Документ взят из кэша поисковой машины. Адрес оригинального документа : http://matematika.phys.msu.ru/files/a_stud_gen/247/matan_examm_09-10.pdf Дата изменения: Fri Dec 3 01:09:02 2010 Дата индексирования: Tue Oct 2 01:37:53 2012 Кодировка:

. ..



.. , .. , .. , .. .



(II )

-2010


1. Rm . 1. .
1.1. 1.2. 1.3. 1.4. 1.5. 1.6. 1.7. 1.8. 1.9. 1.10.

Rm . R m . Rm . D R m . D R m . D Rm . . R m . R m . D Rm .

1.11. Rm. 1.12. Rm. 1.13. Rm.

2. . : . 2.1. , . 2.2. , . 2.3. , , . 2.4. , R m . 2.5. , . 2.6. , . 2.7. , , ? 2.8. , . 2.9. , . 2.10. , . 2.11. , . 2.12. , . 2.13. , , , . 2.14. , . 2.15. , . 2.16. {(x , y ) : x 2 + y 2 < 1} .
2.17. {(x , y ) : x 2 + y 2 < 1} . . 3.1. , 3.2. , . 1

3.


3.3. , R m . 3.4. , . ? 3.5. , . ? 3.6. , . 3.7. cos , sin , n N . n n 3.8. cos , sin , n N . n n 3.9. , .

{ {

} }

1.1. 1.2. 1.3. 1.4. 1.5. 1.6. Rm

R m . Rm . Rm . R m . Rm . . 2. ( ). 2.1. Rm . 2.2. -. 3. . 3.1. , M n (x n , yn ) . 3.2. , M n (x n , yn ) , x n yn . 3.3. , x n yn , M n (x n , yn ) . 3.4. R m . 4. . 4.1. , Rm . 4.2. , x n yn , M n (x n , yn ) . 4.3. , x n yn , M n (x n , yn ) . 4.4. , , , . 4.5. M n cos , sin . n n 5. .

2. Rm . 1. .

2


5.1. M
x
n +1

n

(

x n , yn ) , x 1 = 8 ,

=

1 4 x n + , y n = x 2n , n N . 2 xn

3. , , . 1. .
1.1. u(M),

D Rm .
1.2. u(M), D R m . 1.3. u(M),

D R m .
1.4. u(M), D R m . 1.5. m D Rm . 1.6. m D R m . 1.7. " " u(M ) M 0 Rm . 1.8. " " u(M ) M 0 Rm . 1.9. " " u(M ) M . 1.10. " " u(M ) M . 1.11. u(x , y ) x M 0 (x 0, y 0 ) . 1.12. u(x , y ) M 0 (x 0, y 0 ) . 2. ( ). 2.1. u(M ) M 0 Rm . 2.2. . 2.3. . 2.4. . 2.5. . 2.6. . 2.7. . 2.8. . 2.9. . 3. . 3.1. . 3.2. . 3


3.3. . 3.4. . 3.5. . 3.6. . 3.7. . 3.8. . 4. . 4.1. " " , u(M ) M 0 . 4.2. " " , u(M ) M . 4.3. " " , u(M ) M 0 . 4.4. " " , u(M ) M . 4.5. " " , b u(M ) M 0 . 4.6. 4.6.1. u (x , y ) = xy . y 4.6.2. u ( x , y ) = . x y 4.6.3. u ( x , y ) = 2 . x 4.6.4. u ( x , y ) = x 2 + 2xy + y 2 .
x 2 + y2 . 4.6.5. u ( x , y ) = 2x x 2 + y2 4.6.6. u ( x , y ) = . 2x + 2y 2xy 4.6.7. u ( x , y ) = 2 . x + y2 4.7. , {(x , y ) : x 2 + y 2 1} .

4.8. , {(x , y ) : x 2 + y 2 > 1} . 4.9. , {(x , y ) : x 2 + y 2 > 1} . 4.10. , . 4.11. , . 4.12. , , , . 4.13. , , . 4


