: (, ) - z, z .
z = f(x, y)
: (, ) z, , , .
: z (, ), z .
: 0(0, 0) r (, ), .
: f(x, y) (, ) 0(0, 0), e > 0 r >0, (, ),
.
:
: 0(0, 0) f(x, y). z = f(x, y) 0(0, 0),
(1)
(, ) 0(0, 0) .
(1) , f(x, y). :
1) z = f(x, y) 0(0, 0).
2) .
3) , f(x0, y0).
. f(x, y, )
D,
N(x0, y0, ), ,
f(x0, y0, ) ³ f(x, y, )
N1(x01, y01, ), ,
f(x01, y01, ) £ f(x, y, )
f(x0, y0, ) = M , f(x01, y01, ) = m f(x, y, ) D.
D .
. f(x, y, ) D, M m , m Î [m, M]
N0(x0, y0, ) , f(x0, y0, ) = m.
, D M m. , M m , D .
. f(x, y, ), D, , , .
. f(x, y, ) D, , .. e D > 0, (1, y1) (2, 2) , , D,
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, .
.
. z = f(x, y). (, ) D . Dxz = f(x + Dx, y) f(x, y) .
.
z = f(x, y) .
:
.
( ) , N0(x0, y0, z0) = 0.
.
. f(x, y) Dz = f(x + Dx, y + Dy) f(x, y) .
f(x, y) ,
, .
.. , :
. f(x, y) (, ), a1 a2 D 0 D 0 .
: z = f(x, y) D D Dz (, ).
:
. .
.
.
.
N
j N0
N N0 . NN0. , N0, , NN0 , NN0.
. N0 , N0 .
, , .
z = f(x, y), f(x, y) , 0(0, 0), N0(x0,y0,(x0,y0)) :
.
:
f(x, y) (0, 0) ( z) (0, 0) (0+D, 0+D).
, .
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.
(1, 1, 1).
:
:
.
f(x, y) (, ). :
:
. , x = 1, y = 2, z = 1.
Dx = 1,04 1 = 0,04, Dy = 1,99 2 = -0,01,
Dz = 1,02 1 = 0,02.
u(x, y, z) =
:
u :
: 1,049275225687319176.
.
f(x, y) D, .
.
.
, .
. .. .
. f(x, y) (, ) , :
.
.. .
.
n , .
.
. z = f(x, y), , 0(0, 0)
0 .
. z = f(x, y), , 0(0, 0)
0 .
. ( ).
f(x,y) (0, 0) , , .
(0, 0) .
. ( ).
(0, 0) f(x, y) . :
1) D(x0, y0) > 0, (0, 0) f(x, y) ,
- , - .
2) D(x0, y0) < 0, (0, 0) f(x, y)
, D = 0, .
.
, , u = f(x, y), , ..
j(, ) = 0, .
, .. .
u = f(x, y(x)).
:
=0 (1)
:
(2)
(2) l (1).
l , :
. . .
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u = f(x, y) + lj(x, y) .
.
, , .
.
u(x, y, z) (x, y, z) 1(x + Dx, y + Dy, z + Dz).
1 . , , z a, b, g. .
1 DS.
, :
z
M
M1
y
x
, u(x, y, z) , z. :
,
e1, e2, e3 .
:
, :
;
, s . .
:
: u(x, y, z) (x, y, z).
.
. z = x2 + y2x (1, 2) . (3, 0).
. .
=(3-1; 0-2) = (2; -2) = 2 .
:
=
z :
:
:
=
, , .. .
:
cosa = ; cosb = -
: - .
.
: D u = u(x, y, z) , u
,
u.
, D .
.
: u = u(x, y, z)
.
gradu .
, , u - . , .. .. .
.
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