42

Задание 18.

Найти

циркуляцию

векторного

поля

а = аxi

+ аy j + аz k

вдоль

контура L

двумя

способами

(непосредственно и применив формулу Стокса).

а = аxi + аy j + аz k

L

а = x3i + xyj + (z2 + y)k

2

2

2

1

x

+ y

+ z

= 2,

x =1

2

(x2 + z)i + yj + z3k

y 2

= x2 + z 2 ,

а =

y = 2

3

а = (2x + 4y)i + (7x −3y)j +

x2

+ y2 = (z − 4)2 ,

+ (2xy + z2 )k

z = 2

4

(y + z)i + (x − z)j + (x + y)k

x2

+ y2

= 9,

а =

x + y + z =1

5

(2x + z)i + (2y + x)j + (2z + y)k

z = x2 + y2 ,

а =

z = 4

6

(x + 2z)i + (y + 2x)j + (z + 2y)k

x2

+ z2 = 4y2 ,

а =

y =1

а = zi + (y2 + x)j + (z − x)k

2

2

2

7

x

+ y

+ z

= 4,

z =1

а = (y − 4x)i + (4x + 2y)j + (x − y2 )k

2

2

2

8

x

+ y

= (z

+3)

,

z = −5

9

а = (z + x2 )i + (x + y)j + xk

2

+ y

2

= 4,

x

x + y + z =1

10

z = x2 + y2 ,

а = zxi + xj + zyk

z = 4

11

3xi + (x + y)j + yzk

x2

+ y2 + z2 = 5,

а =

z =1

12

(2x − y)i + (x + 2y)j + zk

4z2 = x2 + y2 ,

а =

z =1

13

x2

+ y2 = 25,

а = x2i + (x + y)j + (y + z)k

x + y + z =1

14

а = (2x +3z)i + (3x − y)j +

x2

+ y2 = (z − 4)2 ,

+ (3y2 − z)k

z = 6

15

z = 4(x2 + y2 ),

а = x3i + (3y + x)j + z3k

z =1

16

2xzi + (x + 2y)j + 2yzk

x2

+ y2 + z2 = 25,

а =

z = 3

а = (x2 + y)i + (x + y2 )j + (y + z2 )k

2

2

2

17

x

= 9(y

+ z

),

x = 2

18

а = (x −3y)i + (5x −3y)j +

x2

+ y2 = (z −1)2 ,

+ (2x − y2 )k

z = −1

19

(2y + z)i + 3xj + 2xzk

z = 9(x2 + y2 ),

а =

z = 4

20

а = (3x + 2y)i + (3y + 2z)j +

x2

+ y2 = 4,

+(2x +3z)k

x + y = z =1

21

а = (2x +5y)i + (8x − y2 )j +

x2

+ y2 = (z +1)2 ,

+ (3xy − z2 )k

z = −3

22

а = (x3 + y)i + (x + y3 )j +

x = y2 + z2 ,

+ (y + z3 )k

x = 9

23

x2

+ y2 + z2 = 2,

а = (x + 2y)i +(3y + x)j + 2xzk

z =1

24

y2

= x2 + z2 ,

а = x2i +3xyj + (2z − x)k

y = 4

25

а = (x2 + 2z)i + (2x + y)j +

x2

+ y2 = 9,

+(x + y)k

x + y + z =1

26

x2

+ y2 = (z +3)2 ,

а = (3x − 2y)i + (x − 2y)j +

z = −1

27

а = (2x − 3y)i + xyj + (3z + x)k

x2

+ y2 + z2

= 5,

+ (z2 − xy)k

y = 2

28

x −1 = 4(y 2 + z 2 ),

а = xyi + (3x − y)j + (2x + 3y)k

x = 3

29

а = (5x − 4y)i + (3y − x2 )j +

x2

+ y2 = (z − 2)2 ,

+(xy − z)k

z = 4

30

а = (3y2 − x)i + (2x + z)j + (x + y)k

2

+ z

2

,

y +1 = x

y = 3

45

Задание

19.

