
Диференціальне числення ФБЗ
.pdf517.2
22.161.1
92
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( 1 1 24.10.2013).

1..
( "
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+ * * , + x . $ "
, , * "
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*, + , .
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1. * "+ " (
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r + :
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' + * ,
+ ( , " ( . "
R = ρ l , S
l – + , S – ( , ρ – (
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3. )!’, + * – (
:
V= πR2 H .
4.& y * + *
+ x1 ( * !) !’,
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( a0 , a1 , a2 – ).
& " * " " . "
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3

2. n .
. / + X , * " , " + ( x y , + *
’, ρ( x, y) , " , * " + x y ,
, :
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 .
% , 2 , 3 + " '.
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z = ( z1 , z2 ,..., zn ) , * + |
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ρ( x, y) = ∑( x j − y j )2 . |
(2.1) |
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& 1), 2) , , . % +,
* 3). " * * 6–
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∑a jbj |
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(2.2) |
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P(x) = ∑(a j + bj x)2 = A + 2Bx + Cx2 , |
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j =1 |
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A = ∑a2j , |
B = ∑a jbj , |
C = ∑b2j |
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j =1 |
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! * " |
x , ( (, |
B2 − AC ≤ 0 , ! |
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, * (2.2). |
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* / *: |
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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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! ' ! ( *,
, * (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 |
j =1 |
j =1 |
2 ’" / *1 ’" ( , " ( , ," ( , * ( , ,
.
3..
* + + "
( ( ), ! , " ! "
+ .
1 / * ( (1864–1909) – * (
5

. {xk } X
, * " * * :
x1, x2 ,..., xk ,... |
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4 a |
, * " {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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k → ∞ |
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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 → ∞ |
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k > N |
1 |
: |
ρ( xk , a) < ε 2 , |
N |
2 |
k > N |
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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2 |
2 |
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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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. r |
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a X |
, * " |
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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 |
) |
a = (a ,...a |
) , |
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k1 |
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lim xkj |
= a j ( j = 1, 2,..., n) . |
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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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6

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n |
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0 ≤| xkj − a j |≤ ∑( xkj − a j )2 = ρ( xk , a) , |
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j =1 |
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,, lim | xkj |
− a j |
|= 0 ( j = 1, 2,..., n) . |
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k → ∞ |
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*. .( |
lim xkj |
= a j |
( j = 1, 2,..., n) . |
# |
lim | xkj |
− a j |
|= 0 |
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k → ∞ |
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k → ∞ |
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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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j =1 |
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% * |
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{xk } |
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X |
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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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lim xk = a . # |
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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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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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k → ∞
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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) . $" ε -
, (! ' Sε ( x) ), ε < min(b − x, x − a) . ), (
+ * (a,b) .
2." , ' + '. (,
" X ' Sr (a) . . ( x0 Sr (a) . " Sε ( x0 ) ,
ε = r − ρ( x0 , a) , + * Sr (a) . (, ( x Sε ( x0 ) . #
ρ( x, a) ≤ ρ( x, x0 ) + ρ( x0 , a) < ε + ρ( x0 , a) = r − ρ( x0 , a) + ρ( x0 , a) = r . 8 ( ,,
x Sr (a) . ) * x – * Sε ( x0 ) , Sε ( x0 ) Sr (a) . & + ' * .
# |
1. |
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# 2. %’ |
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.( G = G1 G2 , |
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+ * ( , + G |
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Sε ( x0 ) G , ! x0 – 6 " + G . & *
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$ . %’ & # #
#.
# 3. # #.
8

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.( |
G = G1 ∩ G2 , |
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– |
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x0 G . # x0 G x0 G . # |
ε > 0 , S |
ε |
(x0 ) G , |
ε |
2 |
> 0 |
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(x0 ) G . % ε = min(ε , ε |
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( x0 ) G , |
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+ S |
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$ . |
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+ , |
x = 0 . + {x} , |
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X , " " x0 , 6 * '.
. ε - x0 X , * " " Sε ( x0 ) .
. # x0 , * " + G X ,
" ! * " x0 ' * + G .
+ G + + + G , + (
+. . , (a,b) , ( . # a b + ( , (a,b)
+ *.
. ' # + G , * "
, +, " ( + + *.
., " (a,b) + , x = a, x = b .
! * " + + G ' * , "
+ G + *, , " + G + * ( . 2).
$ . 2.
. ' # + G , * " +
+ .
9

., " Sr (a) + ' ! X , !
{ x X : ρ(a, x) = r} .
. )!’, " + + , * "
#.
., [a,b] , ' + '. ),
+ + * ( +.
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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x2 + y2 |
)! ' " , , + ,
" * "' * *
z
≤ 1.
x2 + y2
# , !: x2 + y2 > 0 , | z |≤ x2 + y2 . 8 +
, " 6 + | z |= − x2 + y2 | z | =
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