a= −xi + 2yj + zk

5.31.P : x + 2y + 3z =1.

Задача 6. Найти поток векторного поля a через часть плоскости P, расположенную в 1 октанте (нормаль образует острый угол с осью Oz ).

6.1.	a = 7xi + (5π y + 2) j + 4π zk,	6.2.	a = 2π xi + (7 y + 2)j + 7π zk,
	P : x + y 2 + 4z =1.		P : x + y 2 + z 3 =1.
6.3.	a = 9π xi + j− 3zk,		6.4.	a = (2x +1)i − yj+ 3π zk,
	P : x 3 + y + z =1.			P : x 3 + y + 2z =1.
6.5.	a = 7xi + 9π yj + k,		6.6.	a = i + 5yj+11π zk,
	P : x + y 3 + z =1.			P : x + y + z 3 =1.
6.7.	a = xi + (π z −1)k,	6.8.	a = 5π xi + (9y +1)j+ 4π zk,
	P : 2x + y 2 + z 3 =1.		P : x 2 + y 3 + z 2 =1.
6.9.	a = 2i − yj + 3π zk,			a = 9π xi + (5y +1)j+ 2π zk,
	2		6.10.
	P : x 3 + y + z 4 =1.			P : 3x + y + z 9 =1.
	a = 7π xi + 2π yj + (7z + 2)k,			a = π yi + (4 − 2z)k,
6.11. P : x + y + z 2 =1.			6.12. P : 2x + y 3 + z 4 =1.

a= (3π −1)xi + (9π y +1)j+ 6π zk,

6.13.P : 2x + 3y + 9z =1.

6.14.a = π xi + π2 yj+ (4 − 2z)k,

P : x + 3y + 4z =1.

6.15.	a = (5y + 3)j+11π zk,	6.16.	a = 9π yj+ (7z +1)k,
	P : x + y 3 + 4z =1.		P : x + y + z =1.

a= π yj+ (1− 2z)k,

6.17.P : x4 + y2 + z =1.

204

a = (27π −1) xi + (34π y + 3) j+ 20π zk,

6.18. P : 3x +	y	+ z =1.
9

a= π xi + 2j+ 2π zk,

6.19.P : x2 + y3 + z =1.

a= 4π xi + 7π yj+ (2z +1)k,

6.20.P : 2x + y3 + 2z =1.

6.21.	a = 3π xi + 6π yj+10k,	6.22.	a = π xi − 2yj + k,
	P : 2x + y + z 3 =1.		P : 2x + y 6 + z =1.

a= (21π −1)xi + 62π yj+ (1− 2π z)k,

6.23.P : 8x + y2 + z3 =1.

6.24.	a = π xi + 2π yj+ 2k,	6.25.	a = 9π xi + 2π yj+ 8k,
	P : x 2 + y 4 + z 3 =1.		P : 2x + 8y + z 3 =1.

a= 7π xi + (4y +1)j + 2π zk,

6.26.P : x3 + 2y + z =1.

a= 6π xi + 3π yj+10k,

6.27.P : 2x + y2 + z3 =1.

a= (π −1)xi + 2π yj+ (1−π z)k,

6.28.P : 4x + 2y + 3z =1.

6.29.a = π2 xi +π yj+ (4 − 2z)k,

P : x + 3y + 4z =1.

a= 7π xi + 4π yj+ 2(z +1)k,

6.30.P : x3 + y4 + z =1.

a= 5π xi + (1− 2y)j+ 4π zk,

6.31.P : x2 + 4y + z3 =1.

205

Задача 7. Найти поток векторного поля a через замкнутую поверхность S (нормал внешняя).

7.1.a = (ez + 2x)i + ex j + ey k, S : x + y + z =1, x = 0, y = 0, z = 0.

7.2.a = (3z2 + x)i + (ex − 2y)j + (2z − xy)k, S : x2 + y2 = z2 , z =1, z = 4.

7.3.a = (ln y + 7x)i + (sin z − 2y) j + (ey − 2z)k, S : x2 + y2 + z2 = 2x + 2y + 2z − 2.

7.4.a = (cos z + 3x)i + (x − 2y) j+ (3z + y2 )k, S : z2 = 36(x2 + y2 ), z = 6.

7.5.a = (e− z − x)i + (xz + 3y) j + (z + x2 )k, S : 2x + y + z = 2, x = 0, y = 0, z = 0.

