m a t c o n g t r n l p

Cp (p 1) n e u u m o i x S, t o n t a i l a n c a n m V Rn c u a x , t a p m U R2 , v a : U Rn t h u o c l p Cp , (u, v) = (x1(u, v), , xn(u, v)) , s a o c h o :

( 1 ) : U S V l a 1 - 1 . ( 2 ) rank (u, v) = 2 , i . e . D1(u, v), D2(u, v) o c l a p t u y e n t n h , (u, v) U.

K h i o (, U) c g o i l a m o t t h a m s o h o a c u a S t a i x.K h i c o n h m o t b i e n u h a y v , c h o c a c

n g c o n g t o a o . C a c v e c t o r D1(u, v) ,

D2(u, v) g o i l a c a c v e c t o r t i e p x u c c u a S t a i (u, v) . T a c o p h n g t r n h t h a m s o c u a m a t p h a n g t i e p x u c v i

St a i

(u0, v0):

x = (u0, v0) + sD1(u0, v0) + tD2(u0, v0), (s, t) R

2


20/64

I I . 1 . a t a p k h a v i t r o n g Rn

. 2 0

s

E

u

Tv

U

E

s

x

E

SV

T r n g h p n = 3, N(u, v) = D1(u, v) D2(u, v) = (A(u, v), B(u, v), C(u, v)) ,l a v e c t o r v u o n g g o c v i S t a i (u, v) . K h i o p h n g t r n h t o n g q u a t c u a m a t p h a n g t i e p x u c v i S t a i (u0, v0) = (x0, y0, z0):

A(u0, v0)(x x0) + B(u0, v0)(y y0) + C(u0, v0)(z z0) = 0

B a i t a p : X a c n h t o a o v e c t o r p h a p q u a c a c a o h a m r i e n g c u a

.

V d u . T r o n g R

3.

a ) T h a m s o h o a m a t c a u :

x = a cos sin , y = a sin sin , z = a cos , (, ) (0, 2) (0, )

b ) T h a m s o h o a m a t x u y e n :

x = (a+b cos )sin , y = (a+b sin )sin , z = b sin , (, ) (0, 2)(0, 2), (0 < b < a)

B a i t a p : V i e t p h n g t r n h m a t p h a n g t i e p x u c v i c a c m a t t r e n .

B a y g i , t a t o n g q u a t h o a c a c k h a i n i e m t r e n .

1 . 3 a t a p . T a p c o n M Rn c g o i l a a t a p

kc h i e u l p

Cp (p 1) n e u u m o i x M, t o n t a i l a n c a n m V Rn c u a x , t a p m U Rk , v a : U Rn t h u o c l p Cp , s a o c h o :

( M 1 ) : U M V l a 1 - 1 . ( M 2 ) rank (u) = k , i . e . D1(u), , Dk(u) o c l a p t u y e n t n h , v i m o i u U.

K h i o (, U) c g o i l a m o t t h a m s o h o a c u a M t a i x.K h i c o n h

k 1 b i e n t r o n g c a c b i e n , c h o c a c n g c o n g t o a o . C a c v e c t o r D1(u), , Dk(u) g o i l a c a c v e c t o r t i e p x u c c u a M t a i (u) . T a c o p h n g t r n h t h a m s o c u a k - p h a n g t i e p x u c v i M t a i (u0):

x = (u0) + t1D1(u0 + + tkDk(u0), (t1, , tk) Rk

1 . 4 C h o a t a p b i h e p h n g t r n h . C h o t a p m V Rn v a c a c h a m l p Cp

F1, , Fm : V R. X e t t a p c h o b i h e p h n g t r n h

M = {x V : F1(x) = = Fm(x) = 0}


21/64


. 2 1

G i a s rank(DF1, , DFm)(x) = m, x M. K h i o M l a a t a p k h a v i , n mc h i e u , l p

Cp.

C h n g m i n h : a t k = n m . K y h i e u x = (x, y) Rk Rm = Rn , v a F = (F1, , Fm) .

V i m o i a M, b a n g p h e p h o a n v t o a o , c o t h e g i a t h i e t det Fy

(a) = 0. T h e o

n h l y h a m a n , l a n c a n V c u a a = (a, b) , t a c o

M V = {(x, y) V : F(x, y) = 0} = {(x, y) V : y = g(x)},

v i g l a h a m l p Cp m o t l a n c a n U c u a a . V a y : U Rn , (x) = (x, g(x)) l a m o t t h a m s o h o a c u a M t a i a.

V d u . T r o n g R

3.

a ) M a t c a u S2

c h o b i p h n g t r n h : F(x , y , z) = x2 + y2 + z2 1 = 0.

D e k i e m t r a F(x , y , z) = (2x, 2y, 2z) = (0, 0, 0) t r e n S2 . V a y S2 l a a t a p k h a v i 2c h i e u ( = m a t c o n g t r n ) .

b ) n g t r o n C c h o b i h e p h n g t r n h s a u l a a t a p 1 c h i e u F1(x , y , z) = x

2 + y2 + z2 1 = 0F2(x , y , z) = x + y + z = 0

N h a n x e t . N e u (, W) l a t h a m s o h o a k h a c c u a M t a i x , t h t o n t a i c a c l a n c a n W, U c u a 1(x), 1(x) t n g n g s a o c h o t r e n W t a c o = h, t r o n g o h = 1 : W U l a v i p h o i , i . e . s o n g a n h v a h1 k h a v i . C h n g m i n h : R o r a n g h = 1 l a s o n g a n h t 1((W)(U)) l e n 1((W)(U)). T a c a n c h n g m i n h h t h u o c l p Cp .D o rank D = k , h o a n v t o a o , c o t h e g i a t h i e t k d o n g a u c u a D(u) l a o c l a p

t u y e n t n h k h i u

t h u o c m o t l a n c a n U

c u a i e m a n g x e t , i . e .

D(1, , k)

D(u1, , uk)= 0

t r e n U .

K y h i e u x = (x, y) Rk Rnk . G o i i : Rk Rk Rnk l a p h e p n h u n g

i(u) = (u, 0) , v a p = Rk Rnk Rk l a p h e p c h i e u p(x, y) = x .

a t (u, y) = ((u), y) . T g i a t h i e t det D =D(1, , k)

D(u1, , uk)= 0. T h e o n h l y

h a m n g c , t o n t a i 1 Cp a p h n g . T a c o h = 1 = ( i)1 = p 1 . C a c h a m t h a n h p h a n l a t h u o c l p Cp , n e n h t h u o c l p Cp .

1 . 5 K h o n g g i a n t i e p x u c . C h oM Rn

l a a t a p k h a v i k

c h i e u v a x0 M.

C h o : (, ) M l a n g c o n g l p C1 t r e n M, (0) = x0 . K h i o (0) c g o i l a

v e c t o r t i e p x u c v i M

t a i x0 . T a p m o i v e c t o r t i e p x u c v i M t a i x0 c g o i l a

k h o n g g i a n t i e p x u c v i M t a i x0 v a k y h i e u Tx0M.

N e u (, U) l a m o t t h a m s o h o a c u a M t a i x0 = (u0) , t h

Tx0M = {v Rn : v = t1D1(u0) + + tkDk(u0), t1, , tk R} = I m D(u0).


22/64


. 2 2

N e u M c h o b i h e p h n g t r n h F1 = = Fm = 0 , t a i l a n c a n x0 , t h

Tx0M = {v Rn : v grad Fi(x0), i = 1, , m}.

V i e t m o t c a c h k h a c Tx0M c h o b i h e p h n g t r n h

v Rn : < grad F1(x0), v >= =< grad Fm(x0), v >= 0

B a i t a p : T m p h n g t r n h k h o n g g i a n t i e p x u c c h o S2 v a C v d u t r e n .

1 . 6 a t a p c o b . T a s e d u n g c a c k y h i e u :

Hk = {x = (x1, , xk) Rk : xk 0} v a g o i l a n a k h o n g g i a n c u a R

k,

Hk = {x Hk : xk = 0} = Rk1 0 v a g o i l a b c u a Hk ,

Hk+ = {x Hk : xk > 0} v a g o i l a p h a t r o n g c u a H

k.

T a p c o n M Rn

c g o i l a a t a p k

c h i e u l p Cp

c o b n e u u m o i x M

, t o n t a i

l a n c a n m V Rn

c u a x

, t a p m U Rk

, v a : U Rn t h u o c l p Cp , s a o c h o :

( M 1 ) : U Hk M V l a 1 - 1 . ( M 2 ) rank (u) = k , v i m o i u U.

K h i o c a c i e m x = (u), u U, c p h a n t h a n h 2 l o a i : i e m t r o n g c u a M , n e u u Hk+ . i e m b c u a M , n e u u Hk .

K y h i e u M = {x M : x l a i e m b c u a M}, v a g o i l a b c u a M .

N h a n x e t . n h n g h a i e m t r o n g v a i e m b i e n k h o n g p h u t h u o c t h a m s o h o a .

s ER

k

Txk

UH k

E

s

xE

MV

M e n h e . C h o t a p m V Rn v a c a c h a m l p Cp , F1, , Fm, Fm+1 : V R. X e t c a c t a p c h o b i h e p h n g t r n h v a b a t p h n g t r n h

M = {x V : F1(x) = = Fm(x) = 0, Fm+1(x) 0}

M = {x V : F1(x) = = Fm(x) = Fm+1(x) = 0}G i a s rank(DF1, , DFm)(x) = m, x M, v a rank(DF1, , DFm+1)(x) =m + 1, x M. K h i o M l a a t a p k h a v i , n m c h i e u , l p Cp , c o b M.

