матан Бесов - весь 2012

f

(x0, y0)

!

"

lim ( lim f (x, y)) = A

y→0 x→0

§

f : X → R, X Rn,

§

E Rn % E X&

x(0) E

' f x(0)

E

(& ε > 0 δ = δ(ε) > 0) |f (x) − f (x(0))| < ε x E |x −

− x(0)| < δ % ) f (E ∩ Uδ (x(0))) Uε(f (x(0)))&

,

- . .

/ 0

(◦ 1 x(0) 2 E

f x(0) E

lim f (x) = f (x(0))

f x(0) E

f x(0)

E

3

x(0) E !

f E ∩ Uδ (x(0)) δ > 0

x(0) 2 E

x(0) 2 E

f

x(0) E E f (x(0)) = 0

δ > 0

sgn f (x) = sgn f (x(0)) x E ∩ Uδ (x(0)).

n = 1

x(0) E Rn m

f1 fm x(0)

(f1(x), . . . , fm(x)) F Rm x E

g y(0) = (f1(x(0))

fm(x(0))) F

E

h(x) g(f1(x), . . . , fm(x)), h : E → R,

x(0) E

ε > 0 ! η = η(ε) > > 0" |g(y)−g(y(0))| < ε y F ∩Uη (y(0)) #$

δ = δ(η) = δ(η(ε)) = δε > 0

|f1(x) − f1(x(0))| < √ηm , . . . , |fm(x) − fm(x(0))| < √ηm

x E ∩ Uδ (x(0)).

% g f1 fm

&' ( ( ) ( *

+

y = (y1, . . . , ym) = (f1(x), . . . , fm(x)),

i=1

|h(x) − h(x(0))| = |g(f1(x), . . . , fm(x))−

−g(f1(x(0)), . . . , fm(x(0)))| = |g(y) − g(y(0))| < ε.

§

# ε > 0 , h

x(0) E

- !

! & !

-

&' ! + n = m = 1

§

E Rn f ! . E

+

sup f sup f (x) (inf f inf f (x)).

Ex E E x E

/ + f + !

x E Rn E +

E

f

E Rn E

E ! !

!. * ! + + (

0 ! + !

# * ! + + n = 1 E = [a, b]

B sup f +∞ ) ! + ( !

E

' ! {x(m)} x(m) E

m N + lim f (x(m)) = B !

m→∞

{x(m)} E 1

23 2 4 5 6# * !

{x(m)} ( !+' & + ! ! {x(mk )}∞k=1

x(0) = lim x(mk ) x(0) ! E

k→∞

E -! f x(0)

E

*

f (x(mk )) → B, f (x(mk )) → f (x(0)) k → ∞

f (x(0)) = B

f x(0) E

sup f f E

E# & ' ( % !"# x = x(0) x = x

)( * % % n = 1

f (x) = x1 g(x) = sin x1 % E = (0, 1)

+ % E

) E , % %

E ' % '

E +-

f

E Rn f E

. / %

- f %

% E

ε0 > 0 δ > 0 x, y E : |x − y| < δ, |f (x) − f (y)| ε0.

x(m) y(m) +-+ % x y

x(m), y(m) E, |x(m) − y(m)| < m1 , |f (x(m)) − f (y(m))| ε0 > 0.

§

{x(m)}

{x(m)}

! " |x(m) − y(m)| < m1

lim y(mk ) = x(0)! " # x(0) E # # # E # !

k→∞

f # x(0) E

f (x(mk )) → f (x(0)), f (y(mk )) → f (x(0)) k → ∞,

#

|f (x(mk )) − f (y(mk ))| |f (x(mk )) − f (x(0))|+

+|f (y(mk )) − f (x(0))| → 0 k → ∞.

$

|f (x(mk )) − f (y(mk ))| ε0 > 0 k N.

" # !

% & # f ' (a, b) → R

(a, b)! % # f

(a, b)!

% & # f

E Rn! () * E+

& # ω' (0, +∞) → [0, +∞)

ω(δ) = ω(δ; f ) = ω(δ; f ; E) = sup{|f (x) − f (y)| :

x, y E, |x − y| δ}.

, ω - & #

E * ! !

# # #

. # +

ω(δ1 + δ2) ω(δ1) + ω(δ2) δ1, δ2 > 0.

f E R

E

ω(0 + 0; f ; E) = 0.

◦ f E

ε > 0 δ(ε) > 0 : ω(δ; f ) ε < 2ε

0 < δ < δ(ε).

ω(0 + 0; f ) = 0◦ ω(0 + 0; f ) = 0

ε > 0 δ(ε) > 0 : ω(δ(ε); f ) < ε.

! f "

E

G Rn f

G a, b G f (a) < f (b)

C (f (a), f (b)) c G : f (c) = C.

# $ % &

' a, b G

(

Γ = {x = ϕ(t) : α t β}, ϕ(α) = a, ϕ(β) = b, Γ G.

) ' * ! * g(t) = f (ϕ(t)) + "

[α, β] ' , "

! - g(α) = f (a) g(β) = f (b)

-. ' ,

!

ξ [α, β] : g(ξ) = f (ϕ(ξ)) = C.