7.6.a = (6x − cos y)i − (ex + z)j− (2y + 3z)k, S : x2 + y2 = z2 , z =1, z = 2.

7.7.a = (4x − 2y2 )i + (ln z − 4y) j+ (x + 3z4)k, S : x2 + y2 + z2 = 2x + 3.

7.8.a = (1+ z )i + (4y − x )j + xyk, S : z2 = 4(x2 + y2 ), z = 3.

7.9.a = (z − x)i + (x − y)j + (y2 − z)k, S : 3x − 2y + z = 6, x = 0, y = 0, z = 0.

7.10.a = ( yz + x)i + (x2 + y)j + (xy2 + z)k, S : x2 + y2 + z2 = 2z.

7.11.a = (e2 y + x)i + (x − 2y) j+ (y2 + 3z)k, S : x − y + z =1, x = 0, y = 0, z = 0.

7.12.a = (z − 2x)i + (ex + 3y)j + y + xk, S : x2 + y2 = z2 , z = 2, z = 5.

7.13.a = (ez + x4)i + (ln x + y4)j+ 4z k, S : x2 + y2 + z2 = 2x + 2y − 2z − 2.

7.14.a = (3x − 2z)i + (z − 2y)j + (1+ 2z)k, S : z2 = 4(x2 + y2 ), z = 2.

7.15.a = (ey + 2x)i + (x − y)j+ (2z −1)k, S : x + 2y + z = 2, x = 0, y = 0, z = 0.

7.16.a = (x + y2 )i + (xz + y)j+ (x2 +1 + z)k, S : x2 + y2 = z2 , z = 2, z = 3.

7.17.a = (ey + 2x)i + (xz − y) j+ (14)(exy − z)k, S : x2 + y2 + z2 = 2y + 3.

7.18.a = (z + y)i + 3xj + (3z + 5x)k, S : z2 = 8(x2 + y2 ), z = 2.

7.19.a = (8yz − x)i + (x2 −1)j+ (xy − 2z)k, S : 2x + 3y − z = 6, x = 0, y = 0, z = 0.

206

7.20.a = (y + z2 )i + (x2 + 3y)j+ xyk, S : x2 + y2 + z2 = 2x.

7.21.a = (2yz - x)i + (xz + 2y) j+ (x2 + z)k, S : y - x + z =1, x = 0, y = 0, z = 0.

7.22.a = (sin z + 2x)i + (sin x - 3y) j + (sin y + 2z)k, S : x2 + y2 = z2 , z = 3, z = 6.

x

æ z

ö

2

2

2

7.23. a = (cos z + x 4)i + (e

+ y

4)j+ ç

-1÷k, S : x

+ y

+ z

= 2z + 3.

è 4

ø

ì

2

2

2

7.24. a = (

+1+ x)i + (2x + y)j+ (sin x + z)k, S :

íz

= x

+ y

,

z

îz =1.

7.25. a = (

5x - 6y)i + (11x2 + 2y)j + (x2 - 4z)k, S :

ìx + y + 2z = 2,

í

= 0, y

= 0, z = 0.

îx

7.26. a = (y2 + z2 + 6x)i + (ez - 2y + x)j + (x + y - z)k, S :

ì

2

2

2

íx

+ y

= z

,

îz =1, z = 3.

7.27.a = 12 (x + z)i + 14 (x × z + y)j

7.28.a = (3yz - x)i + (x2 - y)j + (

+ (xy - 2)k, S : x2 + y2 + z2 = 4x - 2y + 4z - 8.

ìz2 = 9(x2 + y2 ), 6z -1)k, S : íï

ïîz = 3.

7.29. a = ( yz - 2x)i + (sin x + y)j+ (x - 2z)k, S :	ìx + 2y - 3z = 6,
	í
	îx = 0, y = 0, z = 0.

7.30. a = (8x +1)i + (zx - 4y)j+ (ex - z)k, S : x2 + y2 + z2 = 2y.

				ì2x + 2y - z = 4,
	7.31. a = (2y - 5x)i + (x -1)j+ (2 xy + 2z)k, S :
				í	= 0, y = 0, z = 0.
				îx

Задача 8. Найти поток векторного поля a через замкнутую поверхность S (нормал внешняя).

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

8.1. ìïx2 + y2 = 9,

S : íïîz = x, z = 0 (z ³ 0).

a= 2xi + zk,

8.2.ìïz = 3x2 + 2y2 +1,

S : í

ïîx2 + y2 = 4, z = 0.

