ΑΣΚΗΣΕΙΣ ΔΙΠΛΑ ΤΡΙΠΛΑ ΕΠΙΚΑΜΠΥΛΙΑ ΕΠΙΦΑΝΕΙΑΚΑ...

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  • LUMENENES ASKHSEIS ANALUSH-III

    0.1 DIPLA OLOKLHRWMATA

    'Askhsh 1. Na upologiste to embadn thc lleiyhc

    x2

    a2+y2

    b2= 1

    Lsh: To sunolik embadn thc lleiyhc ja enai to tetraplsio tou embado pou an-

    tistoiqe sto prto tetarthmrio. Mporome na upologsoume to zhtomeno embadn me

    arketoc trpouc. Treic ap autoc enai oi paraktw:

    a' trpoc): 'Ean qrhsimopoisoume kartesianc suntetagmnec to embadn E ja dnetaiapo to dipl oloklrwma

    E = 4

    D

    dxdy =

    ax=0

    b1x2a2

    y=0

    dxdy = 4

    ax=0

    b1x2a2y=0

    dy

    dx= 4b

    ax=0

    1 x

    2

    a2dx = 4b

    api

    4= abpi

    b' trpoc): Ja qrhsimopoisoume polikc suntetagmnec me ton paraktw metasqh-

    matism

    x = cos , y = sin , dxdy = dd

    Ta ria oloklrwshc ja ta brome ap thn profan sunjkh

    x2

    a2+y2

    b2 1 0

    pou me antikatstash kai met ap merikc prxeic brskoume

    abb2 cos2 + a2 sin2

    To embadn ja enai tra

    E = 4

    pi/2=0

    [ = abb2 cos2 +a2 sin2

    =0

    d

    ]d

    = 2a2b2 pi/2=0

    1

    b2 cos2 + a2 sin2 d = 2a2b2

    pi

    2ab= abpi

    Diapistnoume ti me touc do parapnw trpouc h duskola ggeite ston upologism

    tou oloklhrmatoc wc proc x (a' trpoc) kai wc proc (b' trpoc). Efarmzontac mwcton paraktw metasqhmatism o upologismc tou embado aplousteetai shmantik

    g' trpoc): Ja qrhsimopoisoume pli polikc suntetagmnec me ton paraktw mwc

    metasqhmatism

    x = a cos , y = b sin , dxdy = abdd

    Ta ria oloklrwshc ja ta brome ap thn profan sunjkh

    x2

    a2+y2

    b2 1 0

    1

  • pou me antikatstash kai met ap merikc prxeic brskoume

    1To embadn ja enai tra

    E = 4

    pi/2=0

    [ =1=0

    abd

    ]d = 4ab

    1

    2

    pi/2=0

    d = 2abpi

    2= abpi

    Updeixh : 'Opwc ja dexoume sto 5o Keflaio, to parapnw embadn mporome ekola

    na to upologsoume efarmzontac ton tpo tou Green, angwntac tsi to dipl oloklrwmase na kleist epikamplio oloklrwma.

    'Askhsh 2. Na upologisje to dipl oloklrwma

    I =

    D

    (1 x2 y2)1/2dxdy

    tan o tpoc D perikleetai ap thn kamplh x2 + y2 = xLsh: H kamplh pou peribllei ton tpo mpore, met ap merikc prxeic na grafe

    x2 + y2 = x =(x 1

    2

    )2+ y2 =

    (1

    2

    )2'Enac katllhloc metasqhmatismc se polikc suntetagmnec enai o akloujoc

    x = cos , y = sin , 0 cos , pi2 pi

    2

    To dipl oloklrwma, kai lambnontac upyh ti h oloklhrwta sunrthsh enai sum-

    metrik wc proc ton xona twn x ja enai

    I =

    D

    (1 x2 y2)1/2dxdy = 2

    pi/2=0

    [ =cos =0

    (1 2)1/2d]d

    = 2

    pi/2=0

    [(13

    ) =cos =0

    d(1 2)3/2] d

    = 23

    pi/2=0

    ((1 cos2 )3/2 1) d = 2

    3

    pi/2=0

    (| sin |3 1) d= 2

    3

    pi/2=0

    (sin 3 1) d = 2

    3

    1

    6(4 3pi) = 3pi 4

    9

    Updeixh: 'Enac deteroc metasqhmatismc enai o paraktw

    x =1

    2+ cos , y = sin , 0 1

    2, 0 2pi

    kai tte to oloklrwma ja qei thn morf

    I =

    D

    (1 x2 y2)1/2dxdy

    =

    2pi=0

    [ 1/2=0

    (3

    4 2 cos

    )1/2d

    ]d

    2

  • To parapnw mwc oloklrwma, mpore se sqsh me to prohgomeno oloklrwma na qei

    anexrhtec wc proc thn oloklrwsh tic metablhtc kai all exaiteac thc oloklhrwtac

    sunrthshc

    (34 2 cos )1/2 enai duskoltero na upologiste. O upologismc touafnetai san skhsh.

    'Askhsh 3. Na upologiste o gkoc tou stereo pou orzetai ektc tou

    paraboloeidoc x2 + y2 = az, entc tou kulndrou x2 + y2 = 2ax kai pnwap to eppedo xOy.Lsh: 'Opwc fanetai sto sqma ton klindro skepzei h epifneia tou paraboloeidoc,

    en o tpoc oloklrwshc D enai h bsh tou kulndrou. H kamplh pou peribllei tontpo mpore, met ap merikc prxeic na grafe

    x2 + y2 = 2ax = (x a)2 + y2 = a2

    'Enac katllhloc metasqhmatismc se polikc suntetagmnec enai o akloujoc

    x = cos , y = sin , 0 2a cos , pi2 pi

    2

    Sunepc o gkoc tou stereo ja dnetai ap to oloklrwma

    I =

    D

    x2 + y2

    adxdy =

    G

    1

    a2dd

    =1

    a

    pi/2=pi/2

    [ 2a cos =0

    3d

    ]d =

    1

    a

    1

    4(2a)4

    pi/2=pi/2

    cos4 d

    = 16a33pi

    8=

    3pia3

    2

    'Askhsh 4. Na breje o gkoc tou stereo tou prtou ogdou twn axn-

    wn pou orzetai ap thn kulindrik epifneia x2+y2 = a2 kai ap to eppedoz = x+ y.Lsh: O gkoc tou parapnw stereo periorzetai ap pnw ap thn epifneia z = x+ ykai ap ktw ap to eppedo Oxy. Epshc o gkoc problletai ston tpo D pou enai toto kommti tou kukliko dskou x2 + y2 = a2 pou brsketai sto prto ogdohmrio. Jaqrhsimoposoume polikc suntetagmnec me ta antstoiqa ria, dhlad

    x = cos , y = sin , dxdy = dd, 0 a, 0 pi2

    O gkoc tou stereo upologzetai wc exec

    V =

    D

    (x+ y)dxdy =

    G

    ( cos + sin )dd

    =

    a=0

    2d pi/2=0

    (cos + sin ) =1

    3a3 2 = 2a

    3

    3

    'Askhsh 5. Na upologiste to oloklrwma

    D

    (x 2y)dxdy pou D otpoc pou perikleetai ap ton kklo x2 + y2 = R2, thn lleiyh x

    2

    a2+ y

    2

    b2= 1

    3

  • kai touc jetikoc hmixonec Ox, Oy (a, b > R).

    Lsh: a' trpoc: To oloklrwma mpore na grafe

    I = Ie Ic =De

    (x 2y)dxdy

    Dc

    (x 2y)dxdy

    pou De kai Dc oi perioqc pou perikleontai ap thn lleiyh (kai touc jetikoc hmixonec)kai ton kklo (kai touc jetikoc hmixonec). Gia to upologism tou oloklhrmatoc Ieepilgoume thn parametropohsh

    x = a cos , y = b sin , dxdy = abdd, 0 1, 0 pi2

    kai gia to Ic thn

    x = cos , y = sin , dxdy = dd, 0 R, 0 pi2

    Ja qoume tra

    I =

    De

    (x 2y)dxdy

    Dc

    (x 2y)dxdy

    =

    De

    (a cos 2b sin ) abdd

    Dc

    ( cos 2 sin ) dd

    =

    pi/2=0

    1=0

    a2b2 cos dd pi/2=0

    1=0

    2ab22 sin dd

    ( pi/2

    =0

    R=0

    2 cos dd pi/2=0

    R=0

    22 sin dd

    )

    =a2b

    3 2b

    2a

    3(R3

    3 2R

    3

    3

    )=ab

    3(a 2b) + R

    3

    3

    b' trpoc

    1

    : Mporome na upologsoume to oloklrwma efarmzontac katllhla to

    jerhma tou Green. 'Etsi smfwna me ton tpo tou Green qoume

    I =

    D

    (Q

    x P

    y

    )dxdy =

    c

    Pdx+Qdy

    An epilxoume wc kleist kamplh (stp prto tetarhmrio pnta) aut pou apoteletai ap

    tic perifreiec thc lleiyhc kai tou kklou kai ap touc xonec x = 0 kai y = 0 kai epshcepilxoume P = y2 xy kai Q = 0 ja qoume