C h n g m i n h : T n g t 1 . 4

V d u . T r o n g R3

h n h c a u o n g B c h o b i b a t p h n g t r n h : x2 + y2 + z2 1, l a a


23/64


. 2 3

t a p 3 c h i e u c o b l a m a t c a u B c h o b i : x2 + y2 + z2 = 1.

M e n h e . C h oM

l a a t a p k h a v i k

c h i e u . K h i o :

( 1 ) M l a a t a p k h a v i k 1 c h i e u k h o n g b , i . e . (M) = .( 2 ) N e u x M, t h TxM l a k h o n g g i a n c o n k 1 c h i e u c u a TxM.

C h n g m i n h : G o i i : Rk1 Rk, i(u1, , uk1) = (u1, , uk1, 0) . K h i o d e t h a y n e u (, U) l a t h a m s o h o a c u a M t a i x v a x M, t h ( i, i1(U)) l a t h a m s o h o a c u a M t a i x . V i t h a m s o h o a o x l a i e m t r o n g c u a M. V a y (M) = .H n n a TxM l a k h o n g g i a n s i n h b i D1(u), , Dk1(u) n e n l a k h o n g g i a n c o n k 1 c h i e u c u a TxM.

1 . 7 n g d u n g v a o b a i t o a n c c t r i e u k i e n .

C h oF = (F1, , Fm) : V Rm , t h u o c l p C

1t r e n t a p m

V Rn.

G o i M = {x V : F1(x) = = Fm(x) = 0}, v a g i a t h i e t rank F

(x) = m, x M.C h o

f : V R , t h u o c l p C1 .B a i t o a n : T m c c t r c u a h a m h a n c h e f|M. N o i c a c h k h a c l a t m c c t r c u a f v i i e u k i e n r a n g b u o c F1 = = Fm = 0.

N h a n x e t . V M

l a a t a p , n e n v i m o i a M

t o n t a i t h a m s o h o a (, U) c u a M t a i a

, v i a = (b).

i e u k i e n c a n . N e u f a t c c t r v i r a n g b u o c F1 = = Fm = 0, t a i a, t h grad f(a) TaM, i . e . t o n t a i 1, , m R, s a o c h o

grad f(a) = 1grad F1(a) + + mgrad Fm(a)

C h n g m i n h : T h e o n h a n x e t t r e n , r o r a n g f|M a t c c t r t a i a t n g n g v i f a t c c t r t a i b .

S u y r a (f )(b) = f(a)(b) = 0. V a y < grad f(a), v >= 0, v Im(b) = TaM,i . e . grad f(a) TaM. D o rank (grad F1(a), , grad Fm(a)) = m = codimTaM,n e n grad f(a) t h u o c k h o n g g i a n s i n h b i grad F1(a), , grad Fm(a).

P h n g p h a p n h a n t h o a L a g r a n g e . T k e t q u a t r e n , e t m i e m n g h i n g c c t r

c u a f v i i e u k i e n F1 = = Fm = 0, t a l a p h a m L a g r a n g e

L(x, ) = f(x) 1F1(x) mFm(x), x V, = (1, , m) Rm

N e u a l a c c t r i e u k i e n , t h t o n t a i Rm , s a o c h o (a, ) l a n g h i e m h e

Lx

(x, ) = 0

F1(x) = 0.

.

.

Fm(x) = 0

V d u . X e t c c t r f(x , y , z) = x + y + z , v i i e u k i e n x2 + y2 = 1, x + z = 1 .

T r c h e t , t a t h a y i e u k i e n r a n g b u o c x a c n h m o t a t a p ( E l l i p E ) .


24/64

I I . 2 T c h p h a n h a m s o t r e n a t a p . 2 4

L a p h a m L a g r a n g e L(x , y , z , 1, 2) = x + y + z 1(x2 + y2 1) 2(x + z 1) .G i a i h e p h n g t r n h

L

x= 1 21x 2 = 0

L

y = 1 21y = 0L

z= 1 2 = 0

x2 + y2 1 = 0x + z 1 = 0

T a c o c a c i e m n g h i n g c c t r l a (0, 1, 1) . D o t a p i e u k i e n c o m p a c t , n e n f p h a i a t m a x , m i n t r e n t a p o . H n n a , c a c i e m c c t r o p h a i l a m o t t r o n g c a c i e m

n g h i n g c c t r . V a y

max f|E= max{f(0, 1, 1) = 1, f(0, 1, 1) = 0} = f(0, 1, 1) = 1 ,min f|E= min{f(0, 1, 1) = 1, f(0, 1, 1) = 0} = f(0, 1, 1) = 0

T r o n g t r n g h p t a p i e u k i e n k h o n g c o m p a c t , t a c o t h e s d u n g k e t q u a s a u :

i e u k i e n u . G i a s f, F1, , Fm t h u o c l p C

2, v a

grad f(a) = 1grad F1(a) + + mgrad Fm(a) , i . e . L

x(a, ) = 0.

a t HxL(x, a) l a H e s s i a n c u a h a m L a g r a n g e L t h e o b i e n x . K h i o N e u HxL(a, )|TaM x a c n h d n g , t h f|M a t c c t i e u t a i a. N e u HxL(a, )|TaM x a c n h a m , t h f|M a t c c a i t a i a . N e u HxL(a, )|TaM k h o n g x a c n h d a u , t h f|M k h o n g a t c c t r t a i a .

C h n g m i n h : V i c a c k y h i e u p h a n t r e n , b a i t o a n t m c c t r c u a f|M t n g n g b a i t o a n t m c c t r c u a f . D o f(a)(b) = 0, t n h a o h a m c a p 2 , t a c o H(f)(a)(h) =Hf(a)((b)h) ( B a i t a p ) . D o Fi = 0, t a c o H(Fi ) = 0 v a t h e o t n h t o a n t r e n H(Fi )(b)(h) =HFi(a)(

(b)(h) .S u y r a

HxL(a, )|TaM = H(f )(b)|TaM.T i e u k i e n u c u a b a i t o a n c c t r a p h n g t a c o k e t q u a . .

V d u . C h o k N v a a R. T m c c t r f(x1, , xn) = xk1 + + xkn , v i r a n g

b u o c x1 + + xn = an.

2 . T C H P H A N H A M S O T R E N A T A P

2 . 1 o d a i , d i e n t c h , t h e t c h t r o n g R3

. T r o n g R3

, c o t r a n g b t c h v o h n g E u c l i d

< , > , n e n c o k h a i n i e m o d a i v a v u o n g g o c .

o d a i v e c t o r T = (xt, yt, zt) : T =

x2t + y2t + z

2t


25/64


D i e n t c h h n h b n h h a n h t a o b i u = (xu, yu, zu), v = (xv, yv, zv):

d t (u, v) = uv = u v

=

u2 < u, v >< v,u > v2

1

2

=

u2v2 | < u, v > |2.

t r o n g o v = v + v l a p h a n t c h : v l a h n h c h i e u v u o n g g o c v l e n u , v u .

C h n g m i n h : T a c o v = u, < v, u >= 0. S u y r a < u, u > < u, v >< v, u > < v, v > =

< u, u > < u, v > + < u, v >

< v, u > < v, v > + < v, v >

=

< u, u > < u, u >

< v, u > < v, u >

+ < u, u > 0< v,u > v2

= u2v2

T o s u y r a c o n g t h c t r e n

T h e t c h k h o i b n h h a n h t a o b i u , v , w R3 :

t t (u , v , w) = d t (u, v)w= | < u v,w > | = | det(u , v , w)|

=

< u, u > < u, v > < u, w >

< v, u > < v, v > < v, w >

< w, u > < w, v > < w, w >

1

2

t r o n g o w = w + w l a p h a n t c h : w l a h n h c h i e u v u o n g g o c w l e n m a t p h a n g s i n h

b i u, v .

w

Eu

B

v

Tw

C h n g m i n h : T n g t c o n g t h c c h o d i e n t c h . ( B a i t a p )

2 . 2 T h e t c h k c h i e u t r o n g Rn . T r o n g Rn c o t r a n g b t c h v o h n g E u c l i d . T h e t c h

k c h i e u c u a h n h b n h h a n h t a o b i v1, , vk Rn

, c n h n g h a q u i n a p t h e o k :

V1(v1) = v1, Vk(v1, , vk) = Vk1(v1, , vk1)vk

t r o n g o vk = vk+ v

k l a p h a n t c h : v

k l a h n h c h i e u v u o n g g o c c u a vk l e n k h o n g g i a n

s i n h b i v1, , vk1 .

C o n g t h c t n h . G o i G(v1, , vk) = (< vi, vj >)1i,jk l a m a t r a n G r a m m . K h i o

Vk(v1, , vk) =

det G(v1, , vk)


26/64


C h n g m i n h : T n g t c o n g t h c c h o d i e n t c h ( B a i t a p ) .

2 . 3 P h a n t o d a i - o d a i n g c o n g . C h o C R3 l a n g c o n g c h o b i t h a m s o h o a

: I R3, (t) = (x(t), y(t), z(t))

T a c a n t n h o d a i l(C) c u a n g c o n g . P h a n h o a c h I t h a n h c a c o a n c o n Ii = [ti, ti + ti]. K h i o l(C) =

i l((Ii)) .

K h i ti b e , t h l((Ii)) l((ti)ti) =

(ti)ti .

n h n g h a p h a n t o d a i

: dl = (t)dt =

x2t + y2t + z

2tdt

n h n g h a o d a i c u a C:

l(C) =

C

dl =

I

x2t + y

2t + z

2tdt

2 . 4 P h a n t d i e n t c h - D i e n t c h m a t . C h oS R3

l a m a t c o n g c h o b i t h a m s o

h o a

: U R3, (u, v) = (x(u, v), y(u, v), z(u, v))

T a c a n t n h d i e n t c h c u a m a t S.

