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Одесский национальный университет им. И.И. Мечникова

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[НЕСОРТИРОВАННОЕ]

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Диференціальне числення ФБЗ

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. $" :

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

& + ' ϕ( x) , " ':

ϕ( x) = f ( x, y0 + y) − f ( x, y0 ) .

# A = ϕ( x +

x) − ϕ( x ) . ) * u′ ( x , y

) ,

0 0

" ϕ( x) ( [ x0 , x0 +

x] , '

7 + ,:

A = ϕ′( x ) x ,

+ x

x .

ϕ′( x ) = f ′( x , y

+ y) − f ′( x , y

) .

) * u′′

( x , y

) , u′

( y

0 0

[ y0 , y0 +

y] , , + ' 7 +, :

f ′(x , y + y) − f

′( x , y ) = f ′′ (x

, y ) y ,

y + y

y . ) +

A =

f ′′ ( x , y )

y .

(16.1)

% 6 A ":

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

& + ' ψ( y) , " ':

ψ( y) = f ( x0 + x, y) − f ( x0 , y) .

# A = ψ( y0 +

y) − ψ( y0 ) . 2 ' 7 +:

A = ψ′( y ) y ,

+ y

y .

ψ′( y ) = f ′( x + x, y ) − f ′(x , y ) .

# ' 7 +:

ψ′( y ) = f ′′ ( x , y ) x ,

A =

f ′′ (x , y )

y .

(16.2)

2 (16.1) (16.2) ,:

f ′′

( x , y ) = f ′′ (x , y ) .

% (

x → 0,

y → 0 :

lim

f ′′ ( x , y ) =

lim

f ′′

( x , y ) .

x→

x→ 0

y→ 0

) * f

′′

f ′′ ( x , y

) ,

lim

f ′′ ( x , y ) =

′′ ( x , y ),

lim f

′′ ( x , y ) =

f ′′ ( x , y

) .

x→0

x→ 0

0 0

y→0

f ′′	( x , y ) = f ′′		( x , y ) ,
xy	0 0	yx	0 0

( ! ! .

+ , * " ( " ! * "

! ". & u′′	, u′′	( x , y	)
xy	yx	0 0

,. 5 ", + " , . $" :

xy( x

− y

)

, x2 + y2 > 0,

x2 + y2

u =

+ y

= 0.

2 (:

y(x4 − y4 + 4x2 y2 )

∂u

, x2 + y2 > 0,

(x2

+ y2 )

∂x

x2 + y2 = 0,

* u′		(0,0) = lim	u( x,0) − u(0, 0)		= 0 . 2 u′			(0, y) = − y ,
* u′		(0,0) = lim			= 0 . 2 u′			(0, y) = − y ,
	x	x→ 0		x			x
		x→ 0		x
u′′	(0, y) = −1, u′′			(0,0) = −1.
xy			xy
	,, u′′ (0,0) = 1.
				yx
	# " u′′					(0, 0) ≠ u′′	(0, 0) , ! *
					xy	yx

' " + * " ' ". 8 ’"

, u′′	, u′′	(0,0) .
xy	yx

% 2- " + + ' " x , ( y . # , 3- ". " u = f ( x, y) !:

∂3u

∂x3

∂x2∂y ∂x∂y∂x ∂x∂y2

∂y∂x∂y ∂y2∂x ∂y3

&, m - " , 6

(m − 1) - ". . ,

∂mu

, ' m - ":

∂x p ∂y m− p

" u p ', * " x , m − p y .

" ( * "

' * " .

& " " 2- ".

. ( " u = f ( x, y) , + G 2

1- 2- ".

. " 2- u = f ( x, y )

, * " 1- ", ! (

, dx dy , . % , * " 2- " "

d 2u = d (du ) .

$" ' * * 6:

∂

d 2u = d (du ) = d

dx +

∂x

∂y

∂

dx +

dy dx +

dx +

dy dy =

∂x

∂y

∂x

∂y

∂2u

dx2

+ 2

∂2u

dxdy +

∂2u

dy2 .

∂x2

∂x∂y

∂y2

., " u = x2 cos y :

d 2u = 2 cos ydx2 − 4x sin ydxdy − x2 cos ydy2 .