    I =

    D

    (Q

    x P

    y

    )dxdy =

    D

    (x 2y)dxdy =

    c

    (y2 2x)dx

    1 pi Green pi pi 5 . pi pi pi pi Green

    4

  • H kamplh pnw sthn opoa upologzetai to oloklrwma perilambnei tic ce, cc ,x = 0 kaiy = 0. Profanc mwc ta oloklhrmata stouc xonec x = 0 kai y = 0 den suneisfrounafo h posthta P mhdenzetai pnw se autc. Apomnei o uplogismc pnw stic peeifreicth lleiyhc kai tou kklou. Epilgoume antstoiqa tic paraktw parametropoiseic

    ce : x = a cos , y = b sin , 0 pi2

    cc : x = R cos , y = R sin , 0 pi2

    Ja qoume tra

    I =

    c

    (y2 2x)dx =ce

    (y2 2x)dx+cc

    (y2 2x)dx

    =

    pi/2=0

    (b2 sin2 ab cos sin ) (a sin )d

    +

    0=pi/2

    (R2 sin2 2R2 cos sin ) (R sin )d

    = 2b2a

    3+a2b

    3+

    (R

    3

    3+2R3

    3

    )=ab

    3(a 2b) + R

    3

    3

    'Askhsh 6. Na upologiste o gkoc tou stereo pou orzetai ap tic

    epifneiec x2 + y2 2x = 0, z = x, z 0Lsh: O gkoc tou stereo entopzetai sto eswteriko tou kulndrou x2 + y2 2x =0 (x 1)2 + y2 = 1 o opooc perikleetai ap tic epifneiec z1 = z = 0 (ap ktw) kaiz2 = z = x (ap pnw). Profanc h probol tou tpou, D sto eppedo xy enai o kuklikcdskoc (x 1)2 + y2 = 1. Ja qoume sunepc

    V =

    D

    (z2 z1)dxdy =

    D

    xdxdy

    'Enac katllhloc metasqhmatismc se polikc suntetagmnec enai o exc

    x = 1 + cos , y = sin , dxdy = dd

    me profan ria

    0 1, 0 2piO gkoc sunepc ja dnetai apo to dipl oloklrwma

    V =

    D

    xdxdy =

    G

    (1 + cos )dd

    =

    2pi=0

    d

    1=0

    d+

    2pi=0

    cos d

    1=0

    2d = pi + 0 = pi

    Updeixh: Mporome na qrhsimopoisoume kai ton paraktw metasqhmatism se polikc

    suntetagmnec

    x = cos , y = sin , dxdy = dd

    5

  • pou se autn th perptwsh ta ria den enai profan kai upologzontai wc exc

    a) Ap thn exswsh tou kukliko dskou kai apaitntac 0 (x 1)2 + y2 1 ja qoume0 2 cos kai epshcb) apaitntac = 2 cos = 0 ja qoume = pi

    2kai ra pi

    2 pi

    2

    To oloklrwma plon gnetai

    V =

    D

    xdxdy =

    G

    ( cos )dd =

    pi/2=pi/2

    d

    2 cos =0

    2d

    =

    pi/2=pi/2

    1

    3(2 cos )3 cos d =

    8

    3

    pi/2=pi/2

    cos4 d =8

    3

    3pi

    8= pi

    'Askhsh 7. Na upologisje, me thn qrsh diplo oloklhrmatoc, o

    gkoc tou elleiyoeidoc

    x2

    a2+y2

    b2+z2

    c2= 1, (a, b, c > 0) .

    Lsh: O gkoc pou prpei na upologiste enai to diplsio tou gkou pou brsketai sthn

    perioq z 0. H probol epshc tou elleiyoeidoc sto eppedo xy ja enai h lleiyhx2

    a2+y2

    b2= 1. O gkoc pou jloume na upologsoume perikleetai tra ap tic epifneiec

    z1 = z = 0 (ap ktw) kai z2 = c

    1 x

    2

    a2 y

    2

    b2(ap pnw) kai o zhtomenoc gkoc ja

    dnetai apo to dipl oloklrwma

    V = 2

    D

    (z2 z1)dxdy = 2

    D

    c

    1 x

    2

    a2 y

    2

    b2dxdy

    O tpoc oloklrwshc D enai h h lleiyhx2

    a2+y2

    b2= 1 kai qrhsimopointac thn paraktw

    metatrop se polikc suntetagmnec

    x = a cos , y = b sin , dxdy = abdd

    me ria

    0 1, 0 2pija qoume

    V = 2

    D

    c

    1 x

    2

    a2 y

    2

    b2dxdy = 2c

    2pi=0

    1=0

    1 2abdd

    = 2abc

    2pi=0

    d 1=0

    1 2d = 2abc 2pi 1

    3=

    4abcpi

    3

    Updeixh: O gkoc tou parapnw elleiyoeidoc mpore na upologiste ekola me tripl

    oloklrwma kai metatrop se sfairikc suntetagmnec. O upologismc tou ja gnei sto 4o

    Keflaio.

    6

  • 0.2 TRIPLA OLOKLHRWMATA

    'Askhsh 1. Na upologiste to oloklrwma

    I =

    V

    x2 + y2 + z2 dxdydz

    pou V enai o gkoc o opooc perikleetai ap thn epifneia tou kuln-

    drou z =x2 + y2 kai tou epipdou z = 3.

    Lsh: Ja upologsoume to oloklrwma knontac metatrop se kulindrikc suntetagmnec

    pou

    x = cos , y = sin , z = z, dxdydz = ddzd

    To oloklrwma tra grfetai

    I =

    VG

    2 + z2 ddzd

    'Opwc fanetai ap to sqma ta ria tou enai 0

  • To dipl oloklrwma pnw sthn epifneia Dxy pou prokptei, upologzetai jewrntac tita ria tou y kai tou x enai antstoiqa (an epilxoume fusik prta thn oloklrwsh wcproc y) 0 < y < (4 x)/2 kai 0 < x < 4, dhlad enai

    IV =1

    3

    4x=0

    ( (4x)/2y=0

    (4 x 2y) dy)dx

    =1

    3

    4x=0

    (4 2x+ x2

    4)dx =

    1

    3

    16

    3= 16

    'Askhsh 3. Na breje o gkoc tou stereo pou perikleetai ap ta eppe-

    da y + z = 4, x = 0, z = 0 kai ton klindro y = x2.Lsh: O zhtomenoc gkoc fanetai sto sqma.. O gkoc autc perikleetai parpleura

    ap tic epifneiec x = 0 kai y = x2, ap pnw ap thn epifneia z = 4 y kai ap ktwap thn epifneia z = 0. H probol tou gkou sto eppedo xy enai tpoc D o opoocperiblletai ap tic kamplec x = 0, y = 4 kai y = x2. O zhtomenoc gkoc upologzetaiapo to tripl oloklrwma

    I =

    D

    dxdydz =

    D

    ( z=4yz=0

    dy

    )dxdy =

    D

    (4 y)dxdy

    =

    2x=0

    ( 4y=x2

    (4 y)dy)dx =

    2x=0

    (8 4x2 + x

    4

    2

    )dx =

    128

    15

    'Askhsh 4. Na upologiste o gkoc tou elleiyoeidoc

    x2

    a2+y2

    b2+z2

    c2= 1

    Lsh: Gia ton upologism tou gkou ja qrhsimopoisoume tic paraktw sfairikc sun-

    tetagmnec

    x = ar cos sin , y = br sin sin , z = cr cos , dV = abcr2 sin drdd

    me ta antstoiqa ria

    0 x2

    a2+y2

    b2+z2

    c2 1 0 r 1, 0 pi, 0 2pi

    O gkoc upologzetai wc exc

    I =

    V

    dV =

    G

    abcr2 sin drdd

    = abc

    1r=0

    r2dr pi=0

    sin d 2pi=0

    d = abc 13 2 2pi = 4pi

    3abc

    'Askhsh 5. Na breje o gkoc tou stereo pou perikleetai ap to

    paraboloeidc x2 + y2 = 4z kai to eppedo z = y + 3.

    8

  • Lsh: O gkoc pou jloume na upologsoume fanetai sto sqma... Perikleetai ap ktw

    ap thn epifneia z1 =x2 + y2

    4kai ap pnw ap thn z2 = y+3. H probol tou parapnw

    gkou sto eppedo xy perikleetai ap thn kamplh ekenh pou prokptei ap thn apaleiftou z anmesa stic do parapnw epifneiec kai enai o kuklikc dskoc x2+ (y 2)2 = 42.O gkoc upologzetai tra ap to oloklrwma

    V =

    V

    ddxdydz =

    D

    ( y+3z=x

    2+y2

    4

    dz

    )dxdy

    =

    D

    (y + 3 x

    2 + y2

    4

    )dxdy

    Gia ton upologism tou parapnw diplo oloklhrmatoc ja qrhsimopoisoume tic paraktw

    polikc suntetagmnec

    x = cos , y = 2 + sin , dxdy = dd

    me ria

    0 4, 0 2piTra ja qoume

    V =

    G

    (5 + sin 1

    4(2 + 4 + 4 sin )

    )dd

    =

    G

    (4

    2

    4

    )dd =

    2pi=0

    d 4=0

    (4

    2

    4

    )d

    = 2pi 16 = 32pi

    'Askhsh 6. Na upologisje me tripl oloklrwsh o gkoc tou stereo

    pou perikleetai ap tic epifneiec

    x2 + y2 + z2 = 1, x2 + y2 + z2 = 16, z2 = x2 + y2, z > 0

    Lsh: Epeid o gkoc pou jloume na upologsoume enai tmma sfarac enai protimtero

    na qrhsimopoisoume sfairikc suntetagmnec pou

    x = r cos sin , y = r sin sin , z = r cos , dV = r2 sin drdd

    Ta ria twn metablhtn kai enai profan, dhlad

    1 r 4, 0 2pi.