4.14. u(x , y ) M (x , y ) , : x 2 + y2 x 3 + y3 x 2 + y2 1 u (x, y ) = 4 ; u (x, y ) = 4 ; u (x, y ) = 2 ; u ( x , y ) = x y si n 2 ; 4 4 2 x + y2 x +y x +y x + xy + y 1 u ( x , y ) = ( x 2 + y 2 ) sin 2 . x + y2 5. . 5.1. . x 2 - y2 2 2 x 2 + y 2 , x + y 0, (0,0); 5.1.1. u ( x , y ) = 2 2 0, x +y = 0 x 3 + y 3 2 2 x 2 + y 2 , x + y 0, 5.1.2. u ( x , y ) = (0,0); 2 2 0, x +y = 0 2 2 e x +y - 1 , x 2 + y 2 0, 2 x + y2 5.1.3. u ( x , y ) = (0,0); 2 2 0, x +y = 0 sin ( xy ) , xy 0, 5.1.4. u ( x , y ) = xy (0,0) (0,1); 1, xy = 0 sin ( xy ) , x 0, (0,0), (1,0), (0,1); 5.1.5. u ( x , y ) = x 1, x =0 sin ( xy ) 2 2 x 2 + y 2 , x + y 0, 5.1.6. u ( x , y ) = (0,0). 2 2 1, x +y = 0 xy ln( xy ), xy 0, (0,0). 5.1.7. u ( x , y ) = 0, xy = 0 x ln( xy ), xy 0, 5.1.8. u ( x , y ) = (0,0). 0, xy = 0

4. .
1. . 1.1. M (x 1, x 2, ..., x m ) . 1.2. M (x 1, x 2, ..., x 1.3. f (x 1, ..., x m ) f (x 1, ..., x m ) x m). .
k

5


1.4. z = f ( x , y ) M 0 x 0 , y 0 , f ( x 0 , y

(

0

))

.

1.5. n . 1.6. u ( x 1, ..., x m ) . 1.7. n ­ u ( x 1, ..., x m ) . 1.8. f (x , y, z ) M (x 0 , y0 , z 0 ) . r 1.9. l = (cos , cos , cos ) f (x , y, z ) M (x 0 , y0 , z 0 ) . 2. ( ). 2.1. u(x , y ) . 2.2. o o o f (x 1, ..., x m ) M 0 (x 1, x 2, ..., x m ) . 2.3. uxy = uyx . 2.4. . 2.5. . 2.6. . 2.7. f (x , y, z ) . 2.8. f (x , y, z ) . 2.9. . ? 2.10. . n ­ 2.11. . 2.12. . 2.13. o o o f (x 1, ..., x m ) M 0 (x 1, x 2, ..., x m ) . 2.14. o o o f (x 1, ..., x m ) M 0 (x 1, x 2, ..., x m ) . 3. . 3.1. f (x 1, ..., x m ) o o o M 0 (x 1, x 2, ..., x m ) . 3.2. f (x 1, ..., x m ) o o o M 0 (x 1, x 2, ..., x m ) . 3.3. uxy = uyx . 3.4. . 3.5. . 3.6. , M (x 0 , y0 , z 0 ) f (x , y, z ) r r l = (cos , cos , cos ) l f M . 6


3.7. o o o f (x 1, ..., x m ) M 0 (x 1, x 2, ..., x m ) . 4. . 4.1. , u ( x , y ) ux ( x , y ) 1 , uy ( x , y ) 1 , M

N u(M ) - u(N ) < 3 . 4.2. " "? 4.3. " "? 4.4. f (x ) x 1 , g(x ) x 2 . , u(x , y ) = f (x ) g(y ) M = (x 1, x 2 ) . 4.5. f (x ) x 1 , g(x ) x 2 . , u(x , y ) = f (x ) + g(y ) M = (x 1, x 2 ) . 4.6. z = u(x , y ) , , M ( x , y ) , z = u(x , y ) M (x , y, u(x , y )) , . M 0 (x 0, y 0 ) . r L M 0 ( x 0 , y 0 ) . u r 4.6.1. u ( x , y ) = 2x + 3y, M 0 = (3; 2) , L = (3; -2) ; u r 4.6.2. u ( x , y ) = 8x 2 + 2y 2 - x 4 - y 4 , M 0 = (2; 1) , L = (-1; -1) ; u r 4.6.3. u ( x , y ) = xy(3 - x - y ) , M 0 = (1; 1) , L = (-1; -1) ; u r 4.6.4. u ( x , y ) = x 2y 3 (6 - 2x - 3y ) , M 0 = (1; 1) , L = (-1; -1) ; u r 4.6.5. u ( x , y ) = x 3 + y 3 - 3xy M 0 = (1; 1) , L = (-1; -1) ; u r u r y 4.6.6. u ( x , y ) = arctg , M 0 = ( 3, 1) , L1 = (1; - 3 ) , L 2 = ( 3; 1) ; x u r y 4.6.7. u ( x , y ) = x - y x , M 0 = (e;e) , M 1 = (1; 1) , L = (1; -1) ;

