Кратные, криволинейные и поверхностные интегралы: Учебное пособие

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  • ,

    2008

  • 2

    517.373, 514.742.4, 512.623

    ( ),

    .

    .

    ,

    ,

    .

    : . . , .. .

    : ..-.., ...

  • 3

    .

    ,

    ,

    .

    : ( ) ,

    ,

    , , ,

    , -.

    .

    ,

    ,

    ,

    ,

    ,

    .

    , .

    ,

    ,

    , - .

    .

  • 4

    I.

    1. , .

    D,

    L. -

    1 2, , ..., nS S SD D D ( 1 2, , ..., nS S SD D D

    ),

    d1, d2, ..., dn. di

    iSD . iSD i (.1).

    .1.

    D z = f(x, y). f(P1),

    f(P2),, f(Pn)

    f(Pi)Si :

    1

    ( )n

    n i ii

    V f P S=

    = D . (1)

    1. 1

    ( )n

    n i ii

    V f P S=

    = D

    f(x, y) D.

  • 5

    . ( ( , ) 0f x y )

    (1)

    Si f(Pi).

    2.

    (1) n max 0id ,

    D , Pi ,

    f(x, y) D

    max 0 1

    ( , ) lim ( )i

    n

    i id iD

    f x y dxdy f P S =

    = D . (2)

    f (x,y)

    D, D , -

    , dxdy = dS .

    1.

    ,

    D ,

    . ,

    f(x, y)

    , -, D, -,

    max 0

    lim ( ) 0,id

    S st t - = (3)

    , S s

    .

    , .

    2.

    : f(x, y)

    D, .

  • 6

    , :

    1. f(x, y) D, kf(x, y), k = const,

    ,

    ( , ) ( , ) .D D

    kf x y dxdy k f x y dxdy= (4)

    2. D f(x, y) g(x, y),

    f(x, y) g(x, y),

    ( )( , ) ( , ) ( , ) ( , ) .D D D

    f x y g x y dxdy f x y dxdy g x y dxdy = (5)

    3. D f(x, y) g(x, y)

    f(x, y) g(x, y) ,

    ( , ) ( , ) .D D

    f x y dxdy g x y dxdy (6)

    :

    4. D D1 D2

    f(x, y) D,

    1 2

    ( , ) ( , ) ( , ) .D D D

    f x y dxdy f x y dxdy f x y dxdy= + (7)

    .

    D :

    1 2

    ( ) ( ) ( ) ,i i i i i iD D D

    f P S f P S f P SD = D + D

    D , D1 D2

    . max 0id ,

    (7).

    5. D f(x, y)

    | f(x, y) |,

  • 7

    ( , ) | ( , ) | .D D

    f x y dxdy f x y dxdy (8)

    .

    ( ) | ( ) | ,i i i iD D

    f P S f P SD D

    max 0id (8).

    6. ,DD

    dxdy S= SD D.

    , f(x, y) 1.

    7. D f(x, y)

    m f(x, y) M,

    ( , ) .D DD

    mS f x y dxdy MS (9)

    ( ) .D i i DD

    mS f P S MS D

    8 ( ). f (,)

    D, (0 , 0),

    0 01 ( , ) ( , )D D

    f x y dxdy f x yS

    = , (10)

    , ,

    ( , ) , .DD

    f x y dxdy S m Mm m= (10)

    , (9) SD.

    ( -)

    .

  • 8

    V,

    S.

    f(x, y, z). V

    vi , vi ,

    ( )i iV

    f P vD , (11)

    Pi vi .

    V.

    , vi

    , .. .

    3. 0r (11),

    V Pi

    ,

    f(x, y, z) V:

    0

    ( , , ) limV

    f x y z dxdydzr

    = ( )i iV

    f P vD (12)

    1.

    (,

    ..) , ,

    , .

    2.

    .

    3.

    , ,

    - .

  • 9

    V, ,

    z = f(x, y), D

    ,

    , .

    . 2

    , Si D,

    f(Pi), Pi Si.

    max 0iSD , ,

    ( , ) ,D

    V f x y dxdy= (13)

    , z = f(x, y),

    D.

    2.

    D,

    1 2 1 2( ), ( ) ( ( ) ( )),y x y x x xj j j j= = x = a, x = b ( a < b ), 1() 2()

  • 10

    [a, b]. ,

    D,

    : N1 N2 (.1),

    . ,

    . ,

    , . ,

    .3.

    f(x, y) D.