207

a= 2xi + 2yj+ zk,

8.3.ìïy = x2 , y = 4x2 , y =1 (x ³ 0)

S : í

ïîz = y, z = 0.

a= (z + y)i + yj − xk,

8.5.	ì	2	+ y	2	= 2y,
S :	íx
	îy = 2.

a= 2(z − y) j+ (x − z)k,

8.7.ìïz = x2 + 3y2 +1, z = 0,

S : í

ïîx2 + y2 =1.

a= zi − 4yj + 2xk,

8.9. ìz = x2 + y2 ,

S : íîz =1.

a= xi − 2yj + xk,

8.11.ìx + y =1, x = 0, y = 0,

S : íîz = x2 + y2 , z = 0.

a= 6xi − 2yj− zk,

8.13.		ì		= 3 -			2(x	2	+ y	2	),
	S :	ïz
		í		= x2 + y2 (z ³ 0).
		ïz
		î
	a = ( y + 2z)i - yj+ 3xk,
8.15.		ì3z = 27 - 2(x2 + y2 ),
	S :	ï
		í	z	2 = x2 + y2 , (z ³ 0).
		ï
		î
	a = yi + 5yj + zk,
8.17.		ì		2	+ y	2	=1,
	S :	ïx
		í	z	= x, z = 0 (z ³ 0).
		ï
		î

a = 3xi − zj,

ì

2

2

8.4.

ïz = 6 - x

- y

,

S :

í

z2 = x2

+ y2 (z ³ 0).

ï

î

a = xi − (x + 2y) j + yk,

8.6.

S :

ì

2

+ y

2

=1, z = 0,

íx

îx + 2y + 3z = 6.

a = xi + zj− yk,

8.8.

ì

= 4 -

2(x

2

+ y

2

),

S :

ïz

í

= 2(x2 + y2 ).

ïz

î

a = 4xi − 2yj− zk,

8.10.

S :

ì3x + 2y =12, 3x + y = 6, y = 0,

í

îx + y + z = 6, z = 0.

a = zi + xj− zk,

8.12. ì4z = x2 + y2 ,

S : íîz = 4.

a = (z + y)i + (x − z) j+ zk,

8.14.	ì	2	+ 4y	2	= 4,
S :	íx
	î3x + 4y + z =12, z =1.

a = ( y + 6x)i + 5(x + z) j+ 4yk,

8.16.	ìy = x, y = 2x, y = 2,
S :	í	+ y2 , z = 0.
	îz = x2

a = zi + (3y − x) j − zk,

8.18. ìïx2 + y2 =1,

S : í

ïîz = x2 + y2 + 2, z = 0.

208

a = yi + (x + 2y) j + xk,

8.19.	ì	2	+ y	2	= 2x,
	ïx
S :	íz = x2 + y2 ,
	ïz = 0.
	î
a = (x + y + z)i + (2y − x) j + (3z + y)k,
8.20.	ìy = x, y = 2x, x =1,
	ï	= x2 + y2 ,
S :	íz
	ï	= 0.
	îz

a = 7xi + zj + (x − y + 5z)k,

	ì	2		2
	ïz = x		+ y		,
8.21.
S :	íz = x2 + 2y2 ,
	ïy = x, y = 2x, x =1.
	î

a= xi − 2yj+ 3zk,

8.23.ìx2 + y2 = z,

S : íîz = 2x.

a =17xi + 7 yj+11zk,

ìz = x2 + y2 ,

ï

8.22. S : íz = 2(x2 + y2 ),

ïy x2 , y x.

î = =

a = (2x + y)i + ( y + 2z)k,

8.24. S : ìïz = 2 - 4(x2 + y2 ),

íïz = 4(x2 + y2 ).

î

a = (2y − 3z)i + (3x + 2z) j+ (x + y + z)k,

	8.25.	ì	2	+ y	2	=1,
	S :	íx
		îz = 4 - x - y, z = 0.
	a = −2xi + zj+ (x + y)k,
	8.26.	ì	2	+ y	2	= 2y,
	S :	ïx
		í	= x2		+ y2 , z = 0.
		ïz
		î
	a = (2y -15x)i + (z - y) j - (x - 3y)k,
		ì			2			2
	8.27.	ïz = 3x				+ y				+1, z = 0,
	S :	í	2	+ y	2	=	1		.
		ïx					4
		î

209