6 4.7. f (x , y, z ) , , . M 0 (x 0, y 0, z 0 ) . r L M 0 ( x 0, y 0, z 0 ) . r 4.7.1. u (x , y, z ) = x 3 + x + y + xyz , M 0 = (1; 1; 1) , L = (1, 1, 1) ; r 4.7.2. u (x , y, z ) = ln(xyz ), x > 0, y > 0, z > 0, M 0 = (1; 1; 1) , L = (1, 1, 1) ; r 4.7.3. u ( x , y, z ) = xyz (4 - x - y - z ) , M 0 = (1; 1; 1) , L = (1, 1, 1) ; r 4.7.4. u ( x , y, z ) = x 3y 4z 5 (13 - 3x - 4y - 5z ) , M 0 = (1; 1; 1) , L = (1, 1, 1) . r 4.7.5. u ( x , y, z ) = x 3 + x + y + xyz , M 0 = (1; 1; 1) , L = (1, 1, 1) .
4.8. u ( x , y, z ) = e
x 2 +y 2 + z
2

4.6.8. u ( x , y ) = x 3 - x 2y + y 3 - 1,

r L Ox .



3u , x 2 y

3u . x y z

4.9. u, f -- , x y -- : 4.9.1. u = f ( , ), 4.9.2. u = f ( , , ),

= x 2 + y 2, = x 2 - y 2 ;
= xy, = x - y, = x + y .
7


4.10. , , : 1 z 1 z z 2 2 4.10.1. + = 2 , z = y (x - y ) ; x x y y y

z + x 2 2z 4.10.3. a x 2
4.10.2. x

y

y z = xy + z , z = xy + x ; x y 2 z - 2 = 0 , z = (x - ay ) + (x + ay ) . y

4.11. n M0 : y 4.11.1. u ( x , y ) = arctg , M 0 ( 2, 3 ) , n = 2 ; x 4.11.2. u = x y , M 0 (e, e ) , n = 2 ; 4.11.3. u = e x sin y, M 0 ( 0, 0 ) , n = 3 ; 4.11.4. u = ln(1 + x + y ), M 0 ( 0, 0 ) , n = 3 ; 4.11.5. u (x , y, z ) = x 3 + x + y + xyz , M 0 (x 0 , y0 ) , n = 3 . 5. . 5.1. u = f (x , y ) , d 2u M 0 ( x 0 , y 0 ) . , N 0 ( x 0 , y 0 , f ( x 0 , y 0 ) ) u = f (x , y )

N 0 .
5.2. u ( x, y ) (0,0)? , (0,0).

u ( x, y ) = 3 xy ( x + y ) ; u ( x, y ) = 3 xy ( x 2 + y
2

u ( x, y ) = 3 x 2y ;

u ( x, y ) = 3 xy ;

u ( x, y ) = 3 x 3 + y 3 ;

) )

; u ( x, y ) = 3 x 4 - y 4 ; u ( x, y ) = 3 x 5 - y 5 ; u ( x, y ) = 3 yx 4 + xy 4 .

5.3. u ( x, y ) (0,0)?

u ( x, y ) = 3 xy ; u ( x, y ) = 3 x 2y ; u ( x, y ) = xy 3 xy ; u ( x, y ) = 3 x 3 + y 3 ; u ( x, y ) = 3 x 4 - y 4 ; u ( x, y ) = 3 xy ( x 2 + y u ( 0, 0) = 0 ;
2

; u ( x, y ) = 3 xy ( x + y ) ;

u ( x, y ) = xy ln ( x 2 + y 2 ) , x 2 + y 2 > 0 ,

1 u ( x, y ) = xy sin 2 , x 2 + y 2 > 0 , u ( 0, 0) = 0 ; 2 x +y 1 u ( x, y ) = xy 3 x 3 + y 3 sin 2 , x 2 + y 2 > 0 , u ( 0, 0) = 0 . 2 x +y 5.4. u ( x, y ) M 0 ( x 0, y

0

)



N0(x0,y0,u(x0,y0)) . , , . 5.5. , N0(x0,y0,u(x0,y0)) z = u ( x, y ) N0 8


. d 2u M0(x0,y0) ? 5.6. , z = u ( x, y ) M 0 ( x 0, y0 ) u ( x, y ) 5.7. u(x,y) M 0 ( x 0 , y

M0 .

0

)



R3(x, y ) = u(x, y ) - P2(x, y ) ­ , P2 ( x , y ) ­ . , R3 ( x, y ) M 0 .
5.8. u ( x, y ) , M0(x0,y0) u (M
2 0

)

= 0, du
0

M

0

= 0, d 2u

M

= 0.
0

, u ( x , y ) = o (

)

0 , =

(

x -x

0

)

2

+ (y - y

)

2

.

