Диференціальне числення ФБЗ
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3
2. n .
. / + X , * " , " + ( x y , + *
’, ρ( x, y) , " , * " + x y ,
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1)ρ( x, y) = 0 * , x = y ;
2)ρ( x, y) = ρ( y, x) ;
3)z X : ρ( x, y) ≤ ρ( x, z) + ρ( z, y) (* ).
3 ! . 4 ρ( x, y)
, * " + X . %.
1. .( X = – + ( . # ρ( x, y) +
" | x − y | . # – ( .
2. .( X – + ( x = ( x1 , x2 ) , y = ( y1 , y2 ) . &-
:
ρ( x, y) = 
( x1 − y1 )2 + ( x2 − y2 )2 .
# , ( , " ( , * " " 2 . 3. .( X – + " ( ( :
x = ( x1 , x2 , x3 ) , y = ( y1 , y2 , y3 ) . &:
ρ( x, y) = 
( x1 − y1 )2 + ( x2 − y2 )2 + ( x3 − y3 )2 .
) , ( , " ( , * " " 3 .
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ρ( x, y) = ∑( x j − y j )2 .  | 
	(2.1)  | 
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* 3). " * * 6–
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	∑a jbj  | 
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	∑b2j .  | 
	
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	(2.2)  | 
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P(x) = ∑(a j + bj x)2 = A + 2Bx + Cx2 ,  | 
	
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A = ∑a2j ,  | 
	B = ∑a jbj ,  | 
	C = ∑b2j  | 
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! * "  | 
	x , ( (,  | 
	B2 − AC ≤ 0 , !  | 
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, * (2.2).  | 
	
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	∑(a j  | 
	+ bj )2 ≤  | 
	∑a2j +  | 
	
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	∑b2j .  | 
	
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	(2.3)  | 
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2 6–" * ,:  | 
	
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∑(a j + bj )2 = ∑a2j + 2∑a jbj + ∑b2j ≤  | 
	
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j =1  | 
	
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≤ ∑a2j + 2  | 
	
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	∑a2j  | 
	∑b2j + ∑b2j  | 
	=  | 
	∑a2j +  | 
	∑b2j  | 
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j =1  | 
	
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! ' ! ( *,
, * (2.3).
% '  | 
	(2.3) ai = xi  | 
	− zi , bi = zi − yi , , *  | 
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	∑( xi − yi )2 ≤  | 
	∑( xi − zi )2 +  | 
	∑( zi − yi )2 .  | 
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	j =1  | 
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2 ’" / *1 ’" ( , " ( , ," ( , * ( , ,
.
3..
* + + "
( ( ), ! , " ! "
+ .
1 / * ( (1864–1909) – * (
5
. {xk } X
, * " * * :
x1, x2 ,..., xk ,...  | 
	
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	, * " {xk } , "  | 
lim ρ( xk , a) = 0 . * 6: lim xk = a .  | 
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k → ∞  | 
	
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	k → ∞  | 
# ! lim xk  | 
	= a , "  | 
	ε > 0 N k > N : ρ( xk , a) < ε .  | 
k → ∞  | 
	
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" ( * ,
, " , * " " ( .  | 
	
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1. {xk }  | 
	a , M > 0  | 
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, k  | 
	: ρ( xk , a) ≤ M .  | 
	
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	) *  | 
	lim xk = a ,  | 
	
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	ε > 0 N k > N : ρ( xk , a) < ε .  | 
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%:  | 
	M 0 = max(ρ( x1, a),ρ( x2 , a),...,ρ( x N , a)) . # k : ρ( xk , a) ≤ M ,  | 
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M = max(M 0 , ε) .  | 
	
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2. {xk }  | 
	, .  | 
	
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. %, *  | 
	{xk }  | 
	,  | 
	:  | 
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lim xk = a ,  | 
	lim xk = b ,  | 
	
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	a ≠ b ,  | 
	
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	ρ(a,b) > 0 .  | 
	# ε > 0 N  | 
	1  | 
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k → ∞  | 
	
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k > N  | 
	1  | 
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	ρ( xk , a) < ε 2 ,  | 
	N  | 
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	2  | 
	: ρ( xk  | 
	,b) < ε 2 . 2 '  | 
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0 < ρ(a,b) ≤ ρ(xk , a) + ρ( xk ,b) <  | 
	ε  | 
	+  | 
	ε  | 
	= ε .  | 
	
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%:  | 
	ε = ρ(a, b) 2 . #  | 
	,: 0 < ρ(a, b) < ρ(a, b)  | 
	2 ,  | 
	! 1 < 1 2 ,  | 
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+.  | 
	
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+ X :  | 
	
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Sr (a) = {x X : ρ(a, x) < r} .  | 
	
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2, " X = , Sr (a) = (a − r, a + r ) .  | 
	
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3. , {xk }  | 
	
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n , xk = ( x  | 
	,..., x  | 
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	a = (a ,...a  | 
	) ,  | 
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lim xkj  | 
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k → ∞  | 
	
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. .! *. .( lim xk = a . #  | 
	lim ρ( xk , a) = 0 . 2  | 
k → ∞  | 
	k → ∞  | 
' (2.1):  | 
	
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0 ≤| xkj − a j |≤ ∑( xkj − a j )2 = ρ( xk , a) ,  | 
	