" u = f ( x1 , x2 ,...xn ) 2- " , * "

, * ,:

n n	2
d 2u = ∑∑	∂	u	dx j dxk .
d 2u = ∑∑	∂x j ∂xk		dx j dxk .
j =1 k =1	∂x j ∂xk
' * " ":
d 3u = d (d 2u),..., d mu = d (d m−1u) .
"			m - " u = f ( x1 , x2 ,...xn ) ,
:

n n	n
d mu = ∑∑L ∑
k1 =1 k2 =1	km =1

. 2 ( 2 (:

∂u	= a ju,	∂2u
∂x j		∂x j ∂xk

2 ' (16.3)

∂m f ( x , x ,...x )

dx dx

Ldx .

(16.3)

∂xk ∂xk

L∂xk

1 2

d mu , " u = exp(a x + a x

+ ... + a

) .

1 2 2

= a j ak u,...,

∂mu

= ak ak

Lak u .

∂xk ∂xk L∂xk

n n	n
d mu = ∑∑L∑ak		ak	2	Lak	m	u dxk dxk	Ldxk	.
	1		2		m	1	2	m
k1 =1 k2 =1	km =1

17. - .

2«* " ( , » ,,

" F (t) " n + 1 "

+ ! ' # (:

F (t) = F (t

) + F ′(t

)(t − t

) +

F ′′(t

)(t − t

)2 + ... +

F ( n) (t

)(t − t

)n +

1	F (n+1) (t		+ θ(t − t	))(t − t	)n+1	(0 < θ < 1)
		0
(n + 1)!			0	0

(6 ( " 7 +). % ( :

F (t) − F (t0 ) = F (t0 ) .

F (t

) = dF (t

) +

d 2 F (t

) + ... +

d n F (t

) +

d n+1F (t

+ θ t)

(17.1)

(n + 1)!

(0 < θ < 1) .

%, dt , " * "

( , ', t , "

, F (t0 ) ( .

$' ' ! * . "

" " ! + *

u = f ( x, y) .

%, " ( x0 , y0 ) " u = f ( x, y) ,

(n + 1) - " '. . x0 y0

x		y , ! , ’, , ( x0 , y0 )
( x0 + x, y0 +	y) (6 + * . %
" " ,
x = x0 + t	x, y = y0 + t y	(0 ≤ t ≤ 1) .	(17.2)
% (17.2) *			f ( x, y) . # ,
' , t :
F (t ) = f ( x0 + t x, y0 + t y) .
$"		f ( x, y) ( x0 , y0 ) :

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

& ', F (t) :

F (0) = F (1) − F (0) .

9 " F (t) , t = 0 (n + 1) - "

', + # ( (17.1):

F (0) = F (1) − F (0) = dF (0) +	1	d 2 F (0) + ... +	1	d n F (0) +	1		d n+1F (θ)
					(n +
2!			n!			1)!

(0 < θ < 1) . % * dt , " ( * "

, ', t = 1 − 0 = 1 . ' * ' "

, (:

dF (0)

∂f (x0 , y0 )

∂f ( x0

, y0 )

dF (0) =

x +

y =

∂x

∂y

∂f (x0 , y0 )

dx +

∂f ( x0 , y0 )

dy = df (x , y )

∂x

∂y

d 2 F (0) = d (dF (0)) =

d 2 F (0)

t 2 =

d 2 F (0)

dt 2

∂2 f ( x , y

)

dx2 + 2

∂2 f ( x , y

)

∂2

f ( x , y )

dy2 = d 2

0 0

dxdy +

0 0

f ( x , y

) .

∂x2

∂x∂y

∂y2

% + ' , :

d n F (0) = d n f ( x , y

) ,

d n+1F (0) = d n+1 f ( x

+ θ

x, y

+ θ

y) .

# , :

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

2 f ( x , y ) + ... +

d n f ( x , y ) +

d n+1 f (x

+ θ

x, y

+ θ y) . (17.3)

(n +

1)!

