vektoranalízis alapjai (2007, 26 oldal)

8/8/2019 Vektoranalzis alapjai (2007, 26 oldal)

1/26

R 3


2/26

R 2

M

x 1

x 2

M

x 3

t 2

t 1

P

U

t 1 t 2 (t 1 , t 2) P


3/26

k k = 1,2,3,... k k

k k

k

k

n n 1

-1

-0,5

0

0,56

-1 -0,5 01

4

0,5 1

2

0

-2

-4

-6 : R

R 3

(t ) := ( a t, a t,bt ) .

t

(cos t, sin t )

x 1

x 2

U1 := ( 0,2 ) U2 := ( , ) 1(t ) = 2(t ) = ( t, t ) {(U1 , 1), (U2 , 2) R 2

U1 := ( 0,2 ) R U2 := ( , ) R

1(t, s ) = 2 (t, s ) = ( t, t, s )

{(U1 , 1), (U2 , 2)

U = 2 ,2 0,2

(t 1 , t 2) = ( t 1 t 2 , t 1 t 2 , t 1)


4/26

(t 1 , t 2) = ( a + t 1) t 2 , (a + t 1) t 2 , t 1)

U = 2

, 2

0,2 [0, 1 ]

(t 1 , t 2) = ( t 3 t 1 t 2 , t 3 t 1 t 2 , t 3 t 1)

(x1 , x2) R 2 x21 + x

22 = 1

F(x1 , x2 ) = x21 + x22 1

x1 x2 x1 x2

M F : R n

R m M R n F

M := {x R n | F(x) = 0},

x R n M F(x) = 0

xi M M


5/26

F : R n R m

x M F i

x j x n k

M k

F : R 2 R F(x, y ) := x2 + y 2 1

S1 := {(x, y ) R 2 | F(x, y ) = 0} R 2

F : R 3 R F(x,y,z ) := x2 + y2 1 M := {(x,y,z ) R 3 | F(x,y,z ) =

0}

F : R 3 R F(x,y,z ) := x2 + y 2 + z2 1 S2 := {(x,y,z )

R 3 | F(x,y,z ) = 0}

M P

R 3 M = [ x1 , x2 ]

x3

x 1

P = t 0v = t 0

x 3

x 2

M

M : (a, b ) R n

M t (a, b ) (t ) M


6/26

(U, ) p M p x0 = ( x10 , . . . ,x

k0 ) 1 i k

i : ( , ) M, i (t ) = (x0 + te i ) = (x10 ...x i0 + t. . .x k0 ) p

p

P

p M

M p T p M T p M TM

: ( , ) M 0 p M (t 0) = p vp =

ddt t = 0

(t )

p

( , )

0 : (a, b ) M v ( t 0 ) := ddt t = t 0 (t ) t 0 (t 0)

ddt t = 0

i (t ) =ddt t = 0

(x0 + te i ) =x i

(x0 ).


7/26

(t 1 , t 2) = ( t 1 , t 1 , t 2)

t 1 t 2 t 2 t 1

x 1

(x0 ), ...,x k

(x0) T p M

T p M p

: (,) M 0 p M (0)

1 q 1 (q ) = q t (t ) = 1 (t ). t

(0) =ddt t = 0

( 1 )( t ) =n

i = 1

x i

(x0)d i

dt(0) =

n

i = 1

vix i

(x0)

vi 1 i

T p M

T p M


8/26

(t 1 , t 2) := ( t 1 , t 1 , t 2)

T p M

v 1

p

v 2

(t 1 , t 2) p = (t 1 , t 2)

v1 =t 1 = (

t 1 , t 1 , 0),

v2 =t 2

= ( 0,0,1 )

(t 1 , t 2) = ( 0, 0) p = ( 1,0,0 ) v1 = ( 0,1,0 ) v2 = ( 0,0,1 ) p

T p M = (0,a,b ) | a, b R ;

(t 1 , t 2) = ( /2, 5 ) ^ p = ( 0,1,5 ) v1 = ( 1,0,0 ) v2 = ( 0,0,1 )

T p M = (a,0,b ) | a, b R .

p ^ p

M X X p M Xp

p Xp T p M M X( M )

(U, ) M x U p T p M

Xp T p M X

1(x) X

k(x)

Xp U

X

X = X1x 1

+ ... + Xkx k

.


