(1) , . , . .
1. (2) : .
2. f , (1) (2) , Ai, , .
2. , .
. , (1) , , , .
3. .
, , , . .
3. f , f .
. u=f(x,y). , (x0,y0) . , , x . ,
. , .
(x0,y0), , , α β . , : . f (x0,y0). .
4.
, (3)
4. (3) , u=f(x1,x2,,xm) , ,
. . (3) . , , u=f(x1,x2,,xm) . f , . (3)
, :
, . , . , . , .
, , , . .
. , . .
5. . .
. u=f(x1,x2,,xm) , .
1, .
. . , . (4)
|
|
, (4) , x1,x2,,xm t1,t2,,tk. .
4,
. u v .
, , . u v. , u v - .
. .
6. . .
. M0(x0,y0,z0) .
, M0(x0,y0,z0). .
, , M(x,y,z), , l M0M. , . , :
l .
. l, l=0, u=f(x,y,z) M0(x0,y0,z0) , .
4, . (5)
. u=f(x,y,z) M0(x0,y0,z0) . grad u.
, . : . , φ , , . , , .
, :
1. u=f(x,y,z) (x0,y0,z0) , , ;
2. , , , .
u=f(x,y,z). , f(x,y,z)=c=const, .
(x0,y0,z0) f(x,y,z)=c , , .
m , .
. .
7.
u=f(x1,x2,,xm), G, . , x1,,xm, G. xk , f xi, xk : . , .
|
|
, (n-1)- . (n-1)- , n- f , .. .
, .
, n- .
.
.
. n=2,3, n- , (n-1)- (n-1) .
, k, , k (n-k) . 0- , .
, f n- , n- .
n- 3 .
1. u=f(x,y) 2- M0(x0,y0). .
. f 2- M0(x0,y0), - M0 .
, h , M0(x0+h,y0+h) - M0. x. , , M0, , - . , - .
, , : , - . . , . .
.
2. M0(x0,y0) f M0. .
. . y [y0,y0+h], : , . M0(x0,y0) , . , , , . .
.
3. u=f(x1,x2,,xm) n- . n- .
. .
.
. 1 . . .
, n- , - , : .
8.
, u=f(x1,x2,,xm) . . .
|
|
x1,x2,,xm; f 2- M(x1,,xm), .
. , , f ( ) .
, .
dnu n .
, dn-1u n-1 f n- M(x1,,xm), x1,,xm , n- t1,,tk.
.
.
1) x1,,xm . , M(x1,,xm).
,
( , 2- ).
.
. (*)
2) x1,,xm 2- t1,,tk.
(**)
(*) (**), , . .
. , 2- u=f(x1,x2,,xm) (*), x1,x2,,xm.
x1,x2,,xm
t1,,tk: , - .
, . , n- (*).
9.
u=f(x,y) n- (x,y), , .
(x,y) .
, [0,1] ( 4 ).
: . ,
u=f(x1,x2,,xm) :
.
4. u=f(x1,x2,,xm) n- , . (2)
(3)
(2) (3) .
. . .
, : .
(1)
, .
1. , .. , , (2) (4)
, , .
2. : , f
10.
5. u=f(x1,x2,,xm) n- . , (5), . n f x0 (n-1)- x0.
|
|
. ,
.
n=0 n=1 1- f
, i=n-1. i=n.
. , k=1,,n x2,x3,,xm
x2,x3,,xm , f . , n-1, , i=2,,m , 1
- , . , .
. (5) , , .