5. .
1. 1.1. 2. 2.1. . . ( ). M 0 ( x 0 , y 0 )

u ( x , y ) , . 2.2. M 0 ( x 0 , y 0 ) u ( x, y ) . 3. . 3.1. . 3.2. . 4. . 4.1. u ( x , y ) v ( x , y ) M 0 ( x 0 , y 0 ) . , u ( x , y ) + v(x , y ) . 4.2. u ( x, y ) v ( x , y ) , M 0 ( x 0 , y 0 ) , u ( x , y ) v(x , y ) . 4.3. u ( x, y ) v ( x , y ) , M 0 ( x 0 , y 0 ) , u ( x , y ) v(x , y ) . 4.4. u(x , y ) = f (x ) g(y ) M (x1, x 2 ) , f (x ) x 1 , f (x 1 ) 0 , g(x ) x 2 , g(x 2 ) 0 . , f ' (x 1 ) = 0 , g ' (x 2 ) = 0 . 4.5. f (x ) x 1 , f (x 1 ) > 0 , g(x ) x 2 , g(x 2 ) > 0 . , u(x , y ) = f (x ) g(y ) M (x1, x 2 ) . 4.6. :

9


u ( x , y ) = x 2 + xy + y 2 ; u ( x , y ) = x 3 + y 3 - 3xy ; u ( x , y ) = xy +

88 +; xy

u ( x , y ) = ( 5 - 2x + y ) e

x 2 -y

; u ( x , y, z ) = x 2 + y 2 - z 2 ; u (x , y, z ) = xy + xz + yz ;

u (x , y, z ) = xyz (4 - x - y - z ) ; u ( x , y, z ) = x 2 + y 2 + z 2 - 2xy - 2xz - 2yz ; y2 z 2 2 u ( x , y, z ) = x + + +. 4x y z

4.7. u = x cos y + z cos x M ;0; 1 .
2
4.8. : 1 u = xy - x 2y - y 2x , 0 x 1, 0 y 2 . 2 5. . 5.1. , d 2u ( M 0 ) - , u M0.
du = 0, d 2u

5.2. , M 0 ( x 0 , y
M
0

0

)

u(x , y ) ,

M

0

= 0, d 3u

M

0, u M 0 .
0

5.3. f (x ) x 1 , f ' (x 1 ) = 0 , g(x ) x 2 , g ' (x 2 ) = 0 , f (x 1 )g(x 2 )f '' (x 1 )g '' (x 2 ) > 0 . , u(x , y ) = f (x ) g(y ) M (x1, x 2 ) . 5.4. f (x ) x 1 , f (x 1 ) > 0 , g(x ) x 2 , g(x 2 ) > 0 . , u(x , y ) = f (x ) g(y ) M (x1, x 2 ) . 5.5. u ( x , y ) M 0 ( x 0 , y 0 ) , x = (t, s ) y = (t, s ) K 0 (s 0, t0 ) , x 0 = (t0, s 0 ) y 0 = (t0, s 0 ) . , u ((t, s ), (t, s )) K 0 . 5.6. x = (t, s ) y = (t, s ) u K 0 (s 0, t0 ) , M 0 ( x 0 , y 0 ) u ( x , y ) , (M 0 ) > 0 x u (M 0 ) > 0 , x 0 = (t0, s 0 ) y 0 = (t0, s 0 ) . , y u ((t, s ), (t, s )) K 0 .

6. . 1. .

1.1. f1 ( x 2 , ..., x n ) ,..., fk ( x 2 , ..., x n ) . 1.2. f1 ( x 2 , ..., x n ) ,..., fk ( x 2 , ..., x n ) .

( ). 2.1. y = f ( x ) , F ( x , y ) = 0 . 2.2. y = f ( x ) , F ( x , y ) = 0 . 10

2.


2.3. z = f ( x , y ) , F ( x , y, z ) = 0 . 2.4. z = f ( x , y ) , F ( x , y, z ) = 0 . 2.5. y = f (x ) , F ( x , y, z ) = 0, z = g(x ) , G ( x , y, z ) = 0. 2.6. . 2.7. . 3. . 3.1. y = f ( x ) , F ( x , y ) = 0 . 3.2. y = f ( x ) , F ( x , y ) = 0 . 3.3. z = f ( x , y ) , F ( x , y, z ) = 0 . 3.4. y = f (x ) , z = g(x ) , F ( x , y, z ) = 0, G ( x , y, z ) = 0. 3.5. . . 4.1. , x 2 + xy y = y(x ) . 4.2. , xy + ln y = y(x ) . 4.3. y = u ( x ) , z = v

4.