G a s U c o t h e p h a n h o a c h b i c a c h n h c h n h a t b e Ui = [ui, ui+ui][vi, vi+vi] .K h i o d t (S) =

i d t ((Ui)).

K h i ui, vi b e , t h d t ((Ui)) d t (D1(ui, vi)ui, D2(ui, vi)vi) . n h n g h a

p h a n t d i e n t c h :

dS = d t (D1, D2)dudv =

EG F2dudv,

t r o n g o

E = D12 = xu2

+ yu2

+ zu2

G = D22 = xv2 + yv

2 + zv2

F = < D1, D2 > = xuxv + y

uyv + z

uzv

K h i o n h n g h a d i e n t c h c u a S

:

d t (S) =

S

dS =

U

EG F2dudv

2 . 5 P h a n t t h e t c h - T h e t c h h n h k h o i . C h o H l a h n h k h o i c h o b i t h a m s o h o a

: A R3, (u , v , w) = (x(u , v , w), y(u , v , w), z(u , v , w))

e t n h t h e t c h H, b a n g l a p l u a n t n g t n h c a c p h a n t r e n , t a c o c a c n h n g h a :

P h a n t t h e t c h :

dV = t t (D1, D2, D3)dudvdw = | det J|dudvdw

T h e t c h H: V(H) =HdV =

A | det J|dudvdw .

B a y g i t a t o n g q u a t h o a c a c k h a i n i e m t r e n .


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2 . 6 P h a n t t h e t c h t r e n a t a p . C h o M Rn l a a t a p k h a v i k c h i e u . P h a n t t h e t c h t r e n M l a a n h x a

dV : M x dV(x) = t h e t c h k c h i e u h a n c h e t r e n TxM.

G i a s

(, U)l a m o t t h a m s o h o a c u a

Mt a i

x = (u1, , uk). K h i o

dV(x)(D1(x)u1, , Dk(x)uk) = Vk(D1(x), , Dk(x))u1 uk

V a y n e u a t G = (< Di, Dj >)1i,jk , t h q u a t h a m s o h o a

dV =

det G du1 duk

2 . 6 T c h p h a n h a m t r e n a t a p . C h o f : M R l a h a m t r e n a t a p k h a v i k c h i e u .

S a u a y t a x a y d n g t c h p h a n c u a f

t r e n M

( c o n g o i l a t c h p h a n l o a i 1 )

M

f dV

N e u

M = (U)v i

(, U)l a t h a m s o h o a , t h n h n g h a

Mf dV =

U

f

det G, t r o n g o G = (< Di, Dj >)1i,jk.

K h i k = 1 t c h p h a n t r e n g o i l a t c h p h a n n g v a k y h i e u

M

f dl .

K h i k = 2 t c h p h a n t r e n g o i l a t c h p h a n m a t v a k y h i e u

M

f dS.

T r n g h p t o n g q u a t , k h i M c h o b i n h i e u t h a m s o h o a , n g i t a d u n g k y t h u a t p h a n

h o a c h n v s a u a y e d a n c a c t c h p h a n t r e n t n g t h a m s o h o a .

C h o O = {(i, Ui) : i I} l a h o c a c t h a m s o h o a M. H o = {i : i I} g o i l a p h a n h o a c h n v c u a M p h u h p v i h o O n e u u c a c i e u s a u t h o a v i m o i i I:

( P 1 ) i : M [0, 1] l i e n t u c . ( P 2 ) s u p p

i = {x M : (x) = 0} l a t a p c o m p a c t . ( P 3 ) s u p p

i i(Ui) .( P 4 ) M o i x M, t o n t a i l a n c a n V c u a x , s a o c h o c h c o h u h a n c h s o i I

i = 0 t r e n V .( P 5 )

iIi(x) = 1, x M.

T n h c h a t ( P 4 ) g o i l a t n h h u h a n a p h n g c u a h o {supp i, i I}. D o t n h c h a t n a y t o n g ( P 5 ) l a t o n g h u h a n v i m o i x .

n h l y . V i m o i h o O c a c t h a m s o h o a c u a a t a p M, t o n t a i h o p h a n h o a c h n v p h u h p v i O .

C h n g m i n h : G a s M c o m p a c t , k c h i e u . V i m o i x M, t o n t a i (x, Ux) O l a t h a m s o h o a t a i

x. G o i

Bx Ux l a m o t h n h c a u t a n 1x (x) . G a s Bx = B(a, r) .

H a m gx : Rk R

c n h n g h a n h s a u

gx(u) =

e 1

r2ua2 , n e u u a r

0 , n e u u a > r.


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K h i o gx C

( b a i t a p ) . a t gx(y) = gx(1x (y)) , n e u y x(Ux) , v a gx(y) = 0,

n e u y x(Ux). K h i o gx l i e n t u c t r e n M. V M c o m p a c t , t o n t a i h u h a n

x1, , xN M, s a o c h o x1(Bx1), xN(BxN) p h u M a t i =gxi

gx1 + + gxN.

K h i o h o {i : i = 1, N} l a p h a n h o a c h n v c a n t m . K h i M k h o n g c o m p a c t , t o n t a i h o e m c c a c t a p x(Bx), h u h a n a p h n g p h u M. L a p l u a n t n g t n h t r e n c o t h e x a y d n g p h a n h o a c h n v t r o n g t r n g h p

n a y .

G a s a t a p M c t h a m s o h o a b i h o O = {(i, Ui) : i I} . T h e o n h l y t r e n t a c o h o = {i : i I} l a p h a n h o a c h n v c u a M p h u h p v i O . n h n g h a

Mf dV =

iI

i(Ui)

if dV (=iI

Ui

if i

det Gi).

v i g a t h i e t v e p h a i t o n t a i . C h a n g h a n , k h i M c o m p a c t v a f l i e n t u c .

N h a n x e t . n h n g h a t r e n k h o n g p h u t h u o c h o t h a m s o v a p h a n h o a c h n v .

C h n g m i n h : K h i h a i t h a m s o h o a c u a M

t h o a (U) = (W). K h i o = h,

v i h l a v i p h o i . D e k i e m t r a c a c m a t r a n G r a m m q u a n h e v i n h a u t h e o c o n g t h c

G(w) =tJh(w)G(h(w))J h(w). T h e o c o n g t h c o i b i e n , t a c o

Uf

det G =

W

f h| det Jh|

det G h

=

W

f

det tJhG h det Jh =W

f

det G.

V a y n h n g h a k h o n g p h u t h u o c t h a m s o h o a .

N e u = {j : j J} l a m o t p h a n h o a c h n v k h a c c u a M. K h i o j

M

jf =j

M

(i

i)jf =

i,j

M

ijf =

i,j

M

jif =i

M

(j

j)if.

V a y n h n g h a c u n g k h o n g p h u t h u o c p h a n h o a c h n v .

N h a c l a i c a c c o n g t h c t n h :

K h i : I Rn, (t) = (x1(t), , xn(t)) l a t h a m s o h o a n g c o n g C. T a c o C

f dl =

I

f =I

f((t))

(x1)2(t) + + (xn)

2(t)dt.

K h i

: U R3

, (u, v) = (x(u, v), y(u, v), z(u, v))l a t h a m s o h o a m a t

S. T a c o

Sf dS =

U

f

EG F2,

t r o n g o

E = D12 = xu2 + yu

2 + zu2

G = D22 = xv2 + yv

2 + zv2

F = < D1, D2 > = xuxv + y

uyv + z

uzv


29/64


V d u .

a ) o d a i n g x o a n C

:x = a cos t, y = a sin t, z = bt,t [0, h] , l a

C

dl =

h0

a2 sin2 t + a2 cos2 t + b2dt = h

a2 + b2

b ) e t n h d i e n t c h m a t c a u b a n k n h R , t r c h e t t h a m s o h o a , c h a n g h a n

(, ) = (R cos sin , R sin sin , R cos ), (, ) U = (0, 2) (0, )

K h i o c a c v e c t o r t i e p x u c c u a c a c n g t o a o :

D1(, ) = (R sin sin , R cos sin , 0)D2(, ) = (R cos cos , R sin cos , R sin ).

S u y r a E = R2 sin2 , F = 0, G = R2 .

D i e n t c h m a t c a u l a

S

dS = UEG F2dd =

2

0

0

R2 sin dd = 4R2

c ) e t n h t h e t c h h n h c a u b a n k n h R , c o t h e d u n g t h a m s o h o a

(r,,) = (r cos sin , r sin sin , r cos ), (r,,) U = (0, R) (0, 2) (0, )

K h i o

D1(r,,) = (cos sin , sin sin , cos )D2(r,,) = (r sin sin , r cos sin , 0)D3(r,,) = (r cos cos , r sin cos , r sin ) .

T h e t c h h n h c a u l a

B(0,R) dV =

Udet(< Di, Dj >)drdd

=

R0

20

0

1 0 00 r2 sin2 00 0 r2

drdd =4

3R3


30/64


31/64

I I I . D a n g v i p h a n

K h i t n h t c h p h a n t r e n a t a p t a c a n m o t o i t n g b a t b i e n v i p h e p t h a m s o h o a .

