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8/8/2019 Vektoranalzis alapjai (2007, 26 oldal)
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R 3
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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
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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)
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(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
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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
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(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 ).
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(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
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(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
.
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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
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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)
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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
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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
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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 )
.
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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
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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
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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
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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
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=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 )
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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
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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
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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
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: [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
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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 )
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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)
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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.
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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