+ y 2 = 3 (1; 1)

( xy ) = 1 (2; 0.5)
(x )



f ( x , y, z ) = 0, g ( x , y, z ) = 0 . u(x ) .
F ( x , y ) = u, 4.4. x = f (u, v ) , y = g (u, v ) G ( x , y ) = v. x . v F ( x , y, z ) = 0, 4.5. y = f ( x ) , z = g ( x ) . G ( x , y, z ) = 0. dz . dx 4.6. , z ( x , y ) ,

F ( z 2 - y 2, x 2 + (y - z )2 ) = 0 , F ­ , z z (z - y )2 + xz = xy . x y
11


4.7. , z (x,y),

z F x 2 + y 2, = 0 , F ­ , x z z xy - x2 = yz . x y
4.8. , , y = f (x ) , F ( x , y ) = 0 . 4.8.1. F ( x , y ) = x 3 + y 3 - 3xy = 0 ; 4.8.2. F ( x , y ) = 8x 2y - x 4 - y 4 = 0, x > 0, y > 0 ; 4.8.3. F ( x , y ) = y 2 - ay - sin x = 0, 0 x 2 . 4.9. z = z (x , y ), 4.9.1. xyz = x 2 + y 2 + z 2 ; 4.9.2. z cos x + y cos z + x cos y = 3 ; 4.9.3. x 2 + zx + z 2 + y = 0 . 4.10. (x 0 , y 0 , z 0 ) z = z (x , y ) . z = z (x , y ) (x 0 , y0 ) . z z z 2z 4.10.1. arctg = z + x + y ; , , ; x x y x 2 z z 2z 4.10.2. ln(xy + yz ) = z 2 + x 2 + y 2 - 2 , , , . x y x y 4.11. u(x,y) v(x,y), xu + yv = 1, x + y + u + v = 0. 4.12. , - , :

xz

z z y + yz = xy , z 2 = xy + . x y x

4.13. , . 4.13.1. y 2 + (x 2 - xy )

dy = 0, y = tx , y = y(t ) ; dx d 2y dy 4.13.2. x 2 2 + 3x + y = 0, x = et , y = y(t ) . dx dx
2u u u + + = -u, u = ve x y x y
- x -y

4.14. v v(x,y), .

4.15. u v , w u v, 4.15.1.

z z + = 4x , u = x , v = x - y, x x

w = x -y +z;

12


4.15.2.

z 1 2z 1 y + x 2 = , u = , v = y, x 2 x y x

w = zy - x .

4.16. u v ,

2z 2z 1 z + y2 2 + = 0, 2 y x 2 y

u = x , v = 2 y , (y > 0) .

. 5.1. du dv , u = f (x , y ) , v = g(x , y ) , F ( x , y, u, v ) = 0, G ( x , y, u, v ) = 0. . 5.2. du dv , u = f (x , y ) , v = g(x , y ) , x = F (u, v ) , y = G (u, v ) . .

5.

7. .
1. . 1.1. u ( x , y ) f ( x , y ) = 0 . 1.2. u ( x , y, z ) f ( x , y, z ) = 0 . 1.3. u ( x , y, z ) f ( x , y, z ) = 0 , g ( x , y, z ) = 0 . 2. ( ). 2.1. u ( x, y ) f ( x , y ) = 0 . 2.2. u ( x, y ) f ( x , y ) = 0 . 2.3. u ( x , y, z ) f ( x , y, z ) = 0 . 2.4. u ( x , y, z ) f ( x , y, z ) = 0 . 2.5. u ( x , y, z ) f ( x , y, z ) = 0 , g ( x , y, z ) = 0 . 2.6. u ( x , y, z ) f ( x , y, z ) = 0 , g ( x , y, z ) = 0 . 3. . 3.1. u ( x, y ) f ( x , y ) = 0 . 3.2. u ( x , y, z ) f ( x , y, z ) = 0 . 13


3.3. u ( x , y, z ) f ( x , y, z ) = 0 , g ( x , y, z ) = 0 . 4. . 4.1. , u . 4.1.1. u ( x , y ) = x 2 + y 2 x + y = 2 ; 4.1.2. u ( x , y ) = x + y x 2 + y 2 = 2 ; 4.1.3. u ( x , y ) = x + y xy = 1 ; 4.1.4. u ( x , y ) = xy x 3 + y 3 - 2xy = 0 ; 4.1.5. u ( x , y, z ) = x + y + z xyz = 1 ; 4.1.6. u ( x , y, z ) = x 2y 3z 4 2x + 3y + 4z = 9 ; 4.1.7. u ( x , y, z ) = xyz x 2 + y 2 + z 2 = 1, x + y + z = 0 . 5. . 5.1. N 0 ( x 0 , y 0 , ) ( ) u ( x , y ) f (x , y ) = 0 gradu (x 0 , y 0 ) 0 , gradf (x 0 , y 0 ) 0 . , M 0 (x 0 , y 0 ) u ( x, y ) f (x , y ) . 5.2. N 0 ( x 0 , y 0 , ) ( ) u ( x , y ) ax + by = c d 2u
M
0