V d u n g i a n n h a t l a k h i t n h t c h p h a n t r e n R, t h e o c o n g t h c o i b i e n t a c o ba

f(x)dx =

f((t))(t)dt

t r o n g o l a v i p h o i t (, ) l e n (a, b).N g i t a a v a o k h a i n i e m d a n g v i p h a n b a c 1 : = f(x)dxv a p h e p o i b i e n : = f((t))(t)dt .K h i o c o n g t h c t r e n c o t h e v i e t l a i l a

b

a

=

N g o a i r a d a n g v i p h a n c u n g l a k h a i n i e m t h c h h p e t c h p h a n t r n g v e c t o r t r e n

a t a p s e c e c a p e n c h n g s a u .

C h n g n a y x e t e n c a c d a n g v i p h a n v a c a c p h e p t o a n t r e n c h u n g .

1 . D A N G k - T U Y E N T N H P H A N O I X N G .

1 . 1 . n h n g h a . C h o V l a k h o n g g i a n v e c t o r t r e n R. M o t d a n g k - t u y e n t n h p h a n o i

x n g t r e n V l a m o t a n h x a

: V V k l a n R

t h o a c a c i e u k i e n s a u v i m o i v1, , vk V , R v a 1 i < j k :( A 1 ) (v1, , vi + v

i, , vk) = (v1, , vi, , vk) + (v1, , v

i, , vk).

( A 2 ) (v1, , vi, , vk) = (v1, , vi, , vk).( A 3 ) (v1, , vi, , vj , , vk) = (v1, , vj, , vi, , vk).

N h a n x e t . i e u k i e n ( A 1 ) ( A 2 ) c o n g h a l a t u y e n t n h t h e o t n g b i e n

N h a n x e t . i e u k i e n ( A 3 ) t n g n g v i m o t t r o n g c a c i e u k i e n s a u :

( A 3 ) (v1, , vi , vj, , vk) = 0, n e u vi = vj , v i m o i i = j .( A 3 ) (v(1), , v(k)) = ()(v1, , vk) ,v i m o i h o a n v c u a {1, , k}, () l a k y s o ( = s i g n i


32/64

I I I . 1 . D a n g k - t u y e n t n h p h a n o i x n g . 3 2

( A 3 ) ( A 3 ) : A p d u n g ( A 3 ) v i l a c h u y e n v i v a j .

V d u . C h oF

l a m o t v e c t o r t r o n g R

3. K h i o :

a ) WF(v) =< F, v >, v R3

, l a d a n g 1- t u y e n t n h t r e n R3 ( c o n g c u a F d o c t h e o v )b ) F(v1, v2) =< F, v1 v2 >, v1, v2 R

3, l a d a n g 2- t u y e n t n h p h a n o i x n g t r e n

R3 ( t h o n g l n g c u a F q u a h n h b n h h a n h t a o b i v1, v2 )

c ) n h t h c l a d a n g n- t u y e n t n h p h a n o i x n g t r e n Rn . G i a t r det(v1, , vn) l a t h e t c h c o h n g c u a b n h h a n h t a o b i

v1, , vn Rn

.

1 . 2 K h o n g g i a n v e c t o r k(V) . K y h i e u k(V) l a t a p m o i d a n g k - t u y e n t n h p h a n o i x n g t r e n V . T r e n t a p n a y t a n h n g h a 2 p h e p t o a n :

( + )(v1, , vk) = (v1, , vk) + (v1, , vk)()(v1, , vk) = (v1, , vk) , v i ,

k(V), R .D e t h a y (k(V), +, ) l a k h o n g g i a n v e c t o r t r e n R.

V d u .

a ) 1(V) c h n h l a k h o n g g i a n o i n g a u c u a V , i . e . 1(V) = V = L(V, R) .b ) C h o 1, 2 V

. n h n g h a d a n g 2- t u y e n t n h : 1 2 : V V R,

(1 2)(v1, v2) = 1(v1)2(v2) 2(v1)1(v2) = det

1(v1) 1(v2)2(v1) 2(v2)

V e m a t h n h h o c g i a t r t r e n c h n h l a d i e n t c h c o h n g c u a h n h b n h h a n h t r o n g R

2

t a o b i (v1), (v2) , t r o n g o = (1, 2) : V R

2.

1 . 3 T c h n g o a i . C h o1, , k V

. T c h n g o a i c u a c a c d a n g t r e n l a m o t

k- d a n g

1 k k(V), c n h n g h a :

1k(v1, , vk) =

()(1)(v1) (k)(vk) = det(i(vj)), v1, , vk V,

i . e . 1 k =

()(1) (k) .

T n h c h a t . V i m o i 1, , k, i

1(V), , R v a i = 1, , k ,( 1 )

1 (i+i) k = 1 i k +1

i k.

( 2 )(1) (k) = ()1 k, v i l a h o a n v .

C h n g m i n h : S u y t t n h c h a t c u a n h t h c .

1 . 4 B i e u d i e n d a n g k - t u y e n t n h p h a n o i x n g . C h o V l a m o t k h o n g g i a n v e c t o r

t r e n R

. G i a s 1, , n l a m o t c s c u a V

. K h i o m o t c s c u a k(V) l a h e {i1 ik , 1 i1 < < ik n}.

N h v a y m o i k(V) c o b i e u d i e n d u y n h a t d i d a n g

=

1i1


33/64

I I I . 2 D a n g v i p h a n . 3 3

v a dimk(V) = Ckn =n!

(n k)!k!.

C h n g m i n h : G o i {1, , n} l a c s o i n g a u c u a {e1, , en}, i . e . i(ej) = ij( d e l t a K r o n e c k e r ) .

C h o k(V). C h o v1, , vk V . K h i o

v1 =i1

i1(v1)ei1, , vk =ik

ik(vk)eik ,

(v1, , vk) = (i1

i1(v1)ei1, ,ik

ik(vk)eik)

=

i1, ,ik

i1(v1) ik(vk)(ei1, , eik)

=

i1


34/64

I I I . 2 D a n g v i p h a n . 3 4

C h o f : U R l a h a m l p Cp+1 . K h i o v i m o i x U, f(x) : Rn R l a d a n g t u y e n t n h . T a n h n g h a v i p h a n c u a

fl a 1- d a n g v i p h a n

df : U 1(Rn), x df(x) = f(x).

X e t h a m t o a o t h i xi : Rn R, (x1, , xn) xi . T a c o

dxi(x)(v) = xi(x)v = vi, v = (v1, , vn) R

n.

V a y

df(x)(v) = f(x)v =f

x1(x)v1 + +

f

xn(x)vn

=f

x1(x)dx1(x)(v) + +

f

xn(x)dxn(x)(v).

H a y l a df =ni=1

f

xidxi .

2 . 2 B i e u d i e n d a n g v i p h a n . T c h n g o a i c u a c a c 1- v i p h a n 1, , k 1(U) :

(1 k)(x) = 1(x) k(x), x U,

l a m o t k - d a n g v i p h a n t r e n U. D o c a c 1- d a n g dx1, , dxn l a m o t c s c u a 1(U) ,

n e n c a c k - d a n g v i p h a n t r e n U c o b i e u d i e n d u y n h a t d i d a n g

=

1i1


35/64

I I I . 2 D a n g v i p h a n . 3 5

V d u .

a ) C h o : R R2, (t) = (x = cos t, y = sin t) v a (x, y) = xdy ydx .

K h i o (t) = cos td(sin t) sin td(cos t) = dt .

b ) C h o : R2 R2, (r, ) = (x = r cos , y = r sin ) v (x, y) = dx dy . K h i

o

(r, ) = d(r cos ) d(r sin )= (cos dr r sin d) (sin dr + r cos d)= rdr d (do dr dr = d d = 0, d dr = dr d).

T n h c h a t .

( 1 ) (1 + 2) = (1) +

(2), 1, 2 k(V) .

( 2 ) (1 k) = (1)

(k), 1, , k 1(V).

( 3 ) (dxi) = di =m

j=1

i

ujduj .

C h n g m i n h : X e m n h b a i t a p .

B a i t a p : C h o : Rn Rn k h a v i . C h n g m i n h

(f(x)dx1 dxn) = f((u))det (u)du1 dun.

N h a n x e t . C o t h e n h n g h a t o a n t o i b i e n k h o n g q u a b i e u d i e n t r e n t o a o ( i . e .

n h n g h a k h o n g p h u t h u o c h e t o a o ) n h s a u

(u)(v1, , vk) = ((u))((u)v1, ,

(u)vk).

2 . 4 T o a n t v i p h a n . V i m o i k N , t o a n t v i p h a n c n h n g h a n h s a u

d : k(U) k+1(U),

d(

1i1


36/64

I I I . 2 D a n g v i p h a n . 3 6

K h i o

d = (d sin xy) dx + d(ex2+y) dy + d(arctgx) dz

= (y cos xydx + x cos xydy) dx + (2xex2+ydx + ex

2+ydy) dy +1

1 + x2dx dz

= (2xex2+y

x cos xy)dx dy

1

1 + x2 dz dx.

B a i t a p : T n h d (P(x , y , z)dx + Q(x , y , z)dy + R(x , y , z)dz),

v a d (P(x , y , z)dx dz + Q(x , y , z)dz dx + Q(x , y , z)dx dy) .

N h a n x e t . N e u k(Rn) v i k n, t h d = 0.

T n h c h a t .

( 1 )d(1 + 2) = d1 + d2, 1, 2 k(U).

( 2 )d(1 2) = d1 2 1 d2, 1, 2

1(U)..( 3 )

d(d) = 0 , i . e . d d = 0 .( 4 ) d() = (d) , i . e . d = d .

C h n g m i n h : ( 1 ) l a r o r a n g . D o ( 1 ) t a c h c a n c h n g m i n h ( 2 ) k h i 1 = adxi, 2 = bdxj .