    Ta ria thc gwnac upologzontai wc exc: To mgisto thc gwnac antistoiqe sthn

    gwna a tou knou h opoa dnetai ap thn sqsh tan a =

    z=

    x2 + y2

    z= 1 a = pi

    4.

    9

  • Sunepc 0 pi4O gkoc upologzetai plon ap to oloklrwma

    V =

    V

    dV =

    G

    r2 sin drdd

    =

    4r=1

    r2dr pi/4=0

    sin d 2pi=0

    d

    = 21 (1

    2

    2

    ) 2pi = 21pi(4

    2)

    'Askhsh 7. Na upologisje to oloklrwma

    I =

    V

    x2 + y2dxdydz

    pou h stere perioq V perikleetai ap tic epifneiec x2 + y2 = ax kaiz2 = x2 + y2.

    Lsh: O zhtomenoc tpoc oloklrwshc, pwc fanetai sto sqma.., perikleetai plgia

    ap thn kulndrik epifneia x2 + y2 = ax (x a2)2 + y2 =

    (a2

    )2, ap ktw ap thn

    epifneia z = 0 kai ap pnw ap thn z =x2 + y2. H probol tou gkou sto eppedo xy

    enai o kuklikc dskoc (x a2)2 + y2 =

    (a2

    )2. To oloklrwma tra gnetai

    I =

    V

    x2 + y2dxdydz =

    D

    ( x2+y2z=0

    x2 + y2dz

    )dxdy

    =

    D

    (x2 + y2)dxdy

    Gia ton upologism tou diplo oloklhrmatoc ja qrhsimopoisoume polikc suntetagmnec

    pou

    x = cos , y = sin , dxdy = dd

    me antstoiqa ria

    0 x2 + y2 ax 0 a cos , 0 2piTra to oloklrwma gnetai

    I =

    D

    (x2 + y2)dxdy =

    G

    2dd =

    2pi=0

    ( a cos =0

    3d

    )d

    =a4

    4

    2pi=0

    cos4 d =3pia4

    16

    'Askhsh 8. Na upologiste o gkoc tou stereo pou orzetai exwterik

    tou knou x2 + y2 = z2 kai eswterik thc sfarac x2 + y2 + z2 = 2.

    Lsh: O gkoc pou jloume na upologsoume enai, lgw summetrac, to diplsio tou

    10

  • gkou pou uprqei sthn perioq z 0. 'Etsi ja ergastome se aut thn perioq kai osunolikc gkoc ja antistoiqe sto diplsio auto tou gkou. Epeid o gkoc pou jloume

    na upologsoume enai tmma sfarac enai protimtero na qrhsimopoisoume sfairikc sun-

    tetagmnec pou

    x = r cos sin , y = r sin sin , z = r cos , dV = r2 sin drdd

    Ta ria twn metablhtn r kai enai profan, dhlad

    1 r 2, 0 2pi.

    Ta ria thc gwnac upologzontai wc exc: To elqisto (sthn perioq tou gkou pou macendiafrei) thc gwnac antistoiqe sthn gwna a tou knou h opoa dnetai ap thn sqsh

    tan a =

    z=

    x2 + y2

    z= 1 a = pi

    4. Sunepc

    pi4 pi O gkoc upologzetai plonap to oloklrwma

    V =

    V

    dV =

    G

    r2 sin drdd

    =

    2r=0

    r2dr pi=pi/4

    sin d 2pi=0

    d

    =22/3

    32

    2 2pi = 4pi

    3

    O sunolikc gkoc ja enai Vall = 2V =8pi

    3.

    'Askhsh 9. Na upologiste me efarmog tou jewrmatoc tou Green tooloklrwma

    I =

    c

    yx2 + y2

    dx+x

    x2 + y2dy

    pou c perifreia lleiyhc me exswsh x2 + 4y2 = 4

    Lsh: Epeid oi sunartseic

    P =y

    x2 + y2, Q =

    x

    x2 + y2

    orzontai panto sto eppedo 0xy ektc ap to shmeo 0, 0 o tpoc D pou periblletai apthn lleiyh enai enac pollapl sunektikc tpoc. Parathrome epshc ti isqei

    Q

    x=P

    y=

    y2 x2(x2 + y2)2

    .

    Etsi smfwna me thn jewra twn pollapl sunektikn tpwn stouc opoouc isqei h para-

    pnw sqsh mporome na upologsoume to kleist epikamplio oloklrwma kata mkoc

    opoisdhpote kleisthc grammc pou periqei to shmeo 0, 0. Epilgoume gia pardeigma nato upologosume kata mkoc thc perifreiac tou kklou x2 + y2 = a2.Etsi, knontac thn parametroposh tou kklou se polikc suntetagmnec ja qoume

    x = a cos , y = a sin

    11

  • kai epshc

    P =a sin

    a2, Q =

    a cos

    a2

    . To oloklrwma twra ja grfetai

    I =

    2pi0

    [a sin a2

    (a sin ) + a cos a2

    (a cos )

    ]d =

    2pi0

    = 0

    0.3 EPIKAMPULIA OLOKLHRWMATA

    'Askhsh 1. An c enai h perifreia x2 + y2 = a2, na upologiste to olok-lrwma

    x2ydx+ xy2dya) kat' eujean

    b) efarmzontac ton tpo tou Green

    Lsh:

    a) H perifreia tou kklou mpore na parametropoihje me ton gnwst trpo wc exec

    x = a cos t, y = a sin t, x = a sin t, y = a cos t, 0

  • Sto parapnw oloklrwma efarmsame metatrop se polikc suntegmnec, dhlad

    x = cos t, y = sin t, dxdy = ddt

    'Askhsh 2. Na upologiste to oloklrwma

    I =

    ydx+ xdyx2 + y2

    a) kat mkoc thc perimtrou tou trignou AB me korufc ta shmeaA(1, 1), B(1,1) kai (3, 0)

    b) kat mkoc thc perifreiac x2 + y2 = a2

    g) kat mkoc thc perifreiac x2 + y2 + 4x+ 3 = 0

    Lsh:

    a) Oi sunartseic

    P =y

    x2 + y2, Q =

    x

    x2 + y2

    orzontai panto sto eswterik tou trignou ABG. Subepc mporome na efarmsoume to

    jerhma Green. Epiplon qoume

    Q

    x=P

    y=

    y2 x2(x2 + y2)2

    Sunepc enai

    I =

    ydx+ xdyx2 + y2

    =

    Dxy

    (Q

    x P

    y

    )dxdy =

    Dxy

    0 = 0

    b) Sto eswterik tou kukliko dskou uprqei to shmeo (x, y) = (0, 0) pou oi sunart-seic P kai Q den orzonontai. Sthn perptwsh aut den mporome na egarmzoume toJerhma tou Green kai prpei na upologsoume to epikamplio oloklrwma. Epeid trasumbanei na enai

    Qx

    = Pymporome na epilxoume opoiadpote kleist kamplh pou per-

    iqei mwc to anmalo shmeo (x, y) = (0, 0). H perifreia tou kklou x2 + y2 = a2 enaimia dunat perptwsh kai enai ap tic protimterec exaiteac thc ekolhc parametroposhc

    thc all pwc ja doume kai paraktw thc aplopohshc twn sunartsewn P kai Q. 'Etsija qoume

    x = a cos t, y = a sin t, x = a sin t, y = a cos t, 0

  • g) H kamplh x2+y2+4x+3 = 0 grfetai epshc (x+2)2+y2 = 1 paristnei perifreiakklou aktnac 1 me kntro to shmeo (x, y) = (2, 0). 'Opwc ekola diapistnoume oparapnw kuklokc dskoc periqei to anmalo shmeo (x, y) = (0, 0). Me dodomno mwcti isqei

    Qx

    = Pymporome na epilxoume opoiadpote kleist kamplh pou periqei to

    shmeo (x, y) = (0, 0). Mia tteia kamplh enai h perifreia x2 + y2 = a2 thc perptwshcb). Sunepc to oloklrwma ja enai

    c:x2+y2+4x+3=0

    ydx+ xdyx2 + y2

    =

    c:x2+y2=a2

    ydx+ xdyx2 + y2

    = 2pi

    'Askhsh 3. Upologste to oloklrwma

    I =

    (2xy x2)dx+ (x+ y2)dy

    pou c enai h kleist kamplh pou sqhmatzetai ap tic parabolc y = x2

    kai x = y2.