> 0 , M 0 ( x 0 , y 0 ) . ,

M 0 ( x 0 , y 0 ) . 5.3. N 0 ( x 0 , y 0 , ) ( ) u(x , y ) = ax + by f ( x , y ) = 0 d 2 f , M 0 ( x 0 , y .
0 M

)

0

> 0 , M 0 (x 0, y0 ) .



8. . 1. .

1.1. . 1.2. . 2. ( ). 2.1. . 2.2. . 3. . 3.1. . 3.2. . 4. . 4.1. . .


0

1

dy xydx ;
y

1




0

sin x

dx


0

2ydy ;


dx
0 x

1

x

2ydy ;

2


0

1

/2

dx

arcsin x



cos ydy .

14


4.2. 4.2.1. D = {(x , y ) : x + y 1} ;


D

f (x , y )dxdy :

4.2.2. D = (x , y ) : y 2 x + 2, y x . 4.3. 4.3.1. ( x 2 + y 2 ) dxdy, D = {x 2 + y 2 6} ;
G

{

}

4.3.2.

(
G

x 2 - y 2 ) dxdy, D = {1 x 2 + y 2 4} I

{6

arctg

y Ix > 0. x4

}

4.4. (u, v ) ( x , y ) , D (x,y), y 2 = 16x , y 2 = 9x , x = 2y, x = 4y , (u,v). D, . 4.5. (u, v ) ( x , y ) , D (x,y), xe y = 1, xe y = 2, x = e y , x = 2e y , (u, v ) . D, . 4.6. m = dxdy , M x = ydxdy , M y = xdxdy
G

G

G

I x =


G

y dxdy , I y =
2


G

x dxdy = 1 ,
2

4.6.1. 0 x 2, 0 y x ;

4.6.2. 0 x 4, 0 y x (4 - x ); 4.6.3. 0 x , 0 y sin x ; 4.6.4. 10
-3

x 1, 0 y x -1 .
1 3 - 2y

4.7. (x,y) D, :


D

f ( x , y ) dx dy = dy
-1


y

f ( x , y ) dx . .

4.8. (x,y) D, :


D

f ( x , y ) dx dy = dy
-1

1

1-y

-2y



f ( x , y ) dx .

f(x,y) = y. 4.9. , x = 1, x = 2, y = 0, y = x ;
1 . 4.10. , y = cos x , y = sin x ( / 4 x 5 / 4 ) ; 1 . 4.11. Oy , x = 0, x = 1, y = 0, y = arcsin x ; ( x ) 1 . 15


4.12.


G

(x 2 + y 2 )dxdydz , G

x + y = 2z, z = 2.
2 2

4.13.


G

f ( x , y, z ) dxdydz , G - ,

x = 0, y = 0, z = 0, x + y + z = 2 . 4.14. x 2 y2 z 2 + + = 1, z = 0, ( z 0 ) . ( 1 ), a 2 b2 c2 4.15. ( x , y, z ) = x 2 + y 2 + z 2 , x 2 + y 2 + z 2 = 4, x 2 + y 2 = z 2 ( z 0 ) . 4.16. G ­ , x 2 + y 2 + z 2 = 4, z = 1 ( z 1) . m0 , .

9. . 1. .

1.1. I f ( x , y ) . 1.2. II P (x , y ) dx .
AB
AB

1.3. II



Q ( x , y ) dy .

2. ( ). 2.1. f (x, y ) dl L.
L

2.2. P (x, y ) dx .
AB

2.3. Q (x, y ) dy .
AB

2.4. . . 3.1. . 3.2. . 3.3. . 3.4. , P ( x , y ) dx + Q ( x , y ) dy . 3.5. P ( x , y ) Q ( x , y ) , P Q P (x, y ) dx + Q (x, y ) dy . , y = x . AB 16

3.


3.6.

P ( x , y ) Q ( x , y ) , P ( x , y ) dx + Q ( x , y ) dy
y x

. , P = Q . 3.7. . 4. . 4.1.


L

f (x , y ) dl .