9/26

Xi

x0 U p x0

(t, s ) = ( t, t, s )

X = s e 2tt

+ s2 ts

(0, 3) p = ( 1,0,3 ) Xp

Xp = 5e 0

t (0,3 )+ 32 0

s (0,3 )= 5(0,1,0 )p + 9(0,0,1 )p = ( 0,5,9 )p

p p


10/26

M N f : M N f f M N

M f NTM f TN vp : ( ,) M (0) = p (0) = vp f( vp ) = ddt t = 0f ( (t )) ,

f N N p

M

p

f

v pf ( p)

f v p

t

f ( t )

(U, ) k M p M U R k k

: U M p x0 e i,x 0 x0 U t (x0 + te i ) R k e i x0 i

(e i,x 0 ) =ddt t = 0 (x0 + te i ) =

x i

(x0)


11/26

V k l

: V V

l

R

l 2 ( 1)

( v1 , . . . ,v i , ...v j , . . . ,v l ) = ( 1) ( v1 , . . . ,v j , ...v i , . . . ,v l ).

V l l (V )

v (..., v, ...v, .. ) = 0.

l (V )

v1 = 2 v2 + ... + k vk

p ( v1 , v2 , . . . ,v k ) = p ( 2 v2 + ... + k vk , v2 , . . . ,v k )

= 2 p ( v2 , v2 , . . . ,v k ) + ... + k p ( vk , v2 , . . . ,v k )1.= 0

l > k l (V ) = 0 l (V ) v1 , . . . ,v l V p ( v1 , . . . ,v l ) = 0

V k l > k

( v1 , . . . ,v l ) = 0


12/26

l (V )

l (V ) , l (V ) R +

( )( v1 , . . . ,v l ) := ( v1 , . . . ,v l )

( + )( v1 , . . . ,v l ) := ( v1 , . . . ,v l ) + ( v1 , . . . ,v l )

+ l (V )

(V ) k V l l (V ) k

l

l (V ) l (V ) {e1 , . . . ,e k } V

1 i 1 , . . . , i l l i 1 ...i l v1 , . . . ,v l V

i 1 ...i l ( v1 , . . . ,v l ) :=

vi 11 vi 12 . . . v

i 1l

vi 21 v

i 22 . . . v

i 2l

vi l1 vi l2 . . . v

i ll

.

i 1 ...i l {e1 , . . . ,e k } k l j vj i 1 i 2 i l l l

i 1 ...i l l V i 1 ...i l l (V )

{ i 1 ...i l | 1 i 1 < i 2 < ... < i l k} l (V ) l (V )

=1 i 1


13/26

k (V ) k (V ) R = 1...k .

l (M ) m (M )

: V V

l + m

R ,

v1 , . . . ,v l + m V

( v1 , . . . ,v l + m ) :=1

l!m !

( 1) ( ) ( v (1) , . . . ,v ( l ) ) ( v ( l + 1) , . . . ,v ( l + m )) .

{1, ...., l + m } ()

( 1) ( )

l + m (M ).

, 1(M ) 2 (M ) v1 , v2 V

( v1 , v2 ) = ( v1 ) ( v2 ) ( v2) ( v1) = ( v1) ( v2 ) ( v1) ( v2) .

,, 1(M ) 3(M ) v1 , v2 , v3 V

( v1 , v2 , v3) = ( v1) ( v2) ( v3) ( v1) ( v2 ) ( v3)( v1) ( v2) ( v3 )

.

1 , ... l 1 (M ) 1 ... l l (M )

1 ... l ( v1 , . . . ,v l ) =

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

l ( v1 ) l ( v2 ) l ( vl )

.