    Lsh: Oi sunartseic P (x, y) = 2xy x2 kai Q(x, y) = x + y2 tou parapnwoloklhrmatoc plhron tic proupojseic tou jewrmatoc Green kai epiplon qoume Q

    x=

    1 kai Py

    = 2x. To epikamplio oloklrwa metatrpetai se na dipl oloklrwma me tpo

    olklrwshc Dxy thn perioq pou periblletai apo tic parabolc y = x2kai x = y2 kaisunepc me ria 0 < x < 1, x2 < y 0}

    Dxz = {(x, z) R2, x, z 0, 2x+ 6z 12, cos = ~ ~j = 37> 0}

    Dxy = {(x, y) R2, x, y 0, 2x+ 3y 12, cos = ~ ~j = 67> 0}

    19

  • kai to oloklrwma plon upologzetai wc exc

    I =

    4y=0

    2 y2

    z=0

    18zdydz +

    6x=0

    3x3

    z=0

    (12)dxdz

    +

    6x=0

    4 2x3

    y=0

    3ydxdy = 48 72 + 48 = 24

    Updeixh: Prospajste na upologsete to oloklrwma efarmzontac a) ton tpo tou

    Gauss kai b) ton tpo tou Stokes

    'Askhsh 9. Na upologiste to kleist oloklrwma

    S(x2+y2+z2)d pnwsthn epifneia thc sfarac x2 + y2 + z2 = a2

    a) Wc epiepeifneio oloklrwma a' tpou

    b) Me th bojeia tou tpou tou Gauss

    Lsh:

    a) To oloklrwma mpore na upologiste ekola pnw sthn epifneia thc sfarac an

    qrhsimopoisoume sfairikc suntetagmnec pou

    x = a cos sin , y = a sin sin , z = a cos , d = a2 sin dd

    me ria

    0 pi, 0 2pi'Etsi ja qoume

    I =

    S

    (x2 + y2 + z2)d =

    G

    a2a2 sin dd

    = a4 pi=0

    sin d 2pi=0

    d = a4 2 2pi = 4pia4

    b) Smfwna me ton tpo tou Gauss qoume S

    ~F ~d =

    V

    div ~FdV

    Sugkrnontac to prto mloc thc parapnw sqshc me to oloklrwma pou qoume na up-

    ologsoume sumperanoume ti ja prpei h sunrthsh

    ~F na ikanopoie thn sqsh ~F ~ =(x2 + y2 + z2). Epeid tra gia (x, y, z) = x2 + y2 + z2 a2 = 0 ja qoume ti

    ~5 = 2x~i+ 2y~j + 2z~k, | ~5 |= 2a, ~ = 1a

    (x~i+ y~j + z~k

    )ekola sumperanoumeti ja enai

    ~F = a(x~i+ y~j + z~k

    ). Sunepc ap ton parapnw tpo

    tou Gauss, kai qrhsimopointac sfairikc suntetagmnec ja qoume

    I =

    S

    ~F ~d =

    V

    div ~FdV = 3a

    V

    dxdydz

    = 3a

    G

    r2 sin drdd = 3a

    ar=0

    r2dr pi=0

    sin d 2pi=0

    d

    = 3a4

    3a3 2 2pi = 4pia4

    20

  • 'Askhsh 10. Na breje h ro tou dianusmatiko pedou

    ~F = x~i+y~j+z2~k ap

    a) To tmma thc kulindrikc epifneiac x2 + y2 = 4 pou apokptetaiap ta eppeda z = 0 kai z = 1b) Ap to snoro tou stereo kulndrou x2 + y2

  • 0.4 EPIEPIFANEIA OLOKLHRWMATA

    'Askhsh 1. Na epalhjeute o tpoc tou Stokes gia to dianusmatik pedo~F = z~i + x~j + y~k pou h epifneia S enai to nw hmisfario thc sfaracx2 + y2 + z2 = 1.

    Lsh: Ja prpei na dexoume ti

    I =

    c

    ~F d~r =S

    rot ~F ~nd

    a) Upologismc tou epikampliou oloklhrmatoc

    To epikamplio oloklrwma gnetai kat mkoc thc kamplhc c h opoa brsketai panwsto eppedo Oxy kai tautzetai me thn perifreia tou kklou x2 + y2 = 1. Ektelntac thnparametropohsh

    x = cos t, y = sin t,dx

    dt= sin t, dy

    dt= cos t

    kai lambnonatc upyh ti kata mkoc thc c enai z = 0 ja qoume

    I =

    c

    ~F d~r =c

    zdx+ xdy + ydz =

    2pit=0

    [z(t)

    dx

    dt+ x

    dy

    dt+ y

    dz

    dt

    ]dt

    =

    2pit=0

    [0 ( sin t) + cos t (cos t) + sin t 0] dt = 2pit=0

    cos2 tdt = pi

    b) Upologismc tou epiepifneiou oloklhrmatoc

    Lambnontac upyh ti i)

    rot ~F = (1 0)~i+ (1 0)~j + (1 0)~k =~i+~j + ~k

    ii) epshc ti gia (x, y, z) = x2 + y2 + z2 1 = 0 ja qoume~5 = 2x~i+ 2y~j + 2z~k, | ~5 |= 2, ~ = x~i+ y~j + z~k

    iii) kai ti gia z =1 x2 y2

    d =

    1 + (

    z

    x)2 + (

    z

    y)2 =

    1

    zdxdy

    ja qoume sunolik

    I =

    S

    rot ~F ~nd =

    Dxy

    (~i+~j + ~k

    )(x~i+ y~j + z~k

    ) 1zdxdy

    =

    Dxy

    (x+ y + z)

    1

    zdxdy

    =

    Dxy

    (x+ y +

    1 x2 y2

    ) 11 x2 y2dxdy

    22

  • pou sthn parapnw sqsh o tpoc Dxy profanc tautzetai me ton kuklik dsko per-ifreiac x2 + y2 = 1. Stic periptseic autc, pwc sunhjzoume, knoume metatrop sepolikc suntetagmnec kai ja qoume sunepc

    x = cos t, y = sin t, dxdy = ddt

    kai to oloklrwma grfetai plon

    I =

    G

    cos t+ sin t+

    1 2

    1 2 ddt

    =

    G

    cos t+ sin t

    1 2 ddt+G

    ddt

    =

    1=0

    2pit=0

    cos t1 2ddt+

    1=0

    2pit=0

    cos t1 2ddt

    +

    1=0

    2pit=0

    ddt = 0 + 0 + pi = pi

    Sunepc to jerhma tou Stokes epalhjethke.

    'Askhsh 2. Dnetai h sunrthsh

    ~F = 3y~i xz~j + yz2~k kai h epifneia S, meexswsh z = (x2+ y2)/2. 'Estw c h kamplh pou enai h tom thc epifneiacS, me to eppedo z = 2. Na gnei epaljeush tou tpou tou Stokes

    Lsh: Ja prpei na epalhjesoume thn sqshc

    ~F d~r =S

    rot ~F ~nd

    a) Upologismc tou epiepeifneiou oloklrmatoc

    (i) 'Eqoume ti

    rot ~F = (z2 + x)~i+ 0~j + (z 3)~k,(ii) epshc gia (x, y, z) = z (x2 + y2)/2 = 0

    ~5 = x~i y~j + ~k, | ~5 |=x2 + y2 + 1, ~ =

    x~i y~j + ~kx2 + y2 + 1

    iii) kai ti

    d =

    1 + (

    z

    x)2 + (

    z

    y)2 dxdy =

    x2 + y2 + 1dxdy

    23

  • ja qoume sunolik

    I =

    S

    rot ~F ~nd

    = Dxy

    ((z2 + x)~i+ 0~j + (z 3)~k

    )(x~i y~j + ~kx2 + y2 + 1

    )

    x2 + y2 + 1dxdy

    =

    Dxy

    ((z2 + x)x+ (z + 3)

    )dxdy

    =

    Dxy

    ((x2 + y2

    2)2x+ x2 +

    x2 + y2

    2+ 3

    )dxdy

    pou sto parapnw oloklrwma Dxy enai h probol thc epifneiac S sto eppedo Oxy kaih opoa peratnetai ap thn kleist kamplh c pou prokptei ap th apaleif tou z aptic exisseic z = (x2 + y2)/2 kai z = 2. Amswc prokptei ti prkeitai gia perifreiakklou me exswsh x2 + y2 = 22. Knoume metatrop se polikc suntetagmnec kai jaqoume sunepc

    x = cos t, y = sin t, dxdy = ddt

    kai to oloklrwma grfetai plon

    I =

    G

    (4

    4 cos t+ 2 cos2 t+

    2

    2+ 3

    )ddt

    =

    2pit=0

    2=0

    5

    4cos tddt+

    2pit=0

    2=0

    3 cos2 tddt

    +

    2pit=0

    2=0

    3

    2ddt+

    2pit=0

    2=0

    3ddt

    = 0 + 4pi + 4pi + 12pi = 20pi

    b) Upologismc tou epikampliou oloklhrmatoc

    Enai profanc ti h exswsh thc kamplhc c enai x2 + y2 = 4, z = 2 h opoa separametrik morf grfetai

    x = 2 cos t, y = 2 sin t, z = 2, t [0, 2pi]

    kai epshc enai

    dx

    dt= 2 sin t, dy

    dt= 2 cos t, dz = 0

    'Etsi ja qoume tra

    I =

    c

    ~F d~r = 2pit=0

    (3(2 sin t)(2 sin t) 2 cos t2(2 cos t) + 2 sin t220) dt

    = 2pit=0

    (4 sin2 t+ 8

    )=

    2pit=0

    4 sin2 tdt 2pit=0

    8dt

    = 4pi 16pi = 20pi

    24

  • 'Askhsh 3. Na epalhjeute o tpoc tou Stokes gia th dianusmatik

    sunrthsh

    ~F = y2~i + z2~j + x2~k kai thn epifneia S pou enai to t-mma thc epifneiac tou epipdou z = x + 1 pou apokptei o klindrocx2 + y2 = 1.