4.2. , y = f (x ), a x b , . 4.3. , , . 4.4. L , : x = x (t ) , y = y(t ) , a t b ; (t ) . 4.5. L , y = y(x ) , a x b ; (x ) . 4.6. x ­ L , y = y(x ) , a x b . . 4.7. y ­ L , y = y(x ) , a x b ; . 4.8. x ­ L , : x = x (t ) , y = y(t ) , a t b ; . 4.9. y ­ L , : x = x (t ) , y = y(t ) , a t b ; . 4.10. Ox L , : x = x (t ) , y = y(t ) , a t b ; 1. 4.11. Oy L , : x = x (t ) , y = y(t ) , a t b ; 1. 4.12. Ox L , y = f (x ) ,

a x b ; 1. 4.13. Oy L , y = f (x ) ,

a x b ; 1.
4.14. 4.15.
AB



P (x , y ) dx . Q (x , y ) dy .

AB

17


4.16. n ­ L .

L

[x

cos ( n, x ) + y cos ( n, y )] ds , L ­ ,

4.17. , L ­ l ­ ,

L



cos ( l, n ) ds = 0 .

4.18. G ­ L. , G f ( x , y ) dx .
L

4.19. G ­ L S. x ­ G


L

f ( x , y ) dx , 1.

4.20. G ­ L S. y ­ G


L

f ( x , y ) dy , 1.

4.21. D ­ L. D r F ( x , y ) = ( P ( x , y ) ;Q ( x , y ) ) L , D. 4.22. t2 4.22.1. 1ds , L ­ x = t, y = , 0 t 1; 2 L 4.22.2. yds , L ­ y = e x , 0 x 2;
L

4.22.3. xydl, L ­ x + y = 1, x - y = -1, -1 x 1, 0 y 1 .
L

4.22.4. x 2ydl, L = (x , y ) : x = 4 cos t, y = sin 2t, 0 t
L

{


2

}

.

2 x x , 0 x 3. 3 2 4.24. y = x x , 0 x 3 (x ) = 2 1 + x . 3
4.23. y = 4.25. x - x = cos t , y = sin t , 0 t



2

,

. 4.26. Ox x = cos t , y = sin t , 0 t , 1 . 4.27. Ox x = cos t , y = sin t , 0 t ; (t ) = sin t . 4.28. m M R; . 4.29. : 4.29.1. xdx + ydy , AB y = x 2 , A ( 0, 0 ) , B (1, 1) .
AB

18


4.29.2. (2 - y )dx + xdy , L x = t - sin t, y = 1 - cos t, 0 t 2
L

t. 4.29.3.
L



xdy + 2ydx , L y = x,
0
y = 0, y = 1 - x 2 ,
L

4.29.4. xydx - x 3y 3dy , L ­ , x - y + x + y = 1 . 4.29.5. ydx + zdy + xdz , L ­ x = cos t, y = sin t, z = t , 0 t 2 ,
L

t. 4.30. , : 4.30.1. x = a sin t, y = b cos t, 0 t 2 , a > 0, b > 0 ; 4.30.2. (x + y )2 = 2ax
2 2 2

(a > 0 ) Ox .

4.30.3. x 3 + y 3 = a 3 . 4.31. , 2x 2 + y 2 - z 2 = -1 z = x + 1 . 4.32. F = {x - y,2x + y 2 } x = y 2 , A(1, -1) B(1, 1). 4.33. F = {y, x } , 3x 2 + y 2 + z 2 = 4 z = x - 2 , , (0,0,-3). 4. . x2 4.14. l (t ) L , y = , 2

0 x t . lim l (t ) . 2
t

4.15. u (x , y ) , v ( x , y ) G, L. , u v u v u : u v dl = dxdy ( ), u v n L G n n 2u 2u L, u = 2 + 2 , x y . 4.16. I = (x cos + y cos ) dl , L ­ ,
L

t

S; - n L M (x , y ) Ox Oy. 4.17. , u (x , y ) G ,

19


u 2 u 2 u x + y dxdy = - u udxdy + u n dl , L ­ , G G L u G, - L. n 1 4.18. , lim ( F n ) dl , S ­ , d (S ) 0 S L
L, (x 0 , y0 ) , d (S ) - S, n ­ L F {x , y } - , S .

10. . .
1.

1.1. 1.2. 1.3. 2. 2.1.

. . . . ( ). , z = h (x , y ), (x , y ) D ,

. 2.2. , , . 2.3. f (x, y, z )d , S z = h (x, y ), (x, y ) G, G ­
S

(x,y). 2.4. f (x, y, z )d , S .
S

2.5.