14/26

M k M l l p M

p : T p M T p M

l

R

l l l (M ) 0(M ) M

l l X1 Xl M

X1 , . . . ,X l (X1 , . . . ,X l ) p M

p

(X1 , . . . ,X l ) p := p (X1,p , . . . ,X l,p )

l (M ) l p M p l

X (..., X, ...X, .. ) = 0

l > k

l

(M ) = 0

l

(M )

X1 , . . . ,X l (M ) (X1 , . . . ,X l ) = 0


15/26

f : M R

df

XX( M )

df (X),

df (X) p M f Xp T p M

df (X)p := Xp (f ) = t0

f ( (t )) f ( (0))t

,

(t ) Xp (0) = p (0) = Xp (U, ) p

Xp X1p Xkp

df (X)p =k

i = 1

Xip (f )x i (x0) ,

x0 p df p df p : T p M

R

df M

df 1(M ).

df f : M

R

(U, ) M (V, x) V = (U) x := 1 x

p M k x k

x = ( x1 , . . . ,x k ) i xi

dx i (X)p = Xip ,

dx i X p i


16/26

M k (U, ) M x U p = (x) p T p M

{x 1 x , ...,x 1 x

} 1 i 1 , . . . , i l

l

dxi 1 ... dx

i l

l

p M

v1 , . . . ,v l T p M

dx i 1p ... dxi lp ( v1 , . . . ,v l ) :=

vi 11 vi 12 . . . v

i 1l

vi 21 vi 22 . . . v

i 2l

vi l1 vi l2 . . . v

i ll

,

( v1j , . . . ,vlj ) vj {

x 1 x , ...,

x 1 x

} l

k l j vj i 1 i 2 i l

l l

dx i 1 ... dx i l l M dx i 1 ... dx i l l (M )

x U p M

{dx i 1p ... dxi lp | 1 i 1 < i 2 < ... < i l k}

l (T p M ) p l (M )

p =1 i 1


17/26

1 (M ) {dx 1 ,...,dx k } f : M R

df 1(M )

df =k

i = 1

(f x 1 )x i

dx i .

M N f : M N f : (N ) (M ) M f N(M ) f

(N )

l (N ) f l (M )

f ( v1 , ...v l ) = (f v1 , . . . ,f vl ).

M k (U, ) l (M ) = i 1 ...i l dx

i 1 ... dx i l , l (U)

=1 i 1


18/26

=i 1 ...i k

i 1 ...i k ( )i 1 ...i k 12...k .

M k (U1 , 1 ) (U2 , 2)

M

U1

U2

1 2

1 12

1 = 1 12...k , 2 =

2 12...k ,

1 2 U1 U2 = 1 12

2 = x 1

d : (M ) (M ) d : l (M )

l + 1(M )

d ( + ) = d + d

, l

(M )

d ( ) = d + ( 1) l d l (M ) m (M )

d 2 = d d = 0

d : 0(M ) 1(M )


19/26

l (M ) d (V, x) = i 1 ...i l dx

i 1 ... dx i l

d = d ( i 1 ...i l dxi 1 ... dx i l ) = (d i 1 ...i l dx

i 1 ... dx i l )

= d i 1 ...i l dxi 1 ... dx i l =

i 1 ...i lx j

dx j dx i 1 ... dx i l

d

M 1(M ) = 1dx 1 + 2dx 2 d 2(M )

d = 2x 1

1x 2

dx 1 dx 2

M 1(M ) = 1dx 1 + 2dx 2 + 3dx 3 d

2(M )

d = 2x 1

1x 2

dx 1 dx 2 + 3x 1

1x 3

dx 1 dx 3 + 3x 2

2x 3

dx 2 dx 3

M 2(M )

= 12 dx 1 dx 2 + 23 dx 2 dx 3 + 31 dx 3 dx 1 ,

d 3 (M )

d = 12x 3

+ 23x 1

+ 31x 2

dx 1 dx 2 dx 3

V R 3 f : V R

g : V R 3

g g = ( g 1 , g 2 , g 3)

f : V

R 3 , f = ( 1f, 2f, 3f ) ,

g : V

R 3 , g = ( 2g 3 3g 2 , 3 g 1 1 g 3 , 1g 2 2g 1),

g : V R , g = 1g 1 + 2g 2 + 3g 3 .

g : V R 3

f : V R

g = f g g = 0


20/26

R 3

R 3

1 (R 3) 1.