    Lsh:

    Lsh: Ja prpei na epalhjesoume thn sqshc

    ~F d~r =S

    rot ~F ~d

    a) Upologismc tou epiepeifneiou oloklrmatoc

    H epifneia tou epipdou pou apokptetai ap ton klindro problletaisto eppedo Oxysthn epifneia Dxy pou enai o kuklikc dskoc x

    2 + y2 = 1(i) 'Eqoume ti

    rot ~F = 2z~i 2x~j 2y~k,(ii) epshc gia (x, y, z) = z x 1 = 0

    ~5 = ~i+ ~k, | ~5 |=2, ~ =

    ~i+ ~k2

    iii) kai ti

    d =

    1 + (

    z

    x)2 + (

    z

    y)2 dxdy =

    2dxdy

    Sunolik ja qoume

    I =

    S

    rot ~F ~d

    =

    Dxy

    (2z~i 2x~j 2y~k)(~i+ ~k

    2

    )2dxdy

    =

    Dxy

    (2z 2y)dxdy

    =

    Dxy

    (2(x+ 1) 2y)dxdy

    =

    Dxy

    2dxdy +

    Dxy

    2xdxdy

    Dxy

    2ydxdy

    Qrhsimopointac polikc suntetagmnec pou

    x = cos t, y = sin t, dxdy = ddt

    me ria

    0 1, 0 t 2pito parapnw oloklrwma ja grfetai

    I = 2

    1=0

    d

    2pit=0

    dt+ 2

    1=0

    2d

    2pit=0

    cos tdt

    2 1=0

    2d

    2pit=0

    sin tdt = 2pi

    25

  • b) Upologismc tou epikampliou oloklhrmatoc H parametrik exswsh thc kamplhc

    pou peribllei thn dosmnh epifneia enai profanc

    ~r(t) = cos t~i+ sin t~j + (cos t+ 1)~k

    To parapnw prokptei ap to ti h probol thc kamplhc sto epipedo Oxy enai per-ifreia kklou me exswsh x2 + y2 kai profan parametropohsh x = cos t, y = sin t. Hexswsh thc parapnw kamplhc tan metafretai sthn dosmnh epifneia prokptei me thn

    antikatstash z = x+ 1 = cos t+ 1. To epikamplio oloklrwma ja enai tra

    I =

    c

    ~F d~r = (

    y2dx

    dt+ z2

    dy

    dt+ x2

    dz

    dt

    )dt

    =

    2pit=0

    ( sin2 t( sin t) + (1 + cos t)2 cos t+ cos2 t( sin t)

    )dt

    = 2

    2pit=0

    cos2 tdt = 2pi

    'Askhsh 4. Na upologisje, me treic diaforetikoc trpouc, to epi-

    faneiak oloklrwma I =

    Sz cos d, pou S enai h exwterik epifneiathc sfarac x2 + y2 + z2 = 1 kai cos enai h trth sunistsa tou kjetousthn epifneia S monadiaou diansmatoc ~ ( enai h gwna pou sqh-matzei to ~ me ton xona zz)

    Lsh: a' trpoc: Qwrzoume thn sfara sta do hmisfaria S1 kai S2 antstoiqa.Kaoi oi do epifneiac probllontai profanc sto eppedo xy ston kuklik dsko Dxy :x2 + y2 = 1. Ja qoume epshc antstoiqa gia tic do epifneiec

    S1 : cos d = +dxdy, z =1 x2 y2

    S2 : cos d = dxdy, z = 1 x2 y2pou sth perptwsh thc epifneia S1 to prshmo (+) ofeletai sto ti h gwna tou di-

    ansmatoc sqhmatzei oxea gwna me to dinusa ~k en sthn epifneia S2 to prshmo ()ofeletai sto ti h gwna tou diansmatoc ~ sqhmatzei amblea gwna me to dinusa ~k. Tooloklrwma upologzetai plon wc exc

    I =

    S

    z cos d =

    S1

    z cos d +

    S2

    z cos d

    =

    Dxy

    1 x2 y2dxdy

    Dxy

    1 x2 y2dxdy

    = 2

    Dxy

    1 x2 y2dxdy = 2

    G

    1 2dd = 22pi

    3=

    4pi

    3

    b' trpoc: Ja qrhsimopoisoume sfairikc suntetagmnec akoloujntac ton paraktw

    metasqhmatism

    x = cos sin , y = sin sin , z = cos , d = 12 sin dd

    26

  • me ria

    0 pi, 0 2piEpshc ja qoume

    ~ =~|~| = (x

    ~i+ y~j + z~k), ~ ~k = z

    To oloklrwma plen upologzetai wc exc

    I =

    S

    z cos d =

    S

    z2d =

    S

    (cos )2 sin dd

    =

    pi=0

    2pi=0

    cos2 sin dd =

    2pi=0

    d

    pi=0

    cos2 sin d = 2pi2

    3=

    4pi

    3

    g' trpoc: Mporome na upologsoume to oloklrwma efarmzontac ton tpo tou Gaussefson to epiepeifneio oloklrwma upologzetai sthn kleist epifneia thc sfarac. To

    oloklrwma mpore na grafe kai wc exc

    I =

    S

    z cos d =

    S

    z(~ ~k

    )d =

    S

    (z~k) ~d

    An sto parapnw oloklrwma jewrsoume ti

    ~F = z~k, smfwna me ton tpo tou Gaussja qoume

    V

    div ~FdV =

    S

    ~F ~d

    Lambnontac upyh ti div ~F = 1 ja qoume

    I =

    S

    z~k ~d =

    V

    div(z~k)dV =

    V

    dV

    =

    G

    2 sin ddd =

    1=0

    d

    pi=0

    sin d

    2pi=0

    d

    =1

    3 2 2pi = 4pi

    3

    'Askhsh 5. Na upologiste to epifaneiak oloklrwma I =

    Szdxdy,pou S h exwterik epifneia tou elleiyoeidoc

    x2

    a2+y2

    b2+z2

    c2= 1, (a, b, c > 0)

    Lsh: a' trpoc: To oloklrwma, kai lambnontac upyh ti cos d = dxdy grfe-tai

    I =

    S

    z cos d =

    S

    z(~ ~k)d =

    S

    (z~k) ~dEfarmzontac to jerhma tou Gauss kai qrhsimopointac, katllhla epilegmnec, s-fairikc suntetagmnec pou

    x = ar cos sin , y = br sin sin , z = cr cos , dV = abcr2 sin drdd

    27

  • me

    0 r 1, 0 2pi, 0 pija qoume

    I =

    S

    (z~k) ~d =

    V

    div(z~k)dV =

    V

    dV

    =

    1r=0

    2pi=0

    pi=0

    abcr2 sin drdd

    = abc

    1r=0

    r2dr

    2pi=0

    d

    pi=0

    sin d =4pi

    3abc

    Updeixh: Prospajeste na upologsete apeujeac to parapnw oloklrwma ete a) probll-

    wntac p.q. thn epifneia tou elleiyoeidoc sto eppedo xy ete b) parametropointac thnepifneia tou elleiyoeidoc qrhsimopointac katllhla epilegmnec sfairikc suntetag-

    mnec.

    'Askhsh 6. Na gnei epaljeush tou tpou tou Gauss gia th sunrthsh~F = 4x~i 2y2~j + z2~k kai th stere perioq V , pou perikleetai ap thnkulindrik epifneia x2 + y2 = 4 kai ta eppeda z = 0, z = 3.

    Lsh: Ja prpei na dexoume ti V

    div ~AdV =

    S

    ~F ~d

    pou V o gkoc tou stereo kai S h kleist epifneia pou perikleei ton parapnw gko. Hepfneia aut sunstatai ap treic diaforetikc (proc thn paramemetroposhsh) epifneiec,

    a) thn epifneia z = 0 (thn onomzoume S1), b) thn epifneia z = 3 (S2), kai g) thn epifneiax2 + y2 = 4 (S3) me ta antstoiqa ria se kje perptwsh (dec sqma..).a) Ja upologsoume arqik to tripl oloklrwma knonatc qrsh kulindrikn susntetag-

    mnwn pou

    x = cos , y = sin , z = z, dV = dddz

    me ria

    0 2, 0 2pi, 0 z 3en epshc ja qoume

    div ~F = (4 4y + 2z) = (4 4 sin + 2z)To oloklrwma tra upologzetai wc exec

    I =

    V

    div ~FdV =

    V

    (4 4 sin + 2z)dddz

    =

    2pi=0

    2=0

    3z=0

    (4 4 sin + 2z)dddz = 84pi

    b) Ja upologsoume tra to epiepeifneio oloklrwma sthn epifneia pou perikleei ton

    gko V , upologzontac tra epiepeifneia oloklhrmata antstoiqa stic epifneiec S1, S2

    28

  • kai S3.b1) Gia to oloklrwma sthn epifneia S1 ja qoume

    d = dxdy, ~ = ~kkai sunepc

    I1 =

    S1

    ~F ~d =

    D1

    (z2)dxdy = 0

    pou lbame upyh ti sthn epifneia S1 qoume stajer z = 0.b2) Gia to oloklrwma sthn epifneia S2 ja qoume

    d =

    1 + (

    y

    x)2 + (

    y

    z)2dxd = dxdy, ~ = ~k

    kai sunepc

    I2 =

    S2

    ~F ~d =

    D2

    (z2)dxdy = 9

    dxdy = 9pi22 = 36pi

    b3) Gia to oloklrwma sthn epifneia S3 ja qrhsimopoisoume kulindrikc suntetagmnecpou

    x = 2 cos , y = 2 sin , z = z, d = 2ddz

    me ria

    0 2pi, 0 z 3kai epshc

    ~ =~|~| =

    1

    2(x~i+ y~j)

    I3 =

    S3

    ~F ~d =

    S3

    (2x2 y3)d

    =

    D3

    8(cos2 sin3 )2ddz

    = 16

    2pi=0

    3z=0

    (cos2 sin3 )ddz = 48pi

    Sunolik ja qoume

    I =

    S

    ~F ~d = I1 + I2 + I3 + I4 = 0 + 36pi + 48pi = 84pi

    'Askhsh 7. Na epalhjeute o tpoc tou Stokes gia to oloklrwma(z y)dx+ (x z)dy + (y x)dz

    pou c enai h permetroc tou trignou ABC me korufc A(a, 0, 0), B(0, a, 0),C(0, 0, a).

