Документ взят из кэша поисковой машины. Адрес оригинального документа : http://www.students.chemport.ru/materials/matan/m4/l1.pdf Дата изменения: Thu Jan 15 17:56:14 2009 Дата индексирования: Mon Oct 1 22:57:12 2012 Кодировка:

N 1


, .. : D ь , D D1 D2,

n

: R , - . : , , . n : D R , D1 , D2 ­ : D1 D ь, : D2 D1 D2 =ь

.1 D : ­ , ­ . 1 2: 1. 1 2 . 2. 1 2 ­

.2 Pn , Pn­ ; , . Pn (. 2): Sn= S . : S(K) K : S(K)=i nf S(Pn). , .. ­ . S (K): 1. S(K) 0 2. S(1 2 )=0 S(1 2 )= S(K1) + S(K2) : y = f(x) [a,b]: 1. K = {(x,y ): x [a,b] ; y = f(x)}; , S(K) = 0.

.3 [a,b] n , mi = min f(x);
i


b-a i n f(x) [a,b] ( ) > 0 n0 n n0: M i - mi< . K Pn : M i = max f(x); i = S (Pn)=


i= 1

n

( M i - mi ) i <


i =1

n

i =


i =1

n

i = (b - a) 0 0

, S(K)=0.

2. f(x) C[a,b]; f( x)>0; K={(x,y ): x [a,b] ;0 y f(x)}; y = f(x), x [a,b] . , S(K) =


a

b

f ( x )dx

.4 M i = max f(x)
i n b

S (Pn)=


i= 1

Mi i




a

f ( x) dx ( , ..

).

.5 . f ( x, y )dxdy : : ­ : {K =
K


i= 1

n

K i ; Ki: S(i j ) = 0, i j}

( i ,i ), i , i = 1,..., n S (T )=


i= 1

n

f ( i , i ) S ( K i ) ,

: d(Ki)=ma x( ( x, y ), ( x, y ) , ( x , y ) K i ; ( x , y ) K i ), ( x, y ), ( x, y ) = ( x - x ) 2 + ( y - y ) 2 - : d(T ) = maxi di . : f( x,y ) : f ( x, y )dxdy = lim S (T ) , .
K d ( T )0


( , , D( x,y ), 1, ).

(1 -5)
1.


K

f ( x, y )dxdy = S(K), f( x,y ) 1

2. S(K) = 0 3.


K

f ( x, y )dxdy =0, f-


K

( f ( x, y ) + µg( x , y ))dxdy =


K

f ( x, y )dxdy + µ


K

g ( x, y )dxdy

4.S(1 2 )=0

K1 K



f ( x, y )dxdy =
2


K
1

f ( x, y )dxdy +


K
2

f ( x, y )dxdy

5.m f(x,y ) M mS(K)


K

f ( x, y )dxdy M S(K)

6. - f( x,y ) C(K), ( 0 , 0 ) K : f ( 0 , 0 ) S ( K ) = f ( x, y )dxdy 1-5: 1.
K


K

f ( x, y )dxdy = S ( )=

lim S (T )
d ( T )0


i= 1

n

S (K i ) = S ( K )


K

f ( x, y )dxdy =S(K)

2. S(K)=0, , : S(Ki)=0 , i = 1,2,...n S ( )=


i= 1

n

f ( i ,i ) S ( K i ) = 0


K

f ( x , y ) dxdy =

lim S (T )
d ( T )0

=0

3. S ( )=


i= 1 n

n

[ f (i , i )S (K i ) + µg ( i , i )S ( K i )] = S(T ,f) + µ S(T ,g)


K

f ( x, y )dxdy + µ


K

g ( x, y )dxdy .

4. S ( )=

K i K


2

f ( i , i ) S ( K i ) +

K j K



n

f ( i , i ) S (K j ) +
2

K t K1 K



n

f ( it , tt ) S ( K t )
2


K
1

f ( x, y )dxdy +


K
2

f ( x, y )dxdy

5. m f(x,y ) M S ( )= mS(K)= mS(K)

f ( m S f (
i= 1 K

n

i

,i ) S ( K i )

(K i ) S (T ) M



S (K i ) =M S(K)

x , y )dxdy M S(K)

6. m


K

fdxdy M , f

S(K)

M m.


S(K) f(x,y) : (f(x,y)>0) V= f ( x, y )dxdy ­ , .6
K

( 0 , 0 ) K f ( 0 , 0 ) =


K

fdxdy

.6 (-): f( x,y )- ,


K

f ( x, y )dxdy .


K

: = 1 2 S(K2)=0, 2, .. S(K2)=0 f ( x , y ) dxdy = f ( x, y )dxdy
K
1