F = (x2 + y2 )i + y2 j,

F = x2 j,

(x ³ 0, y ³ 0),

10.21. L : отрезок MN,

10.22. L : x2 + y2 = 9

M (2,0), N (0,2).

M (3,0), N (0,3).

F = (y2 - y)i + (2xy + x) j,

F = xyi,

10.23. L : x2 + y2 = 9 ( y ³ 0),

10.24. L : y = sin x,

M (3,0), N (-3,0).

M (π ,0), N (0,0).

F = (xy - y2 )i + xj,

F = xi + yj,

10.25. L : y = 2x2 ,

10.26. L : отрезок MN,

M (0,0), N (1,2).

M (1,0), N (0,3).

F = (xy - x)i +

x2

j,

F = −xi + yj,

2

y2

(x ³ 0, y ³ 0),

10.27. L : y = 2

,

10.28. L : x2 +

=1

x

M (0,0), N (1,2).

9

M (1,0), N (0,3).

F = −yi + xj,

F = (x2 - y2 )i + (x2 + y2 )j,

x2

y2

=1 ( y ³ 0),

10.29. L : y = x3,

10.30. L :

+

4

M (0,0), N (2,8).

9

M (3,0), N (-3,0).

F = (x - y)i + j,

10.31. L : x2 + y2 = 4 ( y ³ 0),

M (2,0), N (-2,0).

Задача 11. Найти циркуляцию векторного поляa вдоль контура Г(в направлении соответствующем возрастанию параметра t ).

215

	a = yi - xj + z2k,
	ì
11.1.			2				2
	ïx =				cost, y =				cost,
	Г : í	2				2
	ï
	îz = sint.

a= ( y - z)i + (z - x) j+ (x - y)k,

11.3.ìx = cost, y = sin t,

Г: íîz = 2(1- cost ).

a= ( y - z)i + (z - x)j+ (x - y)k,

11.5.ìx = 4cost, y = 4sint,

Г: íîz =1- cost.

a= 2zi − xj + yk,

11.7.ìx = 2cost, y = 2sint,

Г: íîz =1.

a= xi + z2 j+ yk,

11.9.ìx = cost, y = 2sint,

Г: íîz = 2cost - 2sint -1.

a= -x2 y3i + 2j+ xzk,

11.11.Г : ìíx = 2 cost, y = 2 sint,

îz =1.

a= zi + y2 j- xk,

	ì
11.13.					2 cost, y = 2sint,
	ïx =
	Г : í
					2 cost.
	ïz =
	î
	a = xi -		1	z2 j+ yk,
11.15.			3
	ìx =		(cost ) 2, y = (sint ) 3,
	ï
	Г : í	z = cost - (sint ) 3 -1 4.
	ï
	î

a= -x2 y3i + j+ zk,

11.2.Г : ìíx = 34 cost, y = 34 sint,

îz = 3.

	a = x2i + yj - zk,
	ì
11.4.		= cost, y = (						2 sint ) 2,
	ïx
	Г : í		(			)
	î	=		2 cost			2.
	ïz

a= 2yi − 3xj+ xk,

11.6.ìx = 2cost, y = 2sint,

Г: íîz = 2 - 2cost - 2sint.

a= yi − xj+ zk,

11.8.ìx = cost, y = sint,

Г: íîz = 3.

a= 3yi − 3xj+ xk,

11.10.ìx = 3cost, y = 3sint,

Г: íîz = 3 - 3cost - 3sin t.

a= 6zi − xj+ xyk,

11.12.ìx = 3cost, y = 3sint,

Г: íîz = 3.

a= xi + 2z2 j+ yk,

11.14.ìx = cost, y = 3sin t,

Г: íîz = 2cost - 3sint - 2.

a= 4yi − 3xj+ xk,

11.16.ìx = 4cost, y = 4sint,

Г: íîz = 4 - 4cost - 4sint.

216

	a = −zi − xj + xzk,		a = zi + xj+ yk,
11.17.	ìx = 5cost, y = 5sint,	11.18.	ìx = 2cost, y = 2sint,
	Г : í		Г : í		= 0.
	îz = 4.		îz
	a = ( y - z)i + (z - x) j+ (x - y)k,			a = 2yi − zj+ xk,
11.19.	ìx = 3cost, y = 3sint,		11.20.		ìx = cost, y = sint,
	Г : í			Г : í
	îz = 2(1- cost ).				îz = 4 - cost - sint.
	a = xzi + xj+ z2k,			a = -x2 y3i + 3j+ yk,
11.21.	ìx = cost, y = sint,		11.22.		ìx = cost, y = sint,
	Г : í			Г : í
	îz = sint.				îz = 5.
	a = 7zi − xj+ yzk,		a = xyi + xj+ y2k,
11.23.	ìx = 6cost, y = 6sint,	11.24.	ìx = cost, y = sint,
	Г : í		Г : í		= sint.
	îz =1 3.		îz
	a = xi - z2 j+ yk,	a = ( y - z)i + (z - x)j+ (x - y)k,
11.25.	ìx = 2cost, y = 3sint, 11.26.	ìx = 2cost, y = 2sint,
	Г : í	Г : í			- cost ).
	îz = 4cost - 3sint - 3.	îz = 3(1

	a = -2zi - xj+ x2k,
11.27.	ìx = (cost ) 3, y = (sint ) 3,
	Г : í
	îz = 8.