T a c o

d(1 2) = d(adxi bdxj) = d(abdxi dxj)= d(ab) dxi dxj = (bda + adb) dxi dxj= bda dxi dxj + adb dxi dxj = (da dxi) bdxj adxi db dxj= d1 2 1 2.

T r c k h i c h n g m i n h ( 3 ) e n g a n g o n t a k y h i e u : dxI = dxi1 dxik ,v i I = (i1, , ik) l a m o t b o k c h s o t h u o c {1, n}.D o ( 1 ) c h c a n c h n g m i n h ( 3 ) k h i = aIdxI. T a c o

d(d) = d(daI dxI) = d

i

aI

xidxi dxI

=i

d

aI

xi

dxi dxI =

i

j

2aI

xjxidxj

dxi dxI

= i

j

2aI

xixjdxi dxj dxI ( d o dxi dxj = dxj dxi)

= d(d) (t h a y o i v a i t r o i, j)

V a y 2d(d) = 0, s u y r a ( 3 ) . C u n g v a y c h c a n k i e m t r a ( 4 ) k h i

= aIdxI k(V). T a c o

d() = d(aI dI) = d(aI ) dI.(d) = (daI dxI) =

(daI) (dyI) =

(daI) dI.

C a n c h n g m i n h d(aI ) = (daI). a n g t h c u n g l a d o :

(daI) =

j

aI

xjdxj

=

j

aI

xjdj =

j

aI

xj(i

j

uidui) = d(aI).


37/64

I I I . 3 B o e P o i n c a r e 3 7

V a y c a c t n h c h a t t r e n a c c h n g m i n h .

N h a n x e t . D o ( 4 ) t o a n t d

k h o n g p h u t h u o c h e t o a o .

3 . B O E P O I N C A R E

3 . 1 D a n g o v a d a n g k h p . C h o d a n g v i p h a n k(U). g o i l a o n g t r e n U n e u u d = 0 t r e n U. g o i l a k h p t r e n U n e u u t o n t a i k1(U) s a o c h o = d .

N h a n x e t . N e u

k h p , t h

o n g v d(d) = 0 .

V d u s a u c h r a d a n g o n g n h n g k h o n g k h p : (x, y) =

ydx xdy

x2 + y2 1(R2 \ 0) .

D a n g l a o n g , v d =x2 y2

(x2 + y2)2dy dx

y2 x2

(x2 + y2)2dx dy = 0.

N h n g k h o n g k h p . T h a t v a y , g i a s t o n t a i h a m f 0(R2 \ 0) , = df.G o i (t) = (sin t, cos t). K h i o

= (df) = d(f) = d(f ) = (f )dt.

M a t k h a c =

cos td(sin t) sin td(cos t)

sin2 t + cos2 t= dt . V a y (f )(t) 1.

S u y r a f (t) = t+ c o n s t . i e u n a y v o l y v f l a h a m c o c h u k y 2 .

K h i m o t d a n g P f a f f = a1dx1 + + andxn 1(U), t o n t a i h a m f 0(U) t h o a

df = , t h f c g o i l a m o t t c h p h a n a u c u a .N o i m o t c a c h k h a c

ft h o a h e p h n g t r n h v i p h a n a o h a m r i e n g c a p m o t

f

x1= a1, ,

f

xn= an.

V a y n e u

c o t c h p h a n a u ( = k h a t c h = k h p ) , t h d = 0, i . e . c a c h a m a1, , an

t h o a h e t h c

aj

xi=

ai

xjv i m o i i, j = 1, , n.

T n h c h a t h n h h o c c u a t a p n h i e u k h i q u y e t n h b a i t o a n g i a i t c h . M o t d a n g o n g

c u n g l a k h p t r e n U

, k h i t a p U

c o t n h c h a t h n h h o c s a u :

3 . 2 T a p c o r u t c . T a p c o n

Ut r o n g

Rng o i l a c o r u t c v e m o t i e m

x0 Un e u u t o n t a i m o t a n h x a l p

C1

h : U [0, 1] U, (x, t) h(x, t)

s a o c h o : h(x, 0) = x0 v a h(x, 1) = x, x U.

V d u . S a u a y l a m o t s o l p t a p c o r u t q u a n t r o n g :


38/64


T a p l o i : t a p U g o i l a l o i n e u u x, y U o a n [x, y] = {x + t(y x) : t [0, 1]} U.C h a n g h a n

Rn, h n h c a u , h n h h o p l a c a c t a p l o i .

T a p h n h s a o : t a p U

g o i l a h n h s a o n e u u x0 U : x U, [x0, x] U.T r o n g c a c v d u t r e n a n h x a

h(x, t) = x0 + t(x x0) t h o a n h n g h a 3 . 2 . B a i t a p : R o r a n g l a t a p l o i l a t a p h n h s a o . T m v d u t a p h n h s a o k h o n g l o i , t a p c o

r u t c k h o n g h n h s a o .

3 . 3 n h l y ( B o e P o i n c a r e ) . G i a s U

l a t a p m t r o n g Rn

, v a U

c o r u t c . K h i

o m o i d a n g o n g t r e n U

l a k h p , i . e .

k(U), d = 0 k1(U), = d.

C h n g m i n h : G o i Jt : U U [0, 1], Jt(x) = (x, t) . C h o k = 1, 2, . T r c h e t t a x a y d n g a n h x a t u y e n t n h

K : k(U [0, 1]) k1(U), t h o a

( ) Kd + dK = J1 J0

M o i p h a n t c u a k(U [0, 1]) l a t o n g c a c d a n g c o m o t t r o n g h a i d a n g s a u : ( 1 )

a(x, t)dxI h a y ( 2 ) b(x, t)dt dxJ, v i I = (i1, , ik), J = (j1, , jk1).V v a y c h c a n n h n g h a

Kc h o t n g d a n g c o d a n g t r e n . T a n h n g h a

K(a(x, t)dxI) = 0

K(b(x, t)dt dxJ) =

10

b(x, t)dt

dxJ

K i e m t r a i e u k i e n ( ) v i d a n g ( 1 ) :

(Kd + dK)(adxI) = K(da dxI) + d(0) = (10

a

tdt)dxI

= (a(x, 1) a(x, 0)dxI = (J1 J

0 )(adxI).

K i e m t r a i e u k i e n ( ) v i d a n g ( 2 ) :

(Kd + dK)(bdt dxJ) = K(db dt dxJ) + d((10

bdt) dxJ)

= K(i

b

xidxi dt dxJ) + d((

10

bdt) dxJ)

= 10

(i

b

xi)dt dxi dxJ + d((

10

bdt) dxJ)

= d((

10

bdt) dxJ) + d((

10

bdt) dxJ) = 0.

(J1 J0 )(bdt dxJ) = b(x, 1)d(1) dxJ b(x, 0)d(0) dxJ = 0.

B a y g i c h o h : U [0, 1] U l a a n h x a c o r u t v e x0 . G i a s k(U) o n g , i . e . d = 0. T a c h n g m i n h = Kh l a (k 1)- d a n g t h o a d = .D o ( ) t a c o

(Kd + dK)h = (J1 J0 )h

.

Kdh + dKh = (h J1) (h J0)

.

Khd + dKh = (idU) (x0)

.

0 + dKh = + 0.


39/64


V a y = Kh l a d a n g c a n t m .

H e q u a . N e u U

l a t a p m c o r u t c , 1, 2 k(U), v a d1 = d2 , t h t o n t a i

k1 s a o c h o d = 1 2 .

V d u . T a p R2 \ 0 l a k h o n g c o r u t c v t o n t a i d a n g v i p h a n o n g m a k h o n g k h p

t r e n o ( x e m v d u 3 . 1 ) .

N h a n x e t . T h e q u a t r e n , t a t h a y t h o a b o e P o i n c a r e l a k h o n g d u y n h a t .

C o t h e d a v a o c h n g m i n h c u a n h l y e x a y d n g e d = : = Kh .

V d u . C h o = (x2 2yz)dx + (y2 2zx)dy + (z2 2xy)dz 1(R3).

D e k i e m t r a d = 0. e t m f s a o c h o df = , n h s a u :

C a c h 1 : V R

3l a t a p c o r u t v e 0 v i h(x , y , z , t) = (tx,ty,tz) . T h e o n h n g h a c u a

c a c t o a n t , t a c o :

h

= t2(x2 2yz)(xdt + tdx) + t2(y2 2zx)(ydt + tdy) + t2(z2 2xy)(zdt + tdz).

Kh =

10

t2(x2 2yz)xdt +

10

t2(y2 2zx)ydt +

10

t2(z2 2xy)zdt.

S u y r a f = Kh =1

3(x3 + y3 + z3 6xyz) l a m o t t c h p h a n a u c u a , i . e . df = .

C a c h 2 : H a m f

t h o a df = , c o t h e v i e t l a i

(1)f

x= x2 2yz

(2)f

y= y2 2zx

(3) fz

= z2 2xy

e t m f, t a l a n l t t c h p h a n t h e o t n g b i e n :

T (1) s u y r a f =x3

3 2xyz + (y, z)

T (2) s u y r a

y= y2 . V a y =

y3

3+ (z).

T (3) s u y r a

z= z2 . V a y =

z3

3+ c o n s t .

S u y r a f =1

3(x3 + y3 + z3) 2xyz+ c o n s t

( C a c h 2 c o t h e l a m c h o c a c m i e n h n h h o p ) .