9 (17.3) , * " " + ! " &

u = f ( x, y) 6 7 +. 5 '

", 6 " ,

! , ! 6 ":

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

+	1		∂2	f ( x0 , y0 )

				∂x		2
	2!			∂x
+	1	∂3 f (x0 , y0 )

	3!			∂x	3
	3!			∂x

∂2

f ( x0 , y0 )

x y +

∂2

f ( x0 , y0 )

∂x∂y

∂y2

+ 3

∂3 f (x , y

)

y + 3

∂3 f ( x , y )

∂x2∂y

∂x∂y

y2 +

x y2 + ∂3 f ( x0 , y0 )

∂y3

y3 + ...

(17.4)

. . " ' # ( u = x y

M (1,1) , ! + 6 * 3- " '

x, y . / ,: x0 = 1, y0 = 1,

f ( x, y)

= xy−1

= 1,

∂f

= y

−1

= 1

∂f

= −xy−2

= −1,

∂x

∂y

∂2 f

= 0 ,

∂2 f

= − y−2

= −1,

∂2 f

= 2xy −3

= 2 ,

∂x

∂x∂y

∂y

∂3 f

= 0 ,

∂3 f

= 0

∂3 f

= 2 y−3

= 2 ,

∂3 f

= −6 xy

∂x

∂y

∂x∂y

∂y

∂x

−4

= −6 .

#, (17.4) ,:
	1 +	x	= 1 +	x − y +	1	(−2	x y + 2 y2 ) +	1	(6 x y2 − 6 y3 ) + ... =
1 +
		y		2!			3!
= 1 +		x − y −		x y + y2 + x			y2 − y3 + ...

18. . .

. ( " u = f ( x) = f ( x1 , x2 ,..., xn ) + G n ,

( x0 = ( x01 , x02 ,...x0 n ) – 6 " + G .

. # x0 , * " u = f ( x) , " , " Sδ ( x0 ) G , " x Sδ ( x0 ) , "

' * ' x0 , *: f ( x) > f ( x0 ) .

. # x0 , * " u = f ( x) , " , " Sδ ( x0 ) G , " x Sδ ( x0 ) , "

' * ' x0 , *: f ( x) < f ( x0 ) .

# ' * "

, .

( " &). * ! x0

" & u = f ( x) , ! !

∂f

(k = 1, 2,..., n) . +

∂xk

∂f (x0 ) = 0 (k = 1, 2,..., n) .

∂xk

. .( x0 = ( x01 , x02 ,...x0 n ) –

u = f ( x) = f ( x1 , x2 ,..., xn ) . " +, x0 –

. # , " Sδ ( x0 ) , x Sδ ( x0 ), x ≠ x0 :

f ( x) < f ( x0 ) .

$" ' ,

ϕ( x1 ) = f ( x1 , x02 ,..., x0 n ) .

# , ( δ - ( x01 − δ, x01 + δ) x01 , " ! * " "

* : ϕ( x1 ) < ϕ( x01 ) , ! x01 , '

ϕ( x ) . 2 ' ,

∂f (x0 )

, + ,

∂x1

. 2 ' ! ,

ϕ′( x1 )

x1 =x01

∂f ( x0 )

∂f (x0 )

,: ϕ′( x )

= 0 . ϕ′(x )

. #

= 0 .

x1 =x01

∂x1

∂f (x0 )

= 0

(k = 2,..., n) . # .

∂xk

$ . x0 " u = f ( x) " !,

∂f ( x

)

df ( x0 ) = ∑

dxk = 0 .

(18.1)

∂xk

k =1

(, * x0

" u = f ( x) (, (

' *

∂f ( x)

(k = 1, 2,..., n) , * x0 –

∂xk

∂f (x0 )

= 0

(k = 1, 2,..., n) , ( , * (18.1).

∂xk

! ". )! + " ,

! , 1- "

'' *

" * ( .

$",

, '

u = x2 − y2 .

/ ,

u′

= 2x , u′ = −2 y , ! '' * ' (0, 0) . $

( .

(, " (

f (0,0) = 0 . 2 6 * , *

+ Ox . ) ,

f (ε, 0) = ε2 > f (0, 0).

" 6 *

Oy ,

,: f (0, ε) = −ε2 < f (0, 0).

, , , ! (

!, (0, 0) – ( .

