1. . .
. - , - , 1) , - , 2) , . .
( ). : 1) , - . 2) , . . . , , - , . . , . . , .
2. . .
. , 1) . 2) . , .
( ). , ; , .. , . . , .. , , , |Δf(x0)|=Δif(x0) . . , . (0,0) : - , - . - , .. .
3. . . .
: - k ; , , , . , , . .
. - k : , ; , . - ( ).
( ). , , . . 1) , , , . , . 2) , , .. - Δ .
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4. . ( )
( ). , , . . . , . , . ( ). , .
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, x2+y2≠0
0, x2+y2=0
, . - (0;0). -, - Δf(0,0) = 0*Δx + 0*Δy + α(Δx, Δy) Δx + β(Δx, Δy) Δy = α(Δx, Δy) Δx + β(Δx, Δy) Δy, ( α(Δx, Δy) β(Δx, Δy) 0 0). , . = + 0 = (.. x0=0; y0=0) = = + = + . .. α , β - 0. , - (0;0) , - - => . (0;0) -, .
( , ).
PS 5 . : Fil McArov
- - . - . x0 G, - , - . - - 2 -. . f:(Gc R 2)-> R; f=f(x,y); x0 =(x0,y0); . 1 3, 2 4 , - -. , - ξ (x0; x0+Δx) η (y0; y0+Δy), Δf(x0) = fx(ξ, y0+Δy)Δx + fy(x0, η)Δy. - fx(ξ, y0+Δy) fy(x0, η) : fx(ξ, y0+Δy) = f(x0, y0) + α(Δx, Δy),
α(Δx,Δy)->0 Δx>0,Δy>0, fy(x0, η) = fy(x0, y0) + β(Δx,Δy), β->0, Δx>0,Δy>0. : f(x0, y0)Δx + fy(x0, y0)Δy + α(Δx,Δy) + β(Δx,Δy) => - - .
. -. - 1)xi = xi(t1...tm), i=1...k - - t0 Rm. 2) f(x1...xk) - x0 G c Rk, - f(x (t)) - t0 Rm : j=1..m; -: Δxi(t0) = BijΔtj + βijΔtj (βij->0, Δt->0); Δf(x0) = AiΔxi + αiΔxi (αi->0, Δx->0); Δf(x (t0)) = Ai ( BijΔtj + βijΔtj) + αi( BijΔtj + βijΔtj) = ( AiBij)Δtj + ( (AiBij + Bijαi + αiβij)Δtj = { cj= AiBij; γj= (AiBij + Bijαi + αiβij) } = cjΔtj + γjΔtj γj->0 Δt->0, , f(x (t)) - t0
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1 -. 1 -. - x0 G, .. - - : Δf(x0) = A Δx + α(Δx)* Δx, α->0 Δx->0. AΔx = df(x0) = - f(x) x0 G . A iΔxi = di f(x0) - - xi. .. - x1...xk -, - Δx1...Δxk dx1...dxk - - - - f(x) . x0 , - A . Ai= - 1)df(x0) = 2) di f(x0)= 1)xi=xi(t1...tm) - . t0 R m. 2)f(x) - . x0=x(t0) G c R k. df(x (t0)) . df(x (t0))= df(x0). -: df(x (t0)) = = )= = .
, . - . x0 G R k e, || e ||=1. lim (t->0) - e . x0 . lim (t->0) = ; (-. - , = lim (t->0)) - . x0 G - - - x0 - = (grad f(x0), e). -: e = (cosα1, cosα2... αk). f(x) - x0 => f(x)-f(x0) = A Δ x + α(Δ x) Δ x; A = gradf(x0) = () Δx->0; x = x0 + t e, , f(x0 +t e)-f(x0) = At e + αt e. limΔx->0( = limt->0 = (A, e) = (grad(f(x0)), e).
\ . -, . : x R nk, y R k, x = (x1,x2... xk), y = (y1, y2.. yk); |(x,y)|≤|| x ||*|| y ||; x ≠0; y ≠0, 0≤ω≤Π; cosω = ω x y. 2 λ≠0 , x = λ y, x y , (x, y)=0, - . x y R k. 1) ω=0; ω= Π , 2) ω=Π\2 . - 1) ω=0 > (x, y) = || x ||*|| y ||; x=λ y; (x- λ y; x-λ y) = (x,x) - 2 λ(x, y) + λ2(y,y) = || x ||2 - 2λ|| x ||*|| y || + λ2||y||2; cos0=1 (x- λ y; x-λ y)=0, λ = +- ||x|| / ||y|| 2) .
. . - - - . , ω = gradf(x0)^ e; .. : . : 1) grad f(x0), - - , ||gradf(x0)||, , -||gradf(x0)||. - f(x) x0. 2) , - , =0, -, , .. -||gradf(x0)||≤ ≤||gradf(x0)||. 3) grad f(x0), (x0 R) , x0 f(x) .
12. n- - xi1, xi2,,xin (i=1,2,k) . :
, (i1,i2,,in=i), : , , .
- n , (n-1)- .
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. f(x,y)(GCR2)→R - (x0y0), G - (x0y0), .
. - f(xy) (xy):
∆xf(x0y0)=f(x0+∆x,y0)-f(x0y0); ∆yf(x0y0)=f(x0,y0+∆y)-f(x0y0), : ∆y(∆xf(x0y0))=∆xf(x0,y0+∆y)-∆xf(x0y0)=
=f(x0+∆x,y0+∆y)-f(x0,y0+∆y)-f(x0+∆x,y0)+f(x0y0), ∆x(∆yf(x0y0): ∆x(∆yf(x0y0))=∆yf(x0+∆x,y0)-∆yf(x0y0)=
=f(x0+∆x,y0+∆y)-f(x0+∆x,y0)-f(x0,y0+∆y)+f(x0y0).
f′(xy) (x0y0) ,
∆y(∆xf′(x0y0))=(f′x(x0,y0+∆y)-f′x(x0y0))∆x=(f′′xy(x0y0)∆y+β1(∆y)∆y)∆x=(f′′xy(x0y0)+β1(∆y))∆y∆x β1(∆y)→0, ∆y→0, , :
∆yf(xy)=f′y(xy)∆y+α1(∆y)∆y, ∆x∆yf(x0y0)=(f″yx(x0y0)+α2(∆x))∆x∆y (α1(∆x)→0 ∆x→0)
.. ∆x∆yf(x0y0)=∆y∆xf(x0y0), f′′xy(x0y0)+β1(∆y)=f″yx(x0y0)+α2(∆x)), ∆x→0 ∆y→0 , .
13. : , - , :
. . (1) aij=ai*aj,
. . =2 (1)=a21x21+a1a2x1x2+a2a1x2x1+a22x22=(a1x1+a2x2)2
, >2, +1:
14. - f:(GCRk)→R (n-1) S(εx→0) n x→0, n-
; , : (aDijp+bDrsq)f=aDijpf+bDrsqf
- f:(GCRk)→R d2f(x→) x→0, :
. ∆xi=dxi . x
, , - , ,
n- -. n≥2 n≥2 n- - , xi (i=1..k) . n . -: , n=2 - . xi (i=1..k) - . d2xi - : d2f(x) = d(df(x))= = . d(dxi)=d2xi≠0, - f(x) xi-.-. , - => -. ..
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16.
:
f(x) = f(x1, x2 xk): (GÌRk)R, f(x) .
1. () l G, l, x1, x2, , x2 x1 l, f(x2)>f(x1) (f(x2)<f(x1))
2. () .xÎG l, G, l, . x , f(x) () l.
.
.
f:(GÌRk)R - G,
1. ÌG l , f(x) l
2. ÌG l , f(x) - l
3. G , f(x)=const
:
1) , x1, x2Î , x2 x1 l, 0£t£1
F(t)=f(x1+t(x2-x1))=f(x1+tl||x2-x1||,
F(t) [0;1] . : F(t) ;
"tÎ(0,1)
. $eÎ(0,1): f(x2)-f(x1)=F(1)-F(0)=Ft(e)(1-0)=Ft(e)=
.. , , f(x2)>f(x1)
2)
3) x1, x2ÎG , G,
(.. 0), , f(x1)=f(x2)
ÌG, , , .
17.
.
S(e,x)=(xÎRk, ||x- x||<e) - e . x Rk
f: S(e,x)R (m+1) - , (1), xÎS(e,x)
:
Dx , .x+DxÎS(e,x) x x+Dx x=x+tDx, 0£t£1
F(t)=f(x+tDx),
F(1)-F(0)=f(x+tDx)-f(x)=Df(x)
f(x) (m+1) - Þ ÞF(t) . , ..