    2

    1

    ( )

    ( )

    ( , )b x

    Da x

    I f x y dy dxj

    j

    = , (14)

    f(x, y) D.

    ( )

    , .

    :

    .3

    2

    1

    ( )

    ( )

    ( ) ( , ) .x

    x

    x f x y dyj

    j

    F =

    b.

    ( ) .b

    Da

    I x dx= F

  • 11

    .

    1. D, ,

    D1 D2 , ,

    D

    D1 D2:

    1 2D D DI I I= + . (15)

    .

    ) = D D1 D2 ,

    .

    2

    1

    ( )

    ( )

    ( , ) ( ) ( ) ( )b x b c b

    Da x a a c

    I f x y dy dx x dx x dx x dxj

    j

    = = F = F + F =

    2

    1

    ( )

    ( )

    ( , )c x

    a x

    f x y dy dxj

    j

    +

    2

    1 2

    1

    ( )

    ( )

    ( , ) .b x

    D Dc x

    f x y dy dx I Ij

    j

    = +

    ) y = h D

    D1 D2 (.2). M1 (a1, h) M2 (b1, h)

    y = h L D.

    .4.

    D1

    1) y = 1(x);

  • 12

    2) 112, y = 1*(x),

    1*() = 2() 1 b1 x b, 1*() = h 1 b1;

    3) x = a, x = b.

    D2 y = 1*(x), = 2(), 1 b1.

    : *

    2 1 2

    *1 1 1

    ( ) ( ) ( )

    ( ) ( ) ( )

    ( , ) ( , ) ( , )b x b x x

    Da x a x x

    I f x y dy dx f x y dy f x y dy dxj j j

    j j j

    = = + =

    *1 2

    *1 1

    ( ) ( )

    ( ) ( )

    ( , ) ( , ) .b x b x

    a x a x

    f x y dy dx f x y dy dxj j

    j j

    = +

    :

    2

    *1

    ( )

    ( )

    ( , )b x

    a x

    f x y dy dxj

    j

    =

    1 2

    *1

    ( )

    ( )

    ( , )a x

    a x

    f x y dy dxj

    j

    +

    1 2

    *1 1

    ( )

    ( )

    ( , )b x

    a x

    f x y dy dxj

    j

    +

    +2

    *1 1

    ( )

    ( )

    ( , )b x

    b x

    f x y dy dxj

    j

    .

    1*() = 2() 1 b1 x b,

    . ,

    ID =

    *1

    1

    ( )

    ( )

    ( , )b x

    a x

    f x y dy dxj

    j

    +

    1 2

    *1 1

    ( )

    ( )

    ( , )b x

    a x

    f x y dy dxj

    j

    ,

    1 2D D DI I I= + .

    . D

    . D

    .

  • 13

    1. 1

    , ,

    :

    mS 2

    1

    ( )

    ( )

    ( , )b x

    a x

    f x y dy dxj

    j

    ,MS (16)

    f(x, y) D, S ,

    ID = f(P)S, (17)

    , D .

    2.

    2

    1

    ( )

    ( )

    ( , )b x

    a x

    f x y dy dxj

    j

    =

    2

    1

    ( )

    ( )

    ( , ) .b x

    a x

    dx f x y dyj

    j (18)

    2. f(x, y)

    D

    ,

    ( , )D

    f x y dxdy = 2

    1

    ( )

    ( )

    ( , )b x

    a x

    f x y dy dxj

    j

    . (19)

    .

    D , ,

    ( ) S1, S2,, Sn.

    1

    1 2

    1...

    n i

    n

    D S S S Si

    I I I I ID D D D=

    = + + + = .

    (16) : 1

    ( ) , ( )i

    n

    S i i D i ii

    I f P S I f P SD=

    = D = D ,

    , f

  • 14

    D, ID .

    max 0iSD , (19).

    1.

    z = x + y ,

    (0,0), (0,1)

    (1,0) (.5).

    . 5

    = 0, b = 1, 1(x) = 0, 2(x) = 1 x.

    1 1

    0 0

    ( ) ( )x

    D

    x y dxdy dx x y dy-

    + = + =

    1 1 22

    0 0

    1 (1 )( (1 )02 2

    x xydx xy x x dx- - = + = - + =

    1 32

    0

    11 1 1(1 ) .02 2 3 3

    xx dx x= - = - =

    3.

    ,

    . ()

    ( ).

  • 15

    . 6 . 7

    (. 6)

    : (,).

    , , , > 0,

    .

    . [0,2] [-,

    ],

    (

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