u = f (x, y, z) D . M(x,y,z) S, cosα, cosβ, cosγ. S Δ s 1(+ Δ , + Δ , z+ Δ z),
f :
Δ s :
.
:
(1)
.
u = f (x, y, z) S .
(1) :
(2)
1. . , :
.
2. , , = 0 = 0. l (0, 0) , O z l.
, u = f (x, y, z) , u = f (x, y, z).
: grad u = .
.
1. S grad u S. . S eS ={cosα, cosβ, cosγ}, (4.7) grad u es, .
2. S , |grad u |, . . S grad u φ. 1 , |grad u |∙cosφ, (4.8) , φ=0 |grad u |.
3. , grad u, .
. (4.8)
4. z = f (x,y) , grad f = f (x,y) = c, .
. . . . . .
1. 0 (0, 0) z = f (x, y), f (xo, yo) > f (x, y) (, ) 0.
2. 0 (0, 0) z = f (x, y), f (xo, yo) < f (x, y) (, ) 0.
|
|
1. .
2. .
1 ( ). 0 (0, 0) z = f (x, y), .
.
, = 0. f (x, y0) , = 0 . , . .
3. , , , .
. , , .
.
- z = x ² + y ². 0 = 0 = 0. , , = = 0 z = 0, z > 0.
- z = xy (0, 0), (z ( 0, 0) = 0, , ).
2 ( ). 0 (0, 0), z = f (x, y), 3- . :
1) f (x, y) 0 , AC B ² > 0, A < 0;
2) f (x, y) 0 , AC B ² > 0, A > 0;
3) , AC B ² < 0;
4) AC B ² = 0, .
. z = x ² - 2 xy + 2 y ² + 2 x. . , (-2,-1). = 2, = -2, = 4. AC B ² = 4 > 0, , , ( A > 0).
.
4. f (x1, x2,, xn) m (m < n):
φ1 (1, 2,, n) = 0, φ2 (1, 2,, n) = 0, , φm (1, 2,, n) = 0, (1)
φi , (1) .
5. f (x1, x2,, xn) (1) .
. : f(x,y) φ (,) = 0, . z = f (x,y), , φ (,) = 0. , , , f(x,y).
|
|
, :
6. L (x1, x2,, xn) = f (x1, x2,, xn) + λ1φ1 (x1, x2,, xn) +
+ λ2φ2 (x1, x2,, xn) ++λmφm (x1, x2,, xn), (2)
λi , , λi .
( ). z = f (x, y) φ (, ) = 0 L (x, y) = f (x, y) + λφ (x, y).
. , , : = (). z , : . (3)
, . (4)
(4) λ (3). :
, .
, :
(5)
: , λ, . (5) λ, , .
1. 2.
2. , f (x1, x2,, xn) (1), (6)
. z = xy + = 1. L(x, y) = xy + λ (x + y 1). (6) :
, -2λ=1, λ=-0,5, = = -λ = 0,5. L (x, y) L (x, y) = -0,5 (x y)² + 0,5 ≤ 0,5, L (x, y) , z = xy .