}, y1(x) y2(x) [a, b]. G Oy.
f(x,y) G.
G , x [a, b] , ( ), , (1.13)
.. .
, Ox, ..
G={(x, y) }, x1(y) x2(y) [c;d],
(2.13)
12.1. .
. x xy . x y y=0 ( Ox) y= ( ).
.1
12.2. .
. : = = = = = = .
= = . : . = = = = = .
12.3. , D: x=1, y=x2, y= .
. .
. 2
x [0, 1] y x2, .. G={(x, y) }. (1.13)
= .
, y, -:
= = = = .
x:
= =
= = .
12.4. : .
. y= , y= x=0 (. 3).
. 3
. , G={(x, y) }, x1(y) x2(y) [c, d].
y= x2 +y2 =1. x y:
x= . ,
x= . y= - (1, 0). x, x= . x= , x= - . . y 0 ½ x G1 x= x= . G2 y ½ 1 x x=0 ( Oy) x= ( x ).
= + .
. . - x y u v x = x(u, v), y=y(u, v), (u, v) . (x, y) G (u, v) , . , .. - . , , x = x(u, v), y=y(u, v), (u, v) xu, xv, yu, yv, J , ( ) 0 .
|
|
= .
, .
, .. u, v , - (x, y) , - , - (x, y) Ox.
12.5. , D: y x, .
. . 4.
.4
, x y : . , -, :
= = .
, .
, y=x : . , y=x , .
, 0 3, . 0 3 .
, = = = .
12.6. , , D: .
. 5.
.5
. , , -, , .. , -, . : =2cos . . ,
, .
.
, -:
= = = = = .
, .
D (x, y) . , , .
12.7. D, : 4y=x2 -4x, x=y+3.
. (2, -1), y=x-3. .
. .
D Ox Oy, , ,
{(x, y) }.
= = = = = = = .
12.8. D, ( ): , x2 +y2=a2, ().
.
, . . a. , . , D .
|
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x y. , , , . , , , , . Ox Oy. , (x, y) , (-x, y), (x, - y), (-x, -y) . , , . . , . . , . , . . . . , , .
= = = = = = .
:
12.9. G :
1) G (2; 1), (6, 2), (4; 6);
2) G (1; 1), (4; 4), (6; 4), (7; 1);
3) G , y=-4x+5; y=x2;
4) G , x=0, x=1, x=y2, y=ex;
5) G , ;
12.10. :
1) ;
2) ;
3) ;
4) ;
5) .
12. 11. , :
1) ;
2) . : 1) 2; 2) 8.
12. 12. :
1) , G x=-1, x=1, y=x, y=2x;
2) , .
: 1) 2ch1-2; 2) 135/4.
12.12. , :
1) ;
2) .
3)
13 . . .
f(x, y, z)
G={(x, y, z) , }, y1(x) y2(x) [a, b],
G1={(x, y) }, G xy.
G Oz, G1 Oy.
. (1.14)
13.1. , (z, y, x), .
. R3:
, .
, xy K a, {(x, y, z) }. (z, y, x) (z, y). , , . (y, z) x .
.
13.2. (x, y, z) ( ), G z=2(x2+y2) z=1+x2+y2.
. z=2(x2+y2) z=1+x2+y2 . , G xy, z . x2+y2=1, .. G K. G : {(x, y, z) }, = .
. x,y,z u,v,w x = x(u, v,w), y=y(u, v,w), z=z(u, v,w), (u, v, w) .
|
|
= , - G.
. .
. M(x,y,z) R3, M- xy. M (, z)
- M xy, z M. (, z) M. (x,y,z) (,z) , z=z. (2.14).
.
.
. M(x,y,z) R3, M- xy. M () r - M , - , OM O M, - M xy. , , z>0, , z<0.
() M. (x,y,z) () . (3.14)
.
.
13.3. , G z=0 ( ).
. z. z=1 . , , , z=1 ( z=1 ).
G={(x, y, z) }.
, , . G 0 2 , 0 1, () z 0 ( z xy) 1 ( z ). f(x,y,z)= , z=1 . .
= = .
13.4. , , f(x,y,z)= , G = .
. 1 2 . (3.14) , G. r =1 r =2 . , , G, . f(x,y,z)= . .
= = .
13.5. V, :
. V . V z=0, z=4-(x2+y2). V. :
,
D xy, V. , D={(x,y,0) }.
= =
. , .
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G . xy, ( ). :
a) m = - ;
b) Mx = , My = - Ox Oy;
c) - .
V R3 , :
d) m = - ;
e) Myz = , Mzx = ,
Mxy = - yz, zx xy;
f) - .
g) - V (x0, y0, z0). r - M(x, y, z) M0(x0, y0, z0), - .
13.6. V , x2 =4z, y2 =4x, x=1, z=0.
. f) V.
m= = = .
, = = .
.
13.7. T R A, d (d>R).
. , x2+y2+z2 =R2. A , Oz, (0,0,d). g) :
.
, , . , , = . , , - m.