/ c = ϕ(ξ) 0 ' *

! G

" G

f (a) < C < f (b) / 1 ε0 > 0 f (a) +

+ ε0 < C < f (b) − ε0 / ! f

0 a b , a(0) b(0) G

|f (a) − f (a(0))| < ε0, |f (b) − f (b(0))| < ε0.

f (a(0)) < C < f (b(0)) 1 "

*

2 "

, 1 $ G "

' 3 4

f

! f (x1, x(0)2 , . . . , x(0)n ) " ! #

! !

& x x(0) ! # ! x ' xx − x(0) $! !

x = (Δx1, . . . , xn) (x1 − x(0)1 , . . . , xn − x(0)n ),

i=1

§

f x(0) #

x(0) ( f

x(0) ! *

f (x(0)) = f (x(0) + x) − f (x(0)) =

n

A1 An . 0 = (0

0)

+,- ! /o ! 0 ! *

! !

x(0)

f x(0)

* #$% +

dxi -

§

/ ( ! )

+ 0 "

∂f

∂xi !! ) / 1

! )

. ! ) ( n 2

$◦ !! ) ! ) f

∂f

( 0 ∂xi #i =

3◦ ! ) "

#n = 2%

1 x = y = 0,

f (x, y) =

0x = y x = y = 0,

-0 ( ∂f∂x (0, 0) = ∂f∂y (0, 0) = 0

-0 ( ( (0, 0)

4 & ! ) !! " ) ( (0, 0) ( "

∂f

0 ∂xi

x(0) ( !! ) ! " ) f (

2◦ ! ) "

( (0, 0) -0 ( "

∂f∂x (0, 0) = ∂f∂y (0, 0) = 0 !! ) (

(0, 0) f

(0, 0)

x = y

f (x, x) = |x| = o(|x|) x → 0.

x x = y,

f (x, y) =

0x = y.

" # "

!

! $!

x(0)

∂f

∂xi i = 1, . . . , n f f

x(0)

$ $ %

! ! n = 2 &

! $! ! (x, y)R2 (x0, y0) " ' !

# Uδ ((x0, y0)) ( (Δx)2 + (Δy)2 < δ2

f (x0, y0) = f (x0 + x, y0 + y) − f (x0, y0) =

= [f (x0 + x, y0 + y) − f (x0, y0 + y)]+

+[f (x0, y0 + y) − f (x0, y0)].

) #

#

) * ' !

! "

f (x0, y0) =

§

! $! #

!

. " f " "

∂f

' $ ∂xi ! i =

= 1, . . . , n $"

/ + + '

'

f # $

#

§

f Uδ (x0, y0) → R

x y δ (x0, y0)

! " S = {(x, y, z) R3: (x, y) Uδ (x0, y0) z = f (x, y)}

# $ "

% f

(x0, y0) "

f (x0, y0, f (x0, y0))

&

* (x0, y0, f (x0, y0))

z = f (x, y) (x, y)

& z (x, y, z ) &

'(( ( +) n = 2 '()

f (x, y) − z = o( (x − x0)2 + (y − y0)2)

'+)

(x, y) → (x0, y0).

% '+)

, & &

" & -

. "

& " "

"

'() f

(x0, y0) , &

" f (x0, y0, f (x0, y0))

" α # " Ox ' , & y = = y0) & Γ (x0, y0, f (x0, y0))

& &0 " -"

- - /

" & & &

∂f∂x (x0, y0)

1 " " &

& & ∂f∂y (x0, y0)

§

m f1 fm

n x(0)

Rn

g m

y(0) = (f1(x(0)), . . . , fm(x(0)))

!

h(x) g(f1(x), . . . , fm(x))

x(0) ! "

!

fk x(0) g y(0)

h

x(0)

x(0)

! ε0(Δy)" εk(Δx) #

% & h"

& ! x" & g(y(0))

yk k = 1, . . . , m & fk(x(0))

fk" & x ! ' ! " | x|

ym=Δfm

#

, ()* " h

x(0) + ∂h (x(0))

∂xi

xi ()*"

(-*

m f1 fm

g m

y(0) = (f1(x(0)), . . . , fm(x(0)))

∂g k = 1, . . . , m

∂yk

! h(x) = g(f1(x), . . . , fm(x))

x(0) ! " #

∂h (x(0)) ! (-*

∂xi

! !

#

(-* #

dg(f1(x(0)), . . . , fm(x(0))) =

g(y) = g(y1, . . . , ym)" !

yk = fk(x) k = 1, . . . , m y(0)

dyk

dg(y)

y1 ym

dy1 dym

! "

#$ % & d(u ± v) = du ± dv,

d(uv) = v du + u dv,

% ' ' u v (

%

' ' )

u v % %( % % u = u(x) v = v(x) x = (x1, . . . , xn)

* "

'

' ' " %

% dyk

' '

§

# % f "

x(0) Rn

# e = (cos α1, . . . , cos αn) "

|e| = 1 + (

e n cos2 αi = 1

i=1

§

, x(0) $ e-

x = (x1, . . . , xn) = (x(0)1 + t cos α1, . . . , x(0)n + t cos αn), t 0,

! .

+ % f x(0)

% "

e

% " Ox Oy Oz

/ % " (x0, y0, z0) f$' ' x y z

(x0, y0, z0)

0 % % %

∂f∂e = (grad f, e)

f e grad f