40/64


41/64

I V . T c h p h a n d a n g v i p h a n

1 . N H H N G

1 . 1 T r n g v e c t o r . C h oM Rn . M o t t r n g v e c t o r t r e n M l a a n h x a

F : M Rn, F(x) = (F1(x), , Fn(x))

V e m a t h n h h o c x e m t r n g v e c t o r n h h o v e c t o r F(x) c o i e m g o c a t t a i x .

1 . 2 n h h n g n g c o n g . n g c o n g t r n C R3 , g o i l a n h h n g

n e u u

: C R3 l a t r n g v e c t o r l i e n t u c v a t i e p x u c v i C, i . e . (x) t i e p x u c v i C t a i

x, v i m o i

x C.

X

'rr

rrr

t(x)

x

C

V d u . n g t r o n n v c o t h e t h a m s o h o a b i (t) = (cos t, sin t), t (0, 2).

K h i o t r n g v e c t o r t i e p x u c (t) = ( sin t, cos t) x a c n h h n g n g c c h i e u k i m o n g h o .

1 . 3 n h h n g m a t . C h oS R3 l a m a t c o n g t r n . T a n o i S l a n h h n g c

n e u u t o n t a i t r n g v e c t o r p h a p l i e n t u c t r e n S

, i . e . t o n t a i N : S R3 , l i e n t u c v a

N(x) TxS, x S.K h i o S g o i l a

n h h n g p h a p N.

sx

N(x)

fffffw

E

S


42/64

I V . 1 . n h h n g . 4 2

V d u .

a ) M a t c a u l a n h h n g c v a c o t h e c h o n m o t t r o n g h a i h n g : h n g p h a p t r o n g

h a y h n g p h a p n g o a i . C u t h e k h i t h a m s o h o a m a t c a u b i

(, ) = (cos sin , sin sin , cos ), (, ) (0, 2) (0, ).

V i t h a m s o h o a o , c a c v e c t o r t i e p x u c v i c a c n g t o a o l a

= ( sin sin , cos sin , 0),

= ( cos cos , sin cos , sin )

D e k i e m t r a h n g p h a p N =

l a h n g p h a p t r o n g .

b ) L a M

o b i u s c h o t a m o t v d u v e m a t k h o n g n h h n g c .

1 . 4 n h h n g k h o n g g i a n v e c t o r .

D a v a o t r c q u a n : t r e n R c o t h e n h h a i h n g ( d n g n e u c u n g h n g v i c h i e u t a n g , a m n e u n g c l a i ) . T r o n g R2 c o t h e n h h a i h n g ( t h u a n h a y n g c c h i e u k i m o n g h o ) . T a c o n h n g h a s a u .

C h o V l a k h o n g g i a n v e c t o r k c h i e u t r e n R. T r o n g a i s o t u y e n t n h t a a b i e t l a n e u (v1, , vk) v a (w1, , wk) l a c a c c s c u a V , t h t o n t a i m a t r a n c h u y e n c s P = (pij )kk s a o c h o wj =

ipij vi .

T a n o i (v1, , vk) v a (w1, , wk) c u n g h n g n e u u det P > 0,(v1, , vk) v a (w1, , wk) n g c h n g n e u u det P < 0.

N h v a y t r e n t a p c a c c s c u a V c c h i a t h a n h h a i l p t n g n g , m o i l p

g o m c a c c s c u n g h n g v i n h a u . L p c u n h h n g v i (v1, , vk) k y h i e u l a [v1, , vk], l p c a c c s n g c h n g k y h i e u l a [v1, , vk] .K h o n g g i a n V g o i l a

a n h h n g n e u t a c h o n m o t h n g = [v1, , vk] .

V d u . T r o n g Rk c s c h n h t a c x a c n h h n g c h n h t a c . T h e o n g o n n g t r c q u a n , h n g c h n h t a c t r o n g R l a h n g d n g , h n g c h n h t a c t r o n g R2 l a h n g n g c c h i e u k i m o n g h o , c o n h n g c h n h t a c t r o n g R3 l a h n g t a m d i e n t h u a n .

E

e

E

e

T

e

$'E

e

T

e

e

'

H n g c h n h t a c c u a R1, R2, R3

1 . 5 n h h n g a t a p . C h o M Rn l a a t a p k h a v i k c h i e u . M o t h o h n g = {x : x l a m o t h n g t r e n TxM, x M} g o i l a t n g t h c h n e u u c h u n g b i e n o i m o t c a c h l i e n t u c t h e o n g h a s a u : v i m o i a M, t o n t a i t h a m s o h o a (, U) t a i a s a o c h o [D1(u), , Dk(u)] = (u) , v i m o i u U.


43/64

I V . 1 . n h h n g . 4 3

M g o i l a n h h n g c

n e u u t o n t a i m o t h o h n g t n g t h c h t r e n M.

Mg o i l a n h h n g

n e u u

M n h h n g c v a h o h n g t n g t h c h

c

c h o n . K h i o m o t t h a m s o h o a n h t r e n g o i l a t h a m s o h o a x a c n h h n g

.

N h a n x e t . o i v i m a t c o n g t r o n g R3 , v i e c x a c n h h n g n h n h n g h a t r e n t n g n g v i v i e c x a c n h t r n g v e c t o r p h a p l i e n t u c . T a c o N = D1 D2 l a t r n g p h a p v e c t o r .

1 . 6 . H n g c a m s i n h t r e n b .

M e n h e . C h o M l a a t a p k h a v i c o b M. N e u M n h h n g c , t h M c u n g

n h h n g c .

C h n g m i n h : G a s O l a h o t h a m s o h o a c u a M x a c n h h n g .V i m o i (, U) O , g o i i : Rk1 Rk, i(u1, , uk1) = (u1, , uk1, 0) . K h i o h o

{( i, i1(U)) : (, U) O , U

H = } l a h o t h a m s o h o a M.V i m o i

x M, v a (, U) O l a h o t h a m s o h o a t a i x , n h n g h a

x = [D1(u), , Dk1(u)], x = (u).

T a s e c h n g m i n h x k h o n g p h u t h u o c t h a m s o h o a (, U) O , v a d o v a y h o M,

= {x : x = (u) M , (, U) O } l a m o t h o h n g t n g t h c h t r e n M.N e u (, U), (, W) O l a c a c t h a m s o h o a t a i x , t h = h v i det h > 0. T o a o t h k c u a h t h o a :

hk(w1, , wk1, 0) = 0, va hk(w1, , wk1, wk) > 0 khi wk > 0.

S u y r a v i w = (w1, , wk1, 0) , d o n g c u o i c u a m a t r a n h

(w) l a

(D1hk(w) = 0 Dk1hk(w) = 0 Dkhk(w) > 0).

D o o det h(w) = det(h i)(w1, , wk1)Dkhk(w) > 0.V a y det(h i)(w1, , wk1) > 0. M a (h i)

(w) c h n h l a m a t r a n c h u y e n c s D1(u), , Dk1(u) s a n g c s D1(w), , Dk1(w) t r o n g k h o n g g i a n TxM(x = (w) = (u)) , n e n

[D1(w), , Dk1(w)] = [D1(u), , Dk1(u)].

D o v a y x c n h n g h a k h o n g p h u t h u o c t h a m s o h o a x a c n h h n g x .

n h n g h a . C h o M l a a t a p n h h n g . K h i o t r e n M t a x a c n h h n g c a m

s i n h n h s a u :

V i m o i x M, g o i (, U) l a t h a m s o h o a t a i x c u a M x a c n h h n g , i . e . x = [D1(u), , Dk(u)] . K h i o n h n g h a

x = (1)k[D1(u), , Dk1(u)].

( D a u (1)k e t h u a n t i e n c h o c o n g t h c S t o k e s s a u n a y )

N h a n x e t . G o i l a t h a m s o h o a n h h n g t a i x = (u). V TxM l a k h o n g g i a n


44/64

I V . T c h p h a n d a n g v i p h a n . 4 4

v e c t o r c o n c u a TxM c o o i c h i e u 1 , n e n v i m o i v TxM \ TxM x a y r a m o t t r o n g h a i t r n g h p :

( 1 )v

h n g v a o t r o n g M

, n e u v (u)(Hk+)

( 2 )v

h n g r a n g o a i M

, n e u n g c l a i t r n g h p ( 1 ) .

V e m a t t r c q u a n , t a n h a n b i e t h n g t r e n M

l a h n g c a m s i n h n h s a u :

C h o v1, , vk1 l a c s TxM. K h i o n e u v TxM l a v e c t o r h n g v a o t r o n g Mv a x a c n h h n g = [v1, , vk1, v], t h h n g c a m s i n h t r e n b l a

x = (1)k[v1, , vk1]

sx

Ev

'

C h a n g h a n , n e u Hk n h h n g c h n h t a c , t h h n g c a m s i n h t r e n Hk = Rk1 0t r u n g v i h n g c h n h t a c t r e n Rk1 n e u k c h a n , v a n g c v i h n g c h n h t a c o n e u k l e .

V d u . T r c q u a n h n n a :

N e u m i e n M t r o n g R2 n h h n g c h n h t a c h a y l a m a t c o n g t r o n g R3 n h h n g p h a p

N, t h h n g c a m s i n h t r e n n g c o n g

Ml a h n g i d o c t h e o o m i e n

p h a t r a i .

N e u

Ml a m i e n t r o n g

R3

n h h n g c h n h t a c , t h h n g c a m s i n h t r e n m a t c o n g

M l a h n g p h a p t u y e n n g o a i .

2 . T C H P H A N D A N G V I P H A N

T r c h e t l a m o t v a i g i y c h o v i e c x a y n g t c h p h a n c u a t r n g v e c t o r h a y c u a

d a n g v i p h a n .