Найти

поток

векторного

поля

а = аxi + аy j + аz k

через

замкнутую

поверхность

(S )

а = аxi + аy j + аz k

S

1

a = (ez + 2x) i + (ex −3xy) j +

x + y + z = 4,

+ (3xz −5z) k

x = 0, y = 0, z = 0

2

a =

(3z2 + x) i + (ex − 2y) j +

x2

= y2 + z2 ,

+ (2z − xy) k

0 ≤ x ≤ 3

(e−z − х) i +(xz +3у) j +

x + y = 2

3

a =

+(z

x = 0, y = 0

+ x2 ) k

z = 0, z = 2

x = 0, x = 3,

( z − х) i +(x − у) j +

4

a =

y = 0, y = 2,

+ (y2 − z) k

z = 0, z = 3

5

a =

(e2 y + х) i +(x −2у) j +

2

+ y

2

= 4,

x

+(y2 + z) k

z = 0, z = 5

6

a =

(e y + 2х2 sin y) i + 4x cos у j +

x − y + z =1,

+ (2z −1) k

x = 0, y = 0, z = 0

2

2

2

2

2

a = (8yzx − х) i + (4y

xz +5) j −

7

y

= x

+ z

,

−12xyz2 k

0 ≤ y ≤ 3

(2yz − х) i + (xz + 2у) j +

3x + 2z = 6,

8

a =

+ (x

y = 0, y = 4,

2 + z) k

x = 0, z = 0

(5x −6y) i +(11x2 + 2у) j +

x = 0, x = 4,

9

a =

+ (x

y = 0, y = 3,

2 −4z) k

0, z =

5

z =

10

a = (yz − 2х) i + (sin x + у) j +

x2 + z2

=1,

+ (x

− 2z) k

y = 0, y = 6

2

2

a = (2y −5х) i + (x −1) j +

4 − x

− y

,

11

z =

+(2

xy + 2z) k

0

z =

x

+

y

+

z

=1,

12

2

3

a = 5x i + 2у j −11z k

−5

0, z = 0

x = 0, y =

13

2x2 i + (3y −2) j − 4xz k

x2

= y2 + z2 ,

a =

0 ≤ x ≤ 4

y + 2z = 6

14

5,

a = x i + y j + z k

x = 0, x =

0

y = 0, z =

x = 0, x =1,

15

(x + z) i + (y + z) k

a =

y = 0, y = 4,

0, z =

6

z =

16

2x i + y j + z k

y2

+ z2

= 4,

a =

3

x = 0, x =

17

a =

(sin y +3х) i + (x2 y −1) j +

x2 + y2 +z2 = 25

+(5

− x2 z) k

2

x

x

y

z

a

= (3z

− х) i +

(e

−

2у) j +

+

+

=

1,

18

2

−3

2

+

(2z − xy) k

0, y = 0, z = 0

x =

z

2

x 2

y2

19

a

= (6x −cos y) i

+ (e

+ z) j −

z

=

+

,

9

9

−

(2y +3z) k

0 ≤ z ≤1

− x + y = 3,

= ( z −2x) i +(ex +3у) j +

a

20

x =

0, y = 0,

+

( y + x

−9z) k

0.z = 4

z =

= (x + y2 ) i +(xz −5у) j +

x = 0, x = −2,

a

21

y =

0, y = 4,

+( x2

+1 −3z) k

0, z = 5

z =

22

a

= (sin z + 2x) i +(sin x −3у) j +

y2

+ y2

=16,

0, z = 4

+

(sin y + 2z) k

z =

− x

2

− y

2

,

a

= ( z +1+ x) i + (2x +8у) j +

z = − 4

23

+

(sin x

0

+ 7z) k

z =

24

a

= (y2 + z2 +6х) i +

4x +3y +12z =12,

+

0, y = 0, z = 0

(ez −2у + x) j

+ (x + y − z) k

x =

25

a = (cos z +3х) i + (x − 2у) j +

y2

= x2 + z2 ,

+

(9z

+

y

2

0 ≤ y ≤ 6

) k

= x2 yz i +3y2 xz j +

3x + 2z = 6,

26

a

0, y = 5,

y =

+

(5z −4xyz2 ) k

0, z = 0

x =

48

x = 0, x = −2,

27

= −

3xi

+

8y j

−

11z k

y = 0, y = 2,

a

z = 0, z = 2

28

a = (x2ez + 2y) i +(xz −4у) j +

2

+ z

2

= 4,

y

+(z −2xez ) k

y = 0, y = 4

29

a = (5x + z) i + (x −3у) j +

x2 + y2 + z2 =1

+ (4y + 2z) k

30

a = z2 i +(xy −5z) j +(8z − xz) k

6x +3y − 2z = 6,

0

x = 0, y = 0, z =

Задание

20.

Решить ДУ P(x, y)dx +Q(x, y)dy = 0

в

полных дифференциалах.