0£t£1, tÎ[0,1]
(2)
q t t
.. t ,
t=1, t=0
(2)
n -
, x=x+qx
dt=Dt=1-0=1
xi=xi+tDxi
dxi=Dx
(2), .
18. . .
Def.1.: (X,d) , f:(EÌX)R,
1) , f ÎE,
2) , f ÎE,
.1.:( )
f:(GÌRk)R / ÎG, . :
1) , Rk;
2)grad(f())=0
3) ,
4)
-: 1) f(x) ÎG .
, f . (1). tÎR ( ). :
, ...
2).. , , -
3)..
, .
4)..
.
19. . . .
Def. 1.:
1) , .
2) , .
3), , ,
. 1.:( )
- k*k, A1=a11, , , - . , , / , , A1>0, A2>0,,Ak>0 / A1<0, A2>0,,sgn(Ak)=(-1)k. (WITHOUT PROVE!)
1.:( )
, , .
-: 1) , - Rk, ..
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. e1,e2,,ek, . , : . . : .
2) - : , : . ( M, .. M>0) . : .
: , -
20. .
. 1.:( )
- , x0 , . :
1) , x0.
2) , x0.
3) , x0 .
: .. ,
, m=1; (1), . .. ( ) .. - , , df(x0)=0 (1) : (, )
-: 1) ,
1 , m>0. ,
, ||Dx||<<1
, .
2) - , ( 1)
, M>0,
.
3) - x - (2), - ,
(3).
(2) , , . , , , (4).
(3) , (5). , , , (4) (5) !!!!!!
21. . .
Def. 1.: y=f(x), F(x,y)=0, (x,y)ÎGÌRk .
. 1.:( )
GÌRk R2 ( ) F(x,y):GR. :
1)F(x,y)ÌC(G)
2)F(x0,y0)=0, x0,y0 G.
3) x, , y /;
F(x,y)=0:
1) x;
2)f(x0)=y0;
3)y=f(x) .
-: .. G , M0 ,
, ÎG.
x=x0 A0B0. F(x,y) F(B0)>0, B0(x0,y0+D`) F(A0)<0, A0(x0,y0-D`). A0,B0: B1B2 A1A2. x: F(x0,y0-D`) F(x0,y0+D`). x $ (x0-d,x0+d), 0<d0<D, .
x - - F(x,y) . .. F(x,y) y [y0-D`,y0+D`] n , F(x,y) y, . .. $ y=f(x)!!!
, y=f(x)
xÎ(x0-d,x0+d). .. , x0 x0. D`, D`=e, d=d(e)=d0. |x-x0|<??? , |f(x)-f(x0)|<D`=e.
22. )
. 1.:( )
F(x,y):(G ÌR2)R :
1)F(x,y) =0 G.
2) G ( )
3) , (x0,y0)ÎG,
, y=f(x) S(d,M0)ÌG, M0(x0,y0).
-: .. y (x0,y0) 0, $S(d,M0)ÌG,
F(x,y) y. . 1. (
).
y=f(x) Dx x.
Dx Dy. (x+Dx,y+Dy)ÎS(d,M0). (x+Dx,y+Dy) 1, .. 0. DF(x,y):
, (1) , .. x. Dx , β0.
(1) Dx0:
G x0, ...
. .
: y=f(x1,x2xk) F(x;y), (x,y) Î G Ì Rk*R=Rk+1 .
( )
G Î Rk+1 F(x;y):GàR :
F(x;y)=0 G
2. ∂F(x;y)/∂y G
0 ∂F(x0;y0)/∂y ¹ 0 (x0;y0) Î G
F(x0;y0)º0
$ S (d, M0), M0(x0,y0)=M0(x10, x20xk0, y0) Î G
F(x;y)=0 - .
F(x;y)=0 y=f(x)=f(x1,x2xk)
y=f(x0)
y=f(x) S (d, M0)
: n- n- yi=fi(x1,x2xn) y=1n G Rn :
( )
( )
xi=xi(t) i=1n t0ÎRn yi=yi(x) i=1n x0=x(t0). yi=yi(x(t)) i=1n :
D(y)/D(t)=D(y)/D(x)*D(x)D(t)
-: An*n Bn*n det An*n n¹0 det Bn*n n¹0 det(A*B)=det(A)*det(B)
D(y)/D(x)*D(x)D(t)= *