C h oF = (F1, F2, F3) l a m o t t r n g v e c t o r t r o n g R

3.

V i

v R3 l a v e c t o r g o c t a i x , g i a t r WF(x)(v) =< F(x), v >, g o i l a c o n g c u a F(x) d o c t h e o v .T a c o 1 - d a n g v i p h a n t n g n g : WF = F1dx1 + F2dx2 + F3dx3 .C h o C l a m o t n g c o n g n h h n g t r o n g R3 . T a c a n x a y n g t c h p h a n c u a t r n g F d o c t h e o C, h a y l a t c h p h a n c u a d a n g v i p h a n WF t r e n C:

CWF =

C

F1dx1 + F2dx2 + F3dx3.

V i v1, v2 R3 l a c a c v e c t o r g o c t a i x , g i a t r F(x)(v1, v2) =< F(x), v1 v2 >,g o i l a t h o n g l n g c u a F(x) q u a m a t b n h h a n h S t a o b i v1, v2 .T a c o 2 - d a n g v i p h a n t n g n g F = F1dx2 dx3 + F2dx3 dx1 + F3dx1 dx2.


45/64


C h o S l a m a t n h h n g t r o n g R3 . T a c a n k h a i n i e m t c h p h a n c u a t r n g v e c t o r Fq u a m a t

S, h a y l a t c h p h a n c u a d a n g v i p h a n

F t r e n S:S

F =

S

F1dx2 dx3 + F2dx3 dx1 + F3dx1 dx2

2 . 1 n h n g h a . C h o U l a t a p m Rk , v a k(U).K h i o = f(u)du1 duk . n h n g h a

U =

U

f(u)du1 duk =

Uf(u)du1 duk.

n e u t c h p h a n v e p h a i t o n t a i .

2 . 2 T c h p h a n d a n g v i p h a n . C h o M l a a t a p k h a v i k c h i e u n h h n g t r o n g

Rn . C h o k(V), v i V l a t a p m c h a M. S a u a y t a x a y d n g t c h p h a n c u a

d a n g t r e n M ( c o n g o i l a t c h p h a n l o a i 2 )

M

N e u M = (U) v i (, U) l a m o t t h a m s o h o a x a c n h h n g , t h n h n g h a M

=

U

.

T r n g h p t o n g q u a t , k h i M c h o b i m o t h o t h a m s o h o a O = {(i, Ui) : i I} x a c n h h n g , t a d u n g k y t h u a t p h a n h o a c h n v . G o i = {i : i I} l a p h a n h o a c h n v c u a M p h u h p v i O . n h n g h a

M =

iI

i(Ui)

i

=iI

Ui

i (i)

,

v i g i a t h i e t v e p h a i t o n t a i . C h a n g h a n k h i M c o m p a c t v a l i e n t u c .

K h ik = 1, t c h p h a n c o d a n g

M

i

Fidxi , v a g o i l a t c h p h a n n g .

K h ik = 2, t c h p h a n c o d a n g

M

i 0. N e u = f(u)du1 duk , t h h(f(u)du1 duk) = h

= ( h) = .T h e o c o n g t h c o i b i e n , t a c o

U =

U

f =

Wf h det Jh =

W

h(f(u)du1 duk) =

W.

V a y n h n g h a k h o n g p h u t h u o c t h a m s o h o a x a c n h c u n g h n g .

N e u = {j : j J} l a m o t p h a n h o a c h n v k h a c c u a M. K h i o j

M

j =

j

M

(

i

i)

j =i,j

M

i

j =i,j

M

ji =

i

M

(

j

j )i

i

M

i.


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V a y n h n g h a c u n g k h o n g p h u t h u o c p h a n h o a c h n v .

2 . 3 T n h c h a t . C h oM

l a a t a p k

c h i e u n h h n g

t r o n g t a p m V

. K h i o

( 1 )

M

: k(V) R l a t u y e n t n h .

( 2 )

M

= M

, v i k y h i e u M e c h M n h h n g .

C h n g m i n h : ( 1 ) s u y t t n h t u y e n t n h c u a

Ui

v a i .

( 2 ) X e t p h e p o i b i e n h(u1, , uk) = (u1, , uk) . K h i o det h

= 1. N e u (, U) l a t h a m s o h o a x a c n h h n g , t h ( h, h1(U)) l a t h a m s o h o a x a c n h h n g . T o s u y r a v i m o i p h a n h o a c h n v p h u h p v i h o t h a m s o h o a , t a c o

M =

h1(U)

( h) =

(

U) =

M

.

V d u .

a ) C h o C l a n g c o n g t r n , c h o b i t h a m s o h o a : I Rn , n h h n g t h e o c h i e u t a n g c u a t h a m s o . K h i o

C

i

Fidxi =

I

i

Fi di =

I(

i

Fi (t)

i(t))dt.

C h a n g h a n , n e u n g t r o n n v n h h n g n g c c h i e u k i m o n g h o , t h x2+y2=1

ydx xdy

x2 + y2=

20

sin td(cos t) cos td(sin t)

cos2 t + sin2 t=

20

dt = 2.

b ) C h o

Sl a m a t c a u n v n h h n g p h a p t r o n g , t h v i t h a m s o h o a x a c n h

h n g t n g n g , t a c o S

xdy dz =[0,2][0,]

cos sin d(sin sin ) d(cos )

=

[0,2][0,]

cos sin (cos sin d + sin cos d) d( sin d)

=

[0,2][0,]

cos2 sin3 d d =?

2 . 4 Q u a n h e g i a t c h p h a n l o a i 1 v a l o a i 2 .

C h oF = (P , Q, R) l a t r n g v e c t o r l p C1 t r e n m o t t a p m V R3 .

( 1 ) C h o C V l a n g c o n g k n , n h h n g b i t r n g v e c t o r t i e p x u c n v

T = (cos , cos , cos ). K h i o C

P dx + Qdy + Rdz =

C

< F, T > dl =

C

(P cos + Q cos + R cos )dl.

( 2 ) C h oS V

l a m a t t r n , n h h n g b i t r n g p h a p v e c t o r n v N =

(cos , cos , cos ) . K h i o S

P dydz+Qdzdx+Rdxdy =

S< F, N > dS=

S

(P cos +Q cos +R cos )dS.


47/64

I V . 3 C o n g t h c S t o k e s 4 7

C h n g m i n h : N h p h a n g i y a u t i e t , t a c o :

( 1 ) V i m o i v R3 , g o i T l a v e c t o r c h p h n g n v c u a v . K h i o 1 - d a n g

WF(v) =< F, v > , c o B i e u d i e n 1 :

WF = P dx + Qdy + Rdz.B i e u d i e n 2 : WF(v) =< F,T > v =< F, T > dl(v).V a y n e u C l a n g c o n g t r o n g R3 n h h n g b i t r n g v e c t o r t i e p x u c n v T,t h

CWF =

C

< F,T > dl.

T o s u y r a ( 1 ) .

( 2 ) V i v1, v2 R3 , g o i N l a v e c t o r n v c h p h n g v1 v2 . K h i o 2 - d a n g v i p h a n F(v1, v2) =< F, v1 v2 >, c o B i e u d i e n 1 : F = P dy dz + Qdz dx + Rdx dy.B i e u d i e n 2 :

F(v1, v2) =< F, N > v1 v2 =< F,N > dS(v1, v2).V a y n e u

Sl a m a t c o n g n h h n g b i t r n g v e c t o r p h a p n v

N, t h

S

F =

< F,N > dS.

T o s u y r a ( 2 ) .

3 . C O N G T H C S T O K E S

3 . 1 n h l y ( C o n g t h c S t o k e s ) . C h o M l a a t a p k h a v i k c h i e u , n h h n g , c o m p a c t

t r o n g t a p m V Rn , v i b M n h h n g c a m s i n h . K h i o

M d = M , k1(V).C h n g m i n h : G i a s M n h h n g v a l a h n g c a m s i n h t r e n M. C h o

{(i, Ui) : i I} l a t h a m s o h o a n h h n g c u a M. K h o n g g i a m t o n g q u a t , g i a s Ui c h a t r o n g m o t h n h h o p Ai .

G o i i : Rk1 Rk, i (u1, , uk1) = (u1, , uk1, 0) . K h i o h o {(i i , i1(Ui)) :

i I}, v i

I = {i I : Ui Hk = }, l a h o t h a m s o h o a M n h h n g (1)k .N e u {i : i I} l a p h a n h o a c h n v p h u h p v i h o a c h o , t h

Md =

M

d(iI

i) =iI

i(Ui k)

di.

M = M(iI i) = iI i(UiHk) i. e c h o g o n , a t = i, U = Ui, A = Ai = [1, 1] [k, k] . T a c a n c h n g m i n h :

( 1 ) N e u U Hk = , i . e . i I \ I , t h

(U)

d = 0.

( 2 ) N e u U Hk = , i . e . i I , t h

(U k)d = (1)k

(UHk)

.


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G o i =k

j=1

aj(u1, , uk)du1 duj duk k1(U) .K h i o x e m

k1(A) b a n g c a c h a t aj (u) = 0 k h i u U. T a c o

( i ) = ak(u1, , uk1, 0)du1 duk1.

(d) =k

j=1

daj du1 duj duk=

kj=1

(1)j1aj

ujdu1 duk.

o i v i t r n g h p ( 1 ) , t a c o

(U)

d =

U

(d) =

A

kj=1

(1)j1aj

ujdu1 duk

= j l=j [l,l](aj( , j , ) aj( , j , ))du1 duj duk= 0.