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,, lim | xkj  | 
	− a j  | 
	|= 0 ( j = 1, 2,..., n) .  | 
	
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*. .(  | 
	lim xkj  | 
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	( j = 1, 2,..., n) .  | 
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( j = 1, 2,..., n) , +  | 
	
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lim ρ( xk , a) = lim ∑( xkj − a j )2  | 
	= 0 .  | 
	
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k → ∞  | 
	k → ∞  | 
	
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, * " ",  | 
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	ε > 0 N  | 
	,  | 
	
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	k , m > N  | 
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: ρ( xk , xm ) < ε .  | 
	
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4.  | 
	{xk }  | 
	
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, ".  | 
	
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	ε > 0 N  | 
	,  | 
	
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	k > N ,  | 
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	k → ∞  | 
	
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m > N : ρ( xk , a) < ε 2 ,  | 
	ρ( xm , a) < ε 2 .  | 
	#  | 
	
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	'  | 
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:  | 
	ρ( xk , xm ) ≤ ρ( xk , a) + ρ( xm , a) < ε 2 + ε 2 = ε ,  | 
	
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	(  | 
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* *.  | 
	
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! ". .  | 
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	+ "  | 
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. # ! * X *  | 
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* + ! ! + '.  | 
	
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. .( X =  | 
	– +  | 
	*  | 
	.  | 
	
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	+  | 
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* p r " | p − r | ( ",
). $" X * {xk } ,
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xk  | 
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&, lim xk = e , ! " * ! +, + *.
k → ∞
e , + X " * ,.
4. # " .
. ( G – + X .
. / + G , * " #, " ,
x0 X r > 0 , G Sr ( x0 ) .
. # x G , * " $ + G , " , ε > 0 , Sε ( x) G .
7
# ! ' x + G ( + + * ( " "
x ( . 1).
$ . 1.
* 6 + G , * " " int G . ), int G G . 5 int G = G , ! + G , * "
* 6 , + G , * " '
X . %.
1. .( X = , G = (a,b) . # ! " + G " , * "
( " (. 7 , , '
+ '. (, ( x – * (a,b) . $" ε -
, (! ' Sε ( x) ), ε < min(b − x, x − a) . ), (
+ * (a,b) .
2." , ' + '. (,
" X ' Sr (a) . . ( x0 Sr (a) . " Sε ( x0 ) ,
ε = r − ρ( x0 , a) , + * Sr (a) . (, ( x Sε ( x0 ) . #
ρ( x, a) ≤ ρ( x, x0 ) + ρ( x0 , a) < ε + ρ( x0 , a) = r − ρ( x0 , a) + ρ( x0 , a) = r . 8 ( ,,
x Sr (a) . ) * x – * Sε ( x0 ) , Sε ( x0 ) Sr (a) . & + ' * .
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. ( X – ( .
. % x0 X , * " ! * " +
X , " " x0 , 6 * '.
. ε - x0 X , * " " Sε ( x0 ) .
. # x0 , * " + G X ,
" ! * " x0 ' * + G .
+ G + + + G , + (
+. . , (a,b) , ( . # a b + ( , (a,b)
+ *.
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., " (a,b) + , x = a, x = b .
! * " + + G ' * , "
+ G + *, , " + G + * ( . 2).
$ . 2.
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+ .
9
., " Sr (a) + ' ! X , !
{ x X : ρ(a, x) = r} .
. )!’, " + + , * "
#.
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+ + * ( +.
5. .
. ( G – + n , ! .
. 5 + ( M ( x1 , x2 ,..., xn ) + G
* ( * ( u , + *, + G " u = f ( x1 , x2 ,..., xn ) .
2 x1 , x2 ,..., xn ' * " # ! ,
u – # , ! ". / + G , * "
, * " D f . / + *
, * " E f .
n = 2 , ' 2- : u = f ( x1 , x2 ) . !, "
6 ! , u = f ( x, y) , ! z = f ( x, y) . )! ' "
2- , " + 2 . n = 3
, ' 3- : u = f ( x1 , x2 , x3 ) , ! u = f ( x, y, z) . )! '
" 3- , " + 3 .
.
1. 2  ( !    * " z = 
4 − x2 − y2 .
)! ' " , , + ,
" * "' * * 4 − x2 − y2 ≥ 0 , ! x2 + y2 ≤ 4 . # '
+ ' , r = 2 . 2. 2 ( ! * "
u = arcsin  | 
	
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z
≤ 1.
x2 + y2
# , !: x2 + y2 > 0 , | z |≤ 
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, " 6 + | z |= −
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 x2 + y2 ,
' ' * 6 – .
10
5 , ! + " , y = f ( x) ,
, , ! " Oxy , ( x, y) +
" ( ’" 6 " y = f ( x) .
$" ' 2- z = f ( x, y) , " + G
Oxy , " Oxyz
3 .
. z = f ( x, y) , * "
3 ( x, y, f ( x, y)) .
2 , " " z = f ( x, y) ,
3 " '. # , ,
" P , " , * " Oxy + G ( . 3).
$ . 3.
., z = x2 + y2 , ! ! " ( . 4).
, * " n u = f ( x1 , x2 ,..., xn )
n > 2 . 8 + n+1 , " ' *
" ( x1 , x2 ,...xn , f ( x1 , x2 ,...xn )) . " '
* +.
11