X( R 3 ) 2.

2 (R 3)

1dx + 2 dy + 3dz 1e1 + 2e2 + 3e3 1dy dz + 2dz dx + 3dx dy

0 (R 3) 3.

3(R 3)

dx dy dz

f : V R

df = f 1 dx + f 2dy + f 3dz f df

g : V R 3

V g 1dx + g 2 dy + g 3 dz

g

g : V R 3

g


21/26

k k

k M Ik = [ 0, 1 ] ... [0, 1 ]

k (Ik ) k Ik =(x) dx 1 ... dx k

I k

= I k

dx 1 ... dx k := I k

(x)dx

M k : Ik

M

k k (M ) k (Ik )

M

:= I k

,

M k i : Ik M i k j 1i k (M )

M

:=i

M i

,

M k R n : I M = i 1 ..i k dx i 1 ... dx i k k R n

M

= I i 1 ...i k

i 1 ...i k ( )i 1 ...i k


22/26

: [a, b ]R n

M = f 1dx 1 + ... + f n dx n M =

ba

. [a, b ] 1 n

t = ( f 1( t ) 1t + ... + f n ( t ) nt ) dt

M

= ba

f 1( t ) 1t + ... + f n ( t ) nt dt

f : V R 3

f =(f 1 , f 2 , f 3)

f = ba

n

i = 1

f i ( t ) it dt = ba

f ( t ) , t dt

, R n

R 3

M R 3 : I2 M =

f 1 dx 2 dx 3 + f 2dx 3 dx 1 + f 3 dx 1 dx 2

= f 1 ( )23 + f 2 ( )31 + f 3 ( )12

= f 1 21

32

21 32

+ f 2 31

12

31 12

+ f 3 11

22

11 22


23/26

n := 1 2 n = (n 1 , n2 , n3)

n1 = 21

32

21 32

, n2 = 11

32

11 32

, n3 = 11

22

11 22

.

= f , n

M

= I 2

f , n

f : V R 3

f (f 1 , f 2 , f 3) 2

M

f = I 2

f , n

f : [a, b ]R

b

a

f = f(b ) f (a ),

f f [a, b ]

k + 1 Ik + 1

Ik + 1 = [ 0, 1 ] ... [0, 1 ]

k + 1,

2(k + 1) k 2 r k + 1 Ikr,1

(x1 , . . . ,x k ) (1

1 xr ,2

x1 , ...,r

1 , ...,k + 1

xk )

Ikr,0

(x1 , . . . ,x k ) (1

xr ,2

x1 , ...,r

0, ...,k + 1

xk )


24/26

r = 0 (x1 , . . . ,x k ) (1

0,2

1 x1 , ...,k + 1

xk )

Ik1,0 (x1 , . . . ,x k ) (1

0 ,2

1 x1 , ...,k + 1

xk )

Ik + 1 k I k

I k := {Ikr,s | 1 r k + 1, s = 0, 1}.

M k + 1 : Ik + 1 M M := (I k + 1 ) M := { (Ikr,s ) | 1 r k + 1, s = 0, 1}.

Rk

Rn

M

I k

M k k 1 M

M

d = M

.

V R n A, B V : [a, b ] R n

A = (a ) B = (b ) F : V

R

F = f

f = F(B) F(A)


25/26

F = f (1F, ..., n F) =(f 1 , . . . ,f n ) f dF

f =

dF(9)=

F = F( (b )) F( (a )) = F(B) F(A).

a b

A

R Rn

B

0

A B

f = ( f 1 , f 2) : V R 2

: I2 M M ( 1f 2 2f 1) = f

f = f 1 dx + f 2dy V d = ( 1f 2 2f 1 )dx dy.

f = M

(9)=

Md =

M(1f 2 2f 1 ).

f = ( f 1 , f 2 , f 3) : V R 3

M V

M

f = M

f.


26/26

f = f 1dx + f 2dy + f 3 dz f d

f = ( f 1 , f 2 , f 3) : V R 3

M

V

M

f = M

f.

f = f 1dy dz + f 2 dz dx + f 1dx dy f d

vektoranalízis alapjai (2007, 26 oldal)

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