( a n g t h c t h b a s u y t c o n g t h c F u b i n i v a c o n g t h c N e w t o n - L e i b n i z , a n g t h c

c u o i l a d o (u1, , j , , uk) , (u1, , j , , uk) U n e n c a c g i a t r c u a aj t a i o t r i e t t i e u ) .

o i v i t r n g h p ( 2 ) , t a c o (U k)

d =

U k

kj=1

(1)j1aj

ujdu1 duk

= A kk

j=1(1)j1aj

uj du1 duk

=

j

(1)j1([1,1][0,k]

aj

ujdu1 duk).

K h i j = k,[j ,j ]

aj

ujduj = aj (u1, , j , , uk) aj(u1, , j , , uk) = 0.

K h i j = k,[0,k]

ak

ukduk = ak(u1, , k) ak(u1, , 0) = ak(u1, , 0) .

V a y t h e o c o n g t h c F u b i n i , t a c o

(U k) d = (1)k

j=k[j ,j ] ak(u1, , 0)du1 duk1.M a t k h a c

(U k) =

A k10

ak(u1, , 0)du1 duk1.

T o s u y r a c o n g t h c c a n c h n g m i n h .

C h u y . N e u M k h o n g c o m p a c t c o n g t h c k h o n g u n g . C h a n g h a n , M l a k h o a n g m


49/64


t r o n g R, (x) = xdx .

3 . 2 C a c c o n g t h c c o i e n . S a u a y l a c a c h e q u a c u a n h l y t r e n :

C o n g t h c N e w t o n - L e i b n i z . C h o V l a t a p m t r o n g Rn , F : V R t h u o c l p C1

v a : [a, b] V l a t h a m s o h o a n g c o n g t r n . K h i o ([a,b])

dF = F((b)) F((a)).

C o n g t h c G r e e n . C h oD R2 l a m i e n c o m p a c t , c o b C = D n h h n g n g c

c h i e u k i m o n g h o . C h o P, Q l a c a c h a m l p C1 t r e n t a p m c h a D . K h i o D

(Q

x

P

y)dxdy =

C

P dx + Qdy.

C o n g t h c S t o k e s c o i e n . C h o

S R3

l a m a t c o n g t r n n h h n g p h a p

N, c o b

S = C l a n g c o n g k n n h h n g s a o c h o m i e n p h a t r a i . C h o P , Q, R c a c h a m l p C1 t r e n m o t t a p m c h a S. K h i o

S(

Q

x

P

y)dxdy+(

R

y

Q

z)dy dz +(

P

z

R

x)dz dx =

C

P dx+Qdy +Rdz.

C o n g t h c G a u s s - O s t r o g r a d s k i . C h oV R3 l a m i e n c o m p a c t , c o b V = S l a m a t

t r n n h h n g p h a p n g o a i . C h o P , Q, R l a c a c h a m l p C1 t r e n m o t m i e n m c h a

V . K h i o

V(P

x+

Q

y+

R

z)dxdydz = S P dy dz + Qdz dx + Rdx dy.

V d u .

a ) D i e n t c h m i e n D

g i i h a n b i n g c o n g k n C

t r o n g R2 :D

dxdy =

C

xdy =

Cydx =

1

2

C

(xdy ydx).

b ) T h e t c h m i e n V

g i i h a n b i m a t c o n g k n S

t r o n g R3 :V

dxdydz =

S

xdy dz =

Sydz dx =

S

zdx dy

=1

3 (S xdy dz + S ydz dx + S zdx dy)3 . 3 M e n h e . G a s U l a t a p m , c o r u t c t r o n g Rn . C h o =

ni=1

aidxi 1(U).

K h i o c a c i e u s a u t n g n g :

( 1 ) l a k h p , i . e . t o n t a i f C1(U), s a o c h o df = .( 2 ) l a o n g , i . e . d = 0.


50/64


( 3 )

ai

xi=

ai

xj, v i m o i i, j .

( 4 )

C

= 0, v i m o i n g c o n g k n C U.

C h n g m i n h : S u y t b o e P o i n c a r e v a c o n g t h c S t o k e s . ( B a i t a p )

V d u . T a p R2 \ {0} k h o n g c o r u t c v t r e n o c o d a n g xdy ydx

x2 + y2 o n g , n h n g

t c h p h a n t r e n n g t r o n l a 2 = 0.B a i t a p : C h n g m i n h Rn \ {0} k h o n g c o r u t c b a n g c a c h x e t d a n g

ni=1

(1)ixi

xn/2dx1 dxi dxn.

( t r o n g o k y h i e u

dxi e c h dxi k h o n g c o m a t t r o n g b i e u t h c . )

3 . 4 n g d u n g v a o g i a i t c h v e c t o r .

C a c t o a n t g r a d , r o t , d i v : T r o n g R3 v i c s c h n h t a c e1, e2, e3 v a U l a t a p m t r o n g R3 .

K y h i e u =

x1e1 +

x2e2 +

x3e3 , g o i l a t o a n t n a b l a .

C h o f : U R l a h a m k h a v i . T r n g g r a d i e n t c u a f, c n h n g h a :

grad f = f =f

x1e1 +

f

x2e2 +

f

x3e3.

C h o F = F1e1 + F2e2 + F3e3 l a t r n g v e c t o r k h a v i t r e n U. T r n g x o a n c u a F, c k y h i e u v a n h n g h a

rot F = F =

e1 e2 e3

x1

x2

x3F1 F2 F3

H a m n g u o n

c u a t r n g F, c k y h i e u v a n h n g h a :

div F =< , F >=F1

x1+

F2

x2+

F3

x3.

Q u a n h e v i t o a n t v i p h a n . n h n g h a c a c a n g c a u :

h1 : X(U) 1(U), h2(F1e1 + F2e2 + F3e3) = F1dx1 + F2dx2 + F3dx3.

h2 : X(U) 2(U), h2(F1e1+F2e2+F3e3) = F1dx2dx3+F2dx3dx1+F3dx1dx2.

h3 : C(U) 3(U), h3(f) = f dx1 dx2 dx3.


51/64


K h i o b i e u o s a u g i a o h o a n

C(U)grad X(U)

rot X(U)

div C(U)

id h1 h2 h3

0(U)d

1(U)d

2(U)d

3(U)

n g h a l a t a c o : h1 grad = d id, h2 rot = d h1, h3 div = d h2.

C h n g m i n h : X e m n h b a i t a p

H e q u a . T d d = 0, s u y r a rot grad = 0, div rot = 0.

3 . 5 C o n g t h c S t o k e s c h o t c h p h a n l o a i 1 . C h o F l a m o t t r n g v e c t o r k h a v i t r o n g

R3 .( 1 ) G i a s S l a m a t c o n g c o m p a c t t r o n g R3 , n h h n g b i t r n g v e c t o r p h a p n v N, c o b S = C l a n g c o n g n h h n g c a m s i n h b i t r n g v e c t o r t i e p x u c n v T s a o c h o m i e n S n a m p h a t r a i . K h i o

C< F, T > dl =

S

< rot F, N > dS.

( 2 ) G i a s V

l a m i e n g i i n o i t r o n g R3 c o b V = S l a m a t c o n g n h h n g b i t r n g v e c t o r p h a p n v

Nh n g r a p h a n g o a i . K h i o

S< F,N > dS=

V

div FdV.

C h n g m i n h : S u y t c o n g t h c S t o k e s v a m o i q u a n h e g i a t c h p h a n l o a i 1 v a l o a i 2 .


52/64


53/64

53

Bi tp gii tch 3

1 Bi tp tich phn ph thuc tham s

1. Tnh cc gii hn

1)limt0

11

x2 + t2dx 2) lim

t0

1+tt

dx

1 + x2 + t23) lim

n

10

dx

1 + (1 + x/n)n

4)limt0

1+t

tln(x + |t|)

ln(x2 +|t2

|5) lim

t0

1

0x

t2ex

2/t2dx 6) limt

/2

0et sinxdx.

2. Kho st tnh lin tc ca hm I(t) =10

tf(x)

x2 + t2, trong hm f(x) lin tc

v d-ng trn on [0, 1].

3.

1) Tm o hm ca cc tch phn eliptic

E(t) =

/2

0

1 t2 sin2 xdx F (t) =

/2

0

dx

1 t2 sin2 x

dx.

2) Hy biu din E, F qua cc hm E, F.3) Chng minh rnh E tha ph-ng trnh vi phn

E(t) +1

tE(t) +

1

1 t2E(t) = 0.

4. Gi s hm f(x, y) c cc o hm ring lin tc. Tnh I(t) nu

1) I(t) =

t

0

f(x + t, x

t)dx 2) I(t) =

t2

0

x+t

xt

sin(x2 + y2

t2)dydx.

5. Chng minh rng hm Bessel vi cc ch s nguyn

In(t) =1

0

cos(nx t sin x)dx,


54/64

54

tha mn ph-ng trnh Bessel

t2y + ty + (t2 n2)y = 0.

6. Cho hm (x) thuc lp C1) trn on [0, a] v I(t) =t0

(x)dxt x . Chng

minh rng, vi mi t (0, a) ta c

I(t) =

t0

(x)dxt x +

(0)t

.

7.

Bng cch ly o hm theo tham s, hy tnh

1) I(t) =

/20

ln(t2 sin2 x + cos2 x)dx 2) I(t) =

0

ln(1 2t cos x + t2)dx.

8. Chng t rng, hm I(t) =0

cos x

1 + (x + t)2dx. kh vi lin tc trn R.

9. Chng minh cng thc Frulanhi

0

f(ax)

f(bx)

xdx = f(0)ln

b

a, (a > 0, b > 0),

tr

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