S

f (x , y, z ) cos d , S z = h (x , y ), (x , y ) G, G ­

(x,y), - Oz . 2.6. f (x, y, z ) cos d , S , -
S

Ox . 2.7. Pdydz + Qdxdz + Rdxdy , S .
S

3. . 3.1. , z = h (x , y ), (x , y ) D .

20


3.2.

, f(x,y,z) S ,




S

f (x , y, z )d . S

. 3.3. , P(x,y,z) S ,


S

P (x , y, z ) cos d . S

. 3.4. . 4. .

4.1. S : 4.1.1. S : z = x 2 + y 2 ; M (3, 4, 25) . 4.1.2. S : x 2 + y 2 + z 2 + xyz = 2 ; M (1, 1, 0) . 4.1.3. S : x = 2uv , y = u + v , z = u 2 + v 2 , M (x (u 0 , v 0 ), y(u 0 , v 0 ), z (u 0 , v 0 )), u 0 = 1, v 0 = -1. 4.2. : 4.2.1. z = 3x + 4y , x 2 + y 2 1 . 4.2.2. z = x 2 + y 2 , x 2 + y 2 1 . 4.2.3. 2z = x 2 + y 2 , x 2 + y 2 1 . 4.2.4. z = xy , x 2 + y 2 a 2 .

1 . 2 4.2.6. x = u cos v,y = u sin v,z = v, 0 u a, 0 v 2.
4.2.5. x 2 + y 2 + z 2 = 1, z 0 , x 2 + y 2 4.3. I . 4.3.1. dS , S : x + y + z = 1 , x [-1; 1] , y [-1; 1] .
S

4.3.2. 4.3.3. 4.3.4. 4.3.5.


S

S
S

(x + y + z )dS , S : x + y + z = 1 , x [-1; 1] , y [-1; 1] .
(x + y + z )dS , S : x 2 + y 2 + z 2 = 1 z 0 .

(x 2 + y 2 )ds , S ­ V ={(x, y, z ): x 2 + y 2 z 1} .

1 (x 2 + y 2 + z - )ds , S ­ 2z = 2 - x 2 - y 2 , z 0 . 2 S 4.4. 2 2 x + y + z 2 = R 2 , x 0, y 0, z 0 .



4.5. : 4.5.1. xdydz + ydzdx + zdxdy , S ­ x + y + z = 1 , x [-1; 1] ,
S

y [-1; 1] , Oz .
4.5.2.


S
2

(y 2 + z 2 )dxdy , S -

z = a - x2 , 0 y b .
21


4.5.3.


S
2

(x 2 + y 2 + z 2 )dxdy , S -

z = x + y 2 , 0 z c ( Oz ).
4.5.4.


S

x 2dydz + y 2dzdx + z 2dxdy , S -

x 2 + y 2 - z 2 = 1, 0 z 3 .
4.5.5.



S

x 2dydz + y 2dzdx + z 2dxdy , S ­ x 2 + y 2 + z 2 = 1 .

5. 5.1.

. , S ­ , , l ­

, n - S, ­ l n ,


S

cos dS = 0 .

2

5.2.


V
2

d d d 1 = r 2
2


S

cos ds , S ­,

V, r = ( - x ) + ( - y ) + ( - z ) , r -- -, (x,y,z), V, (,,), -- r n S . 5.3. , S ­ , V, u(x,y,z) u u D , ds = udxdydz , n n S V

- S, =

2 2 2 + 2+ 2 ­ x 2 y z

. 5.4. , S ­ , V u(x,y,z)

D , u 2 u 2 u 2 u u u n ds = x + y + z dxdydz + uudxdydz , n - S V V
S, - .

11. .
. 1.1. , () . 1.2. . 1.3. . 1.4. . 2. ( ). 2.1. , n. 22


2.2. . 2.3. , y = f(x). 2.4. y = f(x). 2.5. , . 3. . 3.1. , n. 3.2. . 3.3. , y = f (x ) . 3.4. , . 4. . 4.1. Ox : y = 1 - cos x ;

x2 y = e x - 1 + x + ; y = tg x - s in x . 2 4.2. a, b y = ax 2 + bx + c y = e x x = x 0 ? 4.3. ( ­ ): a (a = const ) ; y = Cx - ln C ; 2C 2 (y - Cx ) = 1 ; y 2 = 2Cx + C 2 . y = Cx + C 4.4. y 2 = 2px (x 0 , 2px 0 ) .
. 5.1. , . 5.2. , . 5.3. : x 2 y2 x 2 y2 2 2 + 2 = 1; 2 - 2 = 1. ( x - x 0 ) + (y - y 0 ) = R 2 ; a2 b a b

5.

23