2 6 ! " + ,

1- " ' *. . " u = x2 + y2 ,

( ) (0,0),

u′

x2 + y2

x2 + y

( ' *.

. 5 x0

1- "

u = f ( x)

' * '' * ', x0 , * "

u = f ( x) .

2 ,, (

! ' '. ! + "

, ! + ! '

. # ! "

. / ! + * * ' " .

( ).								* !	x0 n
" & u = f ( x) = f ( x , x							2	,..., x ) . - !		x0
						1	2	n
2-						" & u = f ( x) . +,

n	n	∂2 f (x0 )
d 2 f (x0 ) = ∑ ∑			dx	dx	≠ 0
d 2 f (x0 ) = ∑ ∑			dx	dx	≠ 0
j =1 k =1 ∂x j∂xk				j k
j =1 k =1 ∂x j∂xk
	dx1, dx2 ,..., dxn								(

∑ dx2 > 0 ),

" & u = f ( x) . . ,

j =1

d 2 f ( x

0 ) > 0

d 2 f ( x0 ) < 0

∑ dx2 > 0 , x0

– .

j =1

∑ dx2 > 0 , x0 – .

j =1

dx1, dx2 ,..., dxn

∑ dx2j

> 0

j =1

d 2 f ( x0 ) > 0 ,

dx′

, dx′

,..., dx′

∑ (dx′ )2 > 0

j =1

d 2 f ( x0 ) < 0 , x0 .

	1 2	n		n	j
	1 2	n		∑	j
!	dx′′, dx′′,..., dx′′			∑	(dx′′ )2 > 0		,
			j =1
d 2 f ( x0 ) = 0 , x0	# ,					# !

(! ).

" " , ! *6 ( "

( " & ). * !

! M 0 ( x0 , y0 ) " u = f ( x, y )

f xx′′ ( x0 , y0 ) ( x0 , y0 ) = f xy′′ (x0 , y0 )

f xy′′ ( x0 , y0 )

> 0 ,

f yy′′ ( x0 , y0 )

" u = f ( x, y ) M 0 , ,

fxx′′ (x0 , y0 ) < 0 , , f xx′′ (x0 , y0 ) > 0 .

( x0 , y0 ) < 0 , M 0 .

( x0 , y0 ) = 0 , M 0 # , #

( ! ! ,

#).

1. '

u = x3 + y3 − 9 xy .

1). 2 (

u′	= 3x2 − 9 y , u′	= 3 y2 − 9x .
x	y

'' ", , " *:

x − 3 y = 0,

y2 − 3x = 0.

$ ’" '	'	,	,
M 0 (0, 0), M1 (3, 3).

2). % " M 0 , M1 -. / ,:

u′′	= 6x , u′′	= −9, u′′	= 6 y ,
xx	xy	yy

(x, y ) = 36xy − 81 .

M 0 (0, 0) ,: (0,0) = −81 < 0 – + (

M1 (3,3) ,: (3,3) = 36 3 3 − 81 = 243 > 0 – + (

, , , * u′′xx ( M1 ) = 6 3 = 18 > 0 . 2 "

( f (3,3) = −27 .

u = x4 + y4 .

' , * (0, 0) , * u′

= 4x3 , u′

= 4 y3 .

( :

u′′

= 12x2

= 0 ,

x= y =0

x =0

= 12 y2

u′′

= 0, u′′

= 0 ,

x= y =0

y =0

+ (0, 0) = 0 , , ( , , .

u = x4 + y4 , f (0,0) = 0 , 6 f > 0 , +

(0, 0) , .

19.$& &

" " .

&, " u = f ( x, y ), " (

! + ( + G , " , ( + (! *6

( 6 * (2- & (, 6). 6

+ G ( " + ! * 6

. # !	(
u = f ( x, y ), " + *	!	D ,	!	,	"

1- " ! '' * ' (

), ! ' *. % ! " .

! ( (! *6 ( 6 " 6 +

+ G . & ' " " + (

+ * ! " "), ' " * + " (! *6

( 6 * , " .

! * ! ' * (! *6 ( 6.

1.2 ( (! *6 ( 6 " u = x2 y (2 − x − y )

+ G , " ,' + ' , , " ( "

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