a= xi - 2z2 j+ yk,

11.29.ìx = 3cost, y = 4sint,

Г: íîz = 6cost - 4sint +1.

a= yi3 − 3xj + xk,

11.31.ìx = 2cost, y = 2sint,

Г: íîz =1- 2cost - 2sin t.

a= xi - 3z2 j+ yk,

11.28.ìx = cost, y = 4sin t,

Г: íîz = 2cost - 4sint + 3.

a= -x2 y3i + 4j+ xk,

11.30.ìx = 2cost, y = 2sint,

Г: íîz = 4.

217

Задача 12. Найти модуль циркуляции векторного поля a вдоль контура Г.

a= (x2 - y)i + xj+ k,

12.1.ìx2 + y2 =1,

Г: í

îz =1.

a= yzi + 2xzj+ xyk,

ìx2 + y2 + z2 = 25,

12.3.ï

Г: íïîx2 + y2 = 9 (z > 0).

a= (x − y)i + xj− zk,

12.5.ìx2 + y2 =1,

Г: í

îz =1.

a= yzi + 2xzj+ y2k,

12.7. ï

ìx2 + y2 + z2 = 25,

Г : íïîx2 + y2 =16 (z > 0).

a = yi + (1− x) j− zk,

12.9. ìx2 + y2 + z2 = 4,

Г : ï

íïîx2 + y2 =1 (z > 0).

a = 4xi + 2j− xyk,

ìz = 2(x2 + y2 )+1,

12.11. ï

Г : í

ïîz = 7.

a= -3zi + y2 j+ 2yk,

12.13.ìx2 + y2 = 4,

Г: í

îx - 3y - 2z =1.

a= 2yi + j− 2yzk,

12.15.ìx2 + y2 - z2 = 0,

Г: íîz = 2.

a = xzi − j+ yk,

ìz = 5(x2 + y2 )-1,

12.2. ï

Г : í

ïîz = 4.

a= xi + yzj − xk,

12.4.ìx2 + y2 =1,

Г: íîx + y + z =1.

a = yi - xj+ z2k,

12.6. ï

ìz = 3(x2 + y2 )+1,

Г : í

ïîz = 4.

a = xyi + yzj+ xzk,

12.8.ìx2 + y2 = 9,

Г: íîx + y + z =1.

a= yi - xj+ z2k,

12.10.ìx2 + y2 =1,

Г: í

îz = 4.

a= 2yi - 3xj+ z2k,

12.12.ìx2 + y2 = z,

Г: í

îz =1.

a= 2yi + 5zj+ 3xk,

12.14.ì2x2 + 2y2 =1,

Г: íîx + y + z = 3.

a = (x - y)i + xj+ z2k,

12.16.	ì	2	+ y			2	- 4z	2	= 0,
	ïx
Г :	í			1
	ïz	=			.
				2
	î

218

a = xzi − j+ yk,

12.17.ìx2 + y2 + z2 = 4,

Г: íîz =1.

a= 4xi − yzj+ xk,

12.19.ìx2 + y2 =1,

Г: íîx + y + z =1.

a= yi + 3xj+ z2k,

12.21.ìz = x2 + y2 -1,

Г: í

îz = 3.

a= (2 − xy)i − yzj − xzk,

12.23.ìx2 + y2 = 4,

Г: í

îx + y + z =1.

a= yi − xj + 2zk,

ì

2

2

z2

12.25.

Г :

ïx

+ y

-

= 0,

4

í

ï

= 2.

îz

a = yi - 2xj+ z2k,

ì

(x

2

2

)+ 2,

12.27.

ïz = 4

+ y

Г :

í

= 6.

ïz

î

a= (x + y)i - xj + 6k,

12.29.ìx2 + y2 =1,

Г: í

îz = 2.

a= yzi − xzj+ xyk,

12.31. ìïx2 + y2 + z2 = 9,

Г : í

ïîx2 + y2 = 9.

a= 2yzi + xzj- x2k,

ìx2 + y2 + z2 = 25,

12.18.ï

Г: íïîx2 + y2 = 9 (z > 0).

a= −yi + 2j + k,

12.20.ìx2 + y2 - z2 = 0,

Г: íîz =1.

a = 2yzi + xzj+ y2k,

12.22. ï

ìx2 + y2 + z2 = 25,

Г : íïîx2 + y2 =16 (z > 0).

a = -yi + xj+ 3z2k,

12.24. ï

ìx2 + y2 + z2 = 9,

Г : íïîx2 + y2 =1 (z > 0).

a= x2i + yzj+ 2zk,

12.26.ìx2 + y2 + z2 = 25,

Г: í

îz = 4.

a= 3zi − 2yj+ 2yk,

12.28.ìx2 + y2 = 4,

Г: íî2x - 3y - 2z =1.

a= 4i + 3xj+ 3xzk,

12.30.ìx2 + y2 - z2 = 0,

Г: íîz = 3.

219

220