    29

  • Lsh: a) Arqik ja upologsoume to kleist epikamplio oloklrwma kat mkoc

    thc diadromc A B C. H parametroposhsh twn trin parapnw diadromn enaiantstoiqa:

    A B : x+ y = a, x = t, y = a t, a t 0B C : y + z = a, y = t, z = a t, a t 0C A : x+ z = a, x = t, z = a t, 0 t akai to kleist oloklrwma ja enai

    I =

    ~F d~r =

    (z x)dx+ (x z)dy + (y x)dz

    =

    AB

    () +

    BC

    () +

    CA

    ()

    =

    0t=a

    [(0 (a t)) 1 + (t 0) (1) + (a t t) 0] dt

    =

    0t=a

    [(a t t) 0 + (0 (a t)) 1 + (t 0) (1)] dt

    =

    at=0

    [(a t 0) 1 + (t (a t)) 0 + (0 t) (1)] dt

    =

    0t=a

    (a)dt+ 0t=a

    (a)dt+ at=0

    adt = a2 + a2 + a2 = 3a2

    b) Ja upologsoume tra to antstoiqo epiepeifnio oloklrwma

    S

    rot~F ~d. Ja qoume

    rot~F = 2~i+ 2~j + 2~k

    kai epshc, epeid profanc h exswsh tou epipdou enai x+ y + z = a ja qoume

    ~ =~|~| =

    13(~i+~j + ~k)

    H probol thc epifneiac S sto eppedo Oxy enai h trigwnik epifneia pou perikleetaiap tic kamplec y = 0, x = 0 kai x+ y = a kai epshc enai

    d =

    1 + (

    z

    x)2 + (

    z

    y)2dxdy =

    3dxdy.

    To epiepeifnio oloklrwma grfetai plon

    I =

    S

    rot~F ~d =

    D

    (2~i+ 2~j + 2~k) 1

    3(~i+~j + ~k)

    3dxdy

    =

    ax=0

    ( axy=0

    6dy

    )dx = 6

    ax=0

    (a x)dx = 6a2

    2= 3a2

    'Askhsh 8.Na epalhjeute to jerhma tou Stokes gia to tmma thc k-

    wnikc epifneiac z = 2 x2 + y2 pou brsketai metax twn epipdwnz = 2 kai z = 1 tan ~F = (y z)~i+ (z x)~j + (x y)~k.

    30

  • Lsh: Smfwna me ton tpo tou enai Stokes~F d~r =

    S

    rot~F ~d

    a) Epikamplio oloklrwma : H kamplh c pnw sthn opoa ja upologsoume to kleistopikamplio oloklrwma brsketai sto eppedo z = 1 kai enai perifreia kklou me aktna

    pou ja enai =x2 + y2 = 2 z = 1. Sunepc h parametroposh thc peifreiac ja enai

    x = cos , y = sin , z = 1, 0 1kai to oloklrwma upologzetai wc exc

    I =

    ~F d~r =

    (y z)dx+ (z x)dy + (x y)dz

    =

    2pi=0

    [(sin 1) ( sin ) + (1 cos ) (cos )+ (cos sin ) (0)]d = 2pi

    b) Epiepeifneio oloklrwma : H kwnik epifneia S pnw sthn opoa ja upologsoume tooloklrwma probletai sto eppedo xy sto kuklik dsko x2 + y2 = 1 (prokptei ap thn

    apaleif tou z anmesa sthn exswsh tou knou z = 2 x2 + y2 kai sthn exswsh touepipdou z = 1). Ja qoume epshc

    rot~F = 2~i 2~j 2~k, ~ =~|~| =

    12

    (x

    x2 + y2~i+

    yx2 + y2

    ~j + ~k

    )

    kai epshc

    d =

    1 + (

    z

    x)2 + (

    z

    y)2dxdy =

    2dxdy

    To oloklrwma tra upologzetai wc exc

    I =

    S

    rot~F ~d

    =

    D

    (2~i 2~j 2~k)

    [12

    (x

    x2 + y2~i+

    yx2 + y2

    ~j + ~k

    )]2dxdy

    =

    D

    2(

    xx2 + y2

    +y

    x2 + y2+ 1

    )dxdy

    =

    G

    2(cos + sin + 1)dd

    = 2 2pi=0

    1=0

    cos dd 2 2pi=0

    1=0

    cos dd

    2 2pi=0

    1=0

    dd = 0 + 0 2pi = 2pi

    31

  • pou sto parapnw oloklrwma kname metasqhmatism se polikc suntetagmnec.

    'Askhsh 9. Na gnei epaljeush tou tpou tou Gauss gia th sunrthsh~F = (2x z)~i+x2y~jxz2~k kai gia thn epifneia tou kbou pou orzetai apta eppeda x = 0, x = 1, y = 0, y = 1, z = 0, z = 1.

    Lsh: a' Upologismc tou epiepeifniou oloklhrmatoc O kboc faneta sto sqma....

    o upologismc tou epiepeifneiou oloklhrmatoc

    I =

    S

    ~F ~d

    sunstatai sto upologism 6 epipeifneiwn oloklrwmatwn ma gia kje epifneia toukbou. Prpei se kje perptwsh na kajoriste to kjero dinusma ~, h stoiqeishcepifneia d kai ta oria thc probolc thc kje epifneiac sta eppeda xy, xz, kai yz. 'Etsija qoume se kje perptwsh:

    ABCD : ~ = ~k, d = dxdy, 0 x 1, 0 y 1, z = 0

    HEFG : ~ = ~k, d = dxdy, 0 x 1, 0 y 1, z = 1ADGH : ~ = ~j, d = dxdz, 0 x 1, 0 z 1, y = 0BCFE : ~ = ~j, d = dxdz, 0 x 1, 0 z 1, y = 1ABEH : ~ =~i, d = dydz, 0 y 1, 0 z 1, x = 1DCFG : ~ = ~i, d = dydz, 0 y 1, 0 z 1, x = 0Ta antstoiqa oloklhrmata enai

    I1 =

    DABCD

    (xz2)dxdy = 0

    I2 =

    DHEFG

    (xz2)dxdy =

    1x=0

    1y=0

    (x)dxdy = 12

    I3 =

    DADGH

    (x2y)dxdz = 0

    I4 =

    DBCFE

    (x2y)dxdz =

    1x=0

    1z=0

    x2dxdy =1

    3

    I5 =

    DABEF

    (2x z)dydz =

    1y=0

    1z=0

    (2 z)dydz = 32

    I6 =

    DDCFG

    (2x z)dydz =

    1y=0

    1z=0

    zdydz =1

    2

    Sunolik ja qoume

    I = I1 + I2 + I3 + I4 + I5 + I6 = 0 +1

    2+ 0 +

    1

    3+3

    2+1

    2=

    11

    6

    32

  • b' Efarmog tpou Gauss: Ap ton tpo tou Gauss pou

    I =

    S

    ~F ~d =

    V

    div ~FdV

    kai lambnonatc upyh ti

    div ~F = 2 + x2 2xzja qoume

    I =

    V

    div ~FdV =

    V

    (2 + x2 xz)dxdydz

    =

    1x=0

    1y=0

    1z=0

    (2 + x2 2xz)dxdydz = 116

    'Askhsh 10. Dnetai h epifneia S 2z = 4 x2 y2. Na breje to embadnmetax twn uyn z = 1 kai z =

    2.

    Lsh: a' trpoc: Ja qrhsimopoisoume kulindrikc suntetagmnec pou

    x = cos , y = sin , z = z, 2 = x2 + y2 = 4 2z

    me ria antstoiqa

    1 z 2, 0 2piH parametrik parstash thc thc epifneiac tou paraboloeidoc ja enai

    ~r(, z) =4z 2 cos ~i+4z 2 sin ~j + z~k

    kai sunepc ja qoume

    ~1 =~r

    = 4z 2 sin ~i+4z 2 cos ~j

    ~2 =~r

    = 1

    4z 2 cos ~i 1

    4z 2 sin ~j + ~k

    ~1 ~1 = 4 2z, ~2 ~2 = 5 2z4 2z , ~1 ~2 = 0H stoiqeish epifneia d ja enai sunepc

    d =gddz =

    (~1 ~1) (~2 ~2) (~1 ~2)2 =

    5 2zddz

    H sunolik epifneia ja enai

    =

    S

    d =

    S

    5 2zddz

    2pi=0

    d

    2z=1

    5 2zddz

    = 2pi

    (3 1

    3(5 2

    2)3/2

    )

    33

  • b' trpoc: Mporome na upologsoume thn zhtomenh epifneia probllonatc thn sto

    eppedo xy. H probol thc ja enai kuklikc daktlioc me aktnec pou prokptoun apthn apaleif tou z apo tic exisseic twn epifanein z = 1 kai z =

    2 kai thc epifneiactou paraboloeidoc 2z = 4 x2 y2. 'Etsi oi aktnec twn do kuklikn dskwn ja enaiantstoiqa 1 =

    2 kai 2 =

    4 22. Epiplon ja qoume

    d =

    1 + (

    z

    x)2 + (

    z

    y)2dxdy =

    1 + x2 + y2dxdy

    kai qrhsimopointac polikc suntetagmnec me ria antstoiqa

    x = cos , y = sin , dxdy = dd

    me ria 4 2

    2

    2, 0 2pija qoume telik

    =

    S

    d =

    Dxy

    1 + x2 + y2dxdy =

    G

    1 + 2dd

    =

    2pi=0

    d

    2=

    422

    1 + 2d = 2pi

    (3 1

    3(5 2

    2)3/2

    )g' trpoc: Mporome na upologsoume to embadn thc epifneiac qrhsimopointac to

    jerhma tou Gauss pou S

    ~F ~d =

    V

    div ~FdV.

    Sthn parapnw sqsh V einai o gkoc pou periklietai ap thn to tmma thc epifneiactou paraboloeidoc pou jloume na upologsoume to embadn (S1) kai ap touc kuklikocdskouc twn epifanein z = 1 (epifneia S2) kai z =

    2 (S3). Mporome sunepc nagryoume

    S

    ~F ~d =

    V

    div ~FdV

    S1

    ~F ~d =

    V

    div ~Fdv

    S1

    ~F ~d

    S1

    ~F ~d

    Sthn epifneia S1 to kjeto dinusma enai

    ~ =~|~| =

    11 + x2 + y2

    (x~i+ y~j + ~k)

    kai epilgonantac gia pardeigma thn dianusmatik sunrthsh

    ~F =1 + x2 + y2~k

    tte ja qoume S1

    ~F ~d =

    S1

    d

    34

  • Lambnontac epshc upyh ti oi epifneiac S2 kai S3 probllontai sto eppedo xy se

    antstoiqouc kuklikoc dskouc me aktnec =2 kai =

    4 22 antstoiqa (dec a'kai b' trpouc thc skhshc) kai ti

    div ~F = 0

    ja qoume sunolik

    =

    S1

    d =

    V

    div ~FdV

    S1

    ~F ~d

    S1

    ~F ~d

    = 0D1

    ~F ( ~k)dxdy

    D2

    ~F ~kdxdy

    =

    D1

    1 + x2 + y2dxdy

    D2

    1 + x2 + y2dxdy

    =

    2pi=0

    d

    2=0

    1 + 2d

    2pi=0

    d

    422=0

    1 + 2d

    = 2pi

    (3 1

    3(5 2

    2)3/2

    )Updeixh: Prospajste na upologsete to embadn thc epifneiac efarmzontac katllh-

    la ton tpo tou Stokes.Dete kai th sqetik jewra tou Kefalaou

    'Askhsh 11. Na upologisje to oloklrwmaS

    x2dydz + y2dxdz + z2dxdy

    pou S h epifneia thc sfarac x2 + y2 + z2 = a2

    a) Wc epiepifnio oloklrwma b' tpou

    a) me th bojeia tou jewrmatoc thc apoklsewc

    Lsh: a ' trpoc: To kjeto dinusma pnw sthn sfairik epifneia enai

    ~ =~|~| =

    1

    a(x~i+ y~j + z~k)

    kai epshc enai

    ~F = x2~i+ y2~j + z2~k

    Ean parametropoisoume thn epifneia thc sfarac qrhsimopointac sfairikc suntetag-

    mnec, dhlad

    x = a cos sin , y = a sin sin , z = a cos , d = a2 sin dd

    me ria

    0 2pi, 0 pi

    35

  • To oloklrwma tra grfetai

    I =

    S

    x2dydz + y2dxdz + z2dxdy =

    S

    ~F ~d

    = a

    S

    (x3 + y3 + z3

    )d

    = a4S

    ((cos sin )3 + (sin sin )3 + (cos )3

    )sin dd

    = a4( 2pi

    =0

    cos3 d

    pi=0

    sin4 d +

    2pi=0

    sin3 d

    pi=0

    sin4 d

    +

    2pi=0

    d

    pi=0

    cos3 sin d

    )= 0 + 0 + 0 = 0

    b' trpoc: Ap ton tpo tou Gauss ja qoume

    I =

    S

    ~F ~d =

    V

    div ~FdV =

    V

    (2x+ 2y + 2z)dxdydz

    = 2

    V

    (r cos sin + r sin sin + r cos )r2 sin drdd

    = 2

    ( ar=0

    r3dr

    2pi=0

    cosd

    pi=0

    sin2 d

    +

    ar=0

    r3dr

    2pi=0

    sind

    pi=0

    sin2 d

    +

    ar=0

    r3dr

    2pi=0

    d

    pi=0

    cos sin d

    )= 0 + 0 + 0 = 0

    'Askhsh 12. Na upologiste to oloklrwma

    I =

    c

    yzdx+ xzdy + xydz

    pou c enai h kleist kamplh x2 + y2, z = y2

    a) kateujean san epikamplio oloklrwma b' tpou

    b) qrhsimopointac ton tpo tou Stokes

    Lsh: a' trpoc : H probolc thc tom thc epifneiac z = y2 me thn kulndrikepifneia sto eppedo x2 + y2 = 1 enai nac kuklikc dskoc aktnac = 1. H kamplh cpnw sthn opoa upologzetai to oloklrwma parametropoietai wc exc

    ~r(t) = cos t~i+ sin t~j + sin2 t~k

    36

  • kai sunepc to oloklrwma gnetai

    I =

    c

    yzdx+ xzdy + xydz

    =

    2pit=0

    (y(t)z(t)

    dx

    dt+ x(t)z(t)

    dy

    dt+ x(t)y(t)

    dz

    dt

    )dt

    =

    2pit=0

    (sin t sin2 t( sin t) + cos t sin2 t( cos t)

    + cos t(2 sin t cos t))dt

    =

    2pit=0

    (3 cos2 t sin2 t sin4 t) dt

    =

    2pit=0

    (3 cos2 t sin2 t sin2 t(1 cos2)) dt

    =

    2pit=0

    sin2 2tdt 2pit=0

    sin2 tdt

    =1

    2

    4pix=0

    sin2 xdx 2pix=0

    sin2 xdx = 0

    b' trpoc : Smfwna ton tpo tou Stokes qoume

    I =

    c

    ~F d~r =S

    rot ~F ~nd

    pou S h epifneia pou enai tmma thc epifneiac z = y2 pou peribletai ap thn kleistkamplh c kai pou

    ~F = yz~i+ xzd~j + xy~k

    Ja qoume mwc

    rot~F = (x x)~i+ (y y)d~j + (z z)~k = ~0Sunepc ja qoume

    I =

    S

    rot ~F ~nd = 0

    'Askhsh 13. Na upologiste, me do trpouc, to embadn tou tmmatoc

    pou apokptei ap to nw hmisfario (z 0) thc sfarac x2 + y2 + z2 = 1 oklindroc x2 + y2 y = 0.Lsh: a' trpoc)

    To embadn pou jloume na upologsoume enai tmma sfarac x2 +y2+ z2 = 1 kai h probol tou D sto eppedo xy enai h bsh tou kulndrou x2+ y2 y = 0.Lambnontac upyh ti

    d =

    1 + (

    z

    x)2 + (

    z

    y)2dxdy =

    11 x2 y2dxdy

    to embadn ja dnetai ap to dipl oloklrwma

    I =

    S

    d =

    D

    1

    1 x2 y2dxdy

    37

  • Qrhsimopointac polikc suntetagmnec pou

    x = cos , y = sin , dxdy = dd

    kai lambnontac upyh ti lgw summetrac h epifneia problletai isposa sta do tm-

    mata tou kukliko dskou ta ria gia tic metablhtc kai enai

    x2 + y2 y = 0 0 sin , 0 pi2

    to oloklrwma gnetai

    I =

    D

    1

    1 x2 y2dxdy = 2 pi/2=0

    [ sin =0

    1 2d

    ]d

    = 2

    pi/2=0

    (1

    1 sin2

    )= 2

    pi/2=0

    (1 | cos |) d

    = 2

    pi/2=0

    (1 cos ) d = pi 2

    pou lbame upyh ti

    1 sin2 = cos2 = | cos |Updeixh :

    An sto parapnw oloklrwma parname san ria gia thn gwna , 0 pito oloklrwma ja gintan (met thn oloklrwsh wc proc thn metablht )

    I =

    pi=0

    (1 | cos |) d = pi/2=0

    (1 cos ) d + pi=pi/2

    (1 + cos ) d

    =1

    2(pi 2) + 1

    2(pi 2) = pi 2

    b' trpoc)

    Ja efarmsoume sfairikc suntetagmnec pou

    x = cos sin , y = sin sin , z = cos d = sin dd

    Ta shmea tomc thc sfarac me ton klindro brskontai apalefontac thn posthta x2+y2

    kai qrhsimopointac sfairikc suntetagmnec, ja qoume dhlad (gia thn perioq pou mac

    endiafrei)

    y + z2 = 1 sin sin = 1 cos2 sin = sin =

    To embadn tou tmmatoc thc sfairikc epifneiac ja enai sunepc

    I =

    S

    d = 2

    pi/2=0

    ( pi/2=

    sin d

    )d = 2

    pi/2=0

    (pi2 )sin d

    = pi

    pi/2=0

    sin d 2 pi/2=0

    sin d = pi 2

    38

  • 0.5 EFARMOGES

    'Askhsh 1. Na upologiste h mza tmmatoc sfairiko floio aktnwn a,b, a < b kai puknthtac = xyz

    x2+y2+z2, x, y, z > 0.

    Lsh: H mza tou smatoc dnetai ap thn sqsh

    m =

    V

    (x, y, z)dxdydz

    Epeid to sma tou opoou jloume na upologsoume th mza qei sfairik sqma ja

    qrhsimopoisoume sfairikc suntetagmnec, dhlad

    x = r cos sin , y = r sin, z = r cos, dxdydz = r2 sin drdd

    me ria, efson periorizmaste sto mprto ogdohmrio,

    a r b, 0 pi/2, 0 pi/4To oloklrwma grfetai tra

    m =

    V

    xyzx2 + y2 + z2

    dxdydz =

    br=a

    r4dr

    pi/2=0

    sin d

    pi/4=0

    d

    =1

    5(a5 b5) 1 pi

    4=

    (a5 b5)pi20

    'Askhsh 2. Na upologiste h rop adraneac tou omogenoc kulndrou

    x2 + y2 = a2, me yoc b, wc proc ton xona twn z

    Lsh: H rop adraneac, wc proc ton xona twn z, dnetai ap to tripl oloklrwma

    Iz =

    V

    (x2 + y2)dxdydz

    O kulindroc enai omogenc, sunepc h puknthta tou enai stajer = c. Ja qrhsimopoi-soume kulindrikc suntetagmnec, dhlad

    x = cos, y = sin, z = z, dxdydz = dddz

    me ria

    0 2pi, 0 a, 0 z bTo oloklrwma grfetai tra

    Iz =

    V

    (x2 + y2)dxdydz = c

    a=0

    3d

    2pi=0

    d

    bz=0

    dz

    = c a4

    4 (2pi) b = ca

    4pib

    2

    'Askhsh 3. Na breje h mza tou kwniko kelfouc z2 = 3(x2 + y2) epi-faneiakc puknthtac = x2 + y2, pou perikleetai metax twn epipdwn

    39

  • z = 0 kai z = 3.

    Lsh: H mza tou smatoc enai sugkentrwmnh sthn epifneia tou knou kai sunepc

    ja upologzetai ap to epiepeifneio oloklrwma thc morfc

    m =

    S

    (x, y, z)d =

    S

    (x2 + y2)d

    Ja qrhsimopoisoume kulindrikc suntetagmnec kai afo laboume upyh ti z =3(x2 + y3) =

    3 kai sunepc h gwna a tou knou ja enai tan a = /z =3/3 a = pi/6 ja qoumethn parametropohsh

    x = cos, y = sin, z =3, d =

    sin a= 2

    me ria

    0 2pi, 0 3

    To oloklrwma grfetai tra

    m =

    S

    (x2 + y2)d = 2

    3=0

    3d

    2pi=0

    d = 2 94 2pi = 9pi

    Upxeixh: Ennalaktik h parapnw skhsh mpore na luje jewrntac ti probloume

    thn kwnik epifneia sto eppedo Oxy. H probol pou prokptei enai profanc nackuklikc dskoc Dxy aktnac

    3. Epshc ja qoume ti

    d =

    1 +

    (z

    x

    )2+

    (z

    y

    )2dxdy = 2dxdy

    kai to oloklrwma ja enai

    m =

    S

    (x2 + y2)d =

    Dxy

    (x2 + y2)

    1 +

    (z

    x

    )2+

    (z

    y

    )2dxdy

    =

    Dxy

    2(x2 + y2)dxdy

    kai qrhsimopointac ssthma polikn suntetagmnwn dhlad

    x = cos, y = sin, dxdy = dd

    me ria

    0 2pi, 0 3

    katalgoume pli sto gnwst oloklrwma

    m = 2

    3=0

    3d

    2pi=0

    d = 9pi

    'Askhsh 4. Dnetai h surmtinh lika x = a cos t, y = a sin t, z = t (a > 0, =/0) me puknthta mzac = z. Na brejon gia t (0, 2pi)

    40

  • a) To shmeo tou kntrou mzac thc likac

    b) To mkoc thc likac

    Lsh: Gia profanec lgouc summetrac to kntro mzac ja brsketai ston xona twn zkai ja enai

    zG =1

    m

    c

    zds, m =

    c

    ds

    H mza katarqn upologzetai ap to oloklrwma

    m =

    c

    ds =

    2pit=0

    z

    (dx

    dt

    )2+

    (dy

    dt

    )2+

    (dz

    dt

    )2dt

    = a2 + 2

    2pit=0

    tdt = 2pi2a2 + 2

    Me to dio trpo ja qoume

    I =

    c

    zds =

    2pit=0

    zz

    (dx

    dt

    )2+

    (dy

    dt

    )2+

    (dz

    dt

    )2dt

    = 2a2 + 2

    2pit=0

    t2dt =8

    32pi3

    a2 + 2

    To kntro mzac ja enai sunepc

    zG =1

    mI =

    4

    3pi

    'Askhsh 5. Na upologisje h rop adrneiac omogenoc sfairiko kel-

    fouc wc proc ton xona z an h aktna tou enai a kai h puknthta tou

    Lsh:

    H rop adrneiac tou sfairiko kelfouc wc proc ton xona z dnetai ap to oloklrwma

    Iz =

    S

    (x2 + y2)d

    pou S h epifneia tou sfairiko kelfouc. Ja upologsoume to oloklrwma qrhsi-mopointac sfairikc suntetagmnec pou

    x = a cos sin , y = a sin sin , z = a cos

    To stoiqeidec embadn d se sfairik epifneia enai d = a2 sin ddJa qoume sunepc

    Iz =

    S

    (x2 + y2)d =

    G

    (a sin )2a2 sin dd

    = a4 pi=0

    sin3 d

    2pi=0

    d = a4 43 2pi = 8

    3pia4

    41

  • Lambnontac upyh ti h mza tou kelfouc ja enai

    m =

    S

    d = a2 pi=0

    sin d

    2pi=0

    d = 4pia2

    h rop adrneiac Iz ja dnetai ap thn sqsh

    Iz =2

    3ma2

    'Askhsh 6. Na upologiste h rop adrneiac omogenoc orjo kukliko

    knou wc proc ton xona y an h puknthta tou enai , h aktna thcbshc tou enai a kai to yoc tou h

    Lsh:

    H rop adrneiac tou knou wc proc ton xona y ja dnetai ap th sqsh

    Iy =

    V

    (x2 + z2)dxdydz

    Gia ton upologism tou parapnw oloklhrmatoc ja qrhsimopoisoume kulindrikc sunte-

    tagmnec pou enai

    x = cos , y = sin , z = z, dxdydz = dddz

    Epiplon ap ta moia trgwna OAB kai OOC ja qoume

    a=

    z

    h h

    a

    'Etsi sunolik ja qoume

    Iy =

    V

    (x2 + z2)dxdydz =

    G

    (2 cos2 + z2)dddz

    =

    2pi=0

    a=0

    ( hz=h

    a

    (2 cos2 + z2)dz

    )dd

    =

    2pi=0

    ( a=0

    (2 cos2 (h h

    a) +

    1

    3(h3 (h

    a)3)

    )d

    )d

    =

    2pi=0

    (a2h3

    10+

    1

    20a4h cos2

    )d =

    a2hpi

    20(a2 + 4h2)

    Lambnontac upyh ti h mza tou parapnw knou dnetai ap thn sqsh m = 1/3pia2h(h apdeixh afnetai san skhsh) tte ja qoume

    Iy =3

    20m(a2 + 4h2)

    'Askhsh 7. Ddetai h puramda x + y + z 1, x, y, z 0, puknthtac = x.Na breje to kntro mzac thc.

    42

  • Lsh:

    Upologzoume arqik th mza thc puramdac h opoa enai na oloklrwma gkou miac

    puramdac pou wc proc ton xona twn z periblletai ap tic epifneiec z = 0 kai z =1 x y kai h probol thc opoac ston eppedo Oxy enai h trigwnik epifneia Dxy hopoa periorzetai ap tic eujeec x = 0, y = 0 kai x + y = 1. H mza thc puramdac jadnetai sunepc ap thn sqsh

    m =

    V

    dV =

    V

    xdxdydz =

    Dxy

    ( z=1xyz=0

    xdz

    )dxdy

    =

    Dