. z = z(x, y); (x, y) G . , .. {x, y, z(x, y)│ (x, y) G } . . y = y(x, z) x = x(y, z).
x= (u, v), y= (u, v), z= (u, v), (u, v) , (1.18)
, , - . (u, v) (x, y, z) , . , : , (x, y) G.
, , , , , .
- S. , , M0. , , .
, S, (1.18), .
, A= ; B= ; C= . u, v.
. E= ; G= ; F= . A2+B2+C2=EG-F2. .
. . . .
S f(M)=f(x, y, z).
=
= . (2.18)
, .. z = z(x, y); (x, y) G, = , (3.18)
z x y .
17.1. , S - , = 0.
.
x=f(y, z), . . yz. , .
(3.18). = = .
dS x=1-y2-z2. .
G - yz. f(x, y, z) y, z, .. .
G: = = .
: , , ). : .
17.2. , S - .
. , , , : . G (x,y) S xy ( ). G . , x,y. : .
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17.3. , S x= ucosv, y=usinv, z=v, .
. , . (2.18), . = = = .
17.4. 5.
. x2+y2+z2 =52, . , R=5. , , , , . = = . .
. S M(x, y, z). M= . .
17.5. , =0, =4, .
. . , . , . dS . , . 16.
. S x= (u, v), y= (u, v), z= (u, v), (u, v) , (u, v), (u, v), (u, v) , . . . , M . . {A; B; C} . z=f(x; y; z), .
, S. - , , P(M), Q(M), R(M).
, , . , = . (4.18)
S z=f(x, y), (x, y) ( , ),
= . , , S x=x(y, z), (y,z) y=y(z, x), (z, x) .
18.6. , S - , z=0.
. S : , , , z=0, : . , , .
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. , , S1 - , , O , S2 - , , Ox, . S1 S2 yz, Oy . S1 S2 .
= = =0.
. ? = , S3 - . S3 Oy, S4 - , : , Oy . S3 S4 zx. Ox: .
= . . 8 .
. . . yz, zx. x, .. x y, z, yz . y . x, z.
, ?
, . , = . y=y(x, z), x=x, y=y(x, z), z=z. , . , .
. ? , : , . ? z=0, Ox Oy, , , = = = 0.
, . , - z =4- x 2- y 2, O z , - z = 0, O z . , ( . , , 8 .
,
= 16 .
. S ( -) , L. , , ( ..). L , .
P(x,y,z), Q(x,y,z), R(x,y,z) G, S. : = = .
, S, , - , . , , , xy. , ( z ) = .
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17.7. , L= , (0; 0; 2). S , .
. . S , .
(4.18). A, B, C. .
=
, , .
:
L ,
; . = .
.
-. D .
G= Oz. Ox Oz. , G, , .
G , , - S, P(x,y,z), Q(x,y,z), R(x,y,z) . = ,
.
17.8. , S- , z=0.
. : . =2x, = 1, =1. :
=
= = . . , , , z=0. .
= = .
: , , P(x, y, z), Q(x, y, z), R(x, y, z) .
. G R3. , G , u(M), .. G . .
(M), , G . (M). Oxyz, : u= u(x, y, z),
(M) = {P(x;y;z), Q(x;y;z), R(x;y;z)}.
(), () , .
u(x,y,z)
grad u= grad u= .
- , , M .
u M u(M) , () .
, . Oxyz . grad u= .
() (M) , div div = = .
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(M) rot , [ ] . rot .
G, AB - G. AB .
(A=B), .
- S G .
(M) , u(M): = grad u. u(M) (M). (M)= (P,Q,R), .
(M) G , div =0.
17.9. u=x2+2y2+3z2+4xy 3x 2y z M1(0, 0, 0), M2(-1, 2, 1). xOz M1(0, 0, 0), M2(-1, 2, 1).
. grad u= . (0, 0, 0)
grad u(M1)=(-3; -2; -1), (-1, 2, 1) grad u(M2)=(3; 2; 5).
() = .
. M1 .
, xOz . .
17. 10. ={x; y2; z3} M(-2, 4, 5).
. div = = 1+2y+3z2. div (M) = 84.
17. 11. ={x2; y2; z3}.
. rot =
=i(xz 3y 2z2) j(yz x2) +k(0 0).
, rot ={xz 3y 2z2; (yz x2); 0}.
:
17. 12. , , . : 7 /3.
17. 13. , , .
: ( + ln(1+ )).
17. 14. , 1) S- ; 2) S- .
: 1) ; 2) .
17. 15. 1) , 2) ,
S , .
: 1) ; 2) 0.
17. 16. , S , . : 128/3.
17. 17. , . : .
17. 18. - , S , . : 3abc/2.
17. 19. - , S: 1) , ; 2) . : 1) ; 2) .
18 . ,
1) .
2) ?
3) .
4) , .
5) ? .
6) ?
7) ? .
8) , .
9) , .
10) - , .
11) ?
12) , C x = t 2, (0 ≤ t ≤ 3).
13) : , C , , (0 ≤ t ≤ 2).
14) P =xy, Q=x2/2 1/(2-y), G={(x, y), x2 +y2 <4}. , Pdx+Qdy u(x, y), . , A(-1; -2), B(1; 1).
15) : , P(x,y)= -x2 y, Q(x,y)= xy2, C: .
.
1. . , , ,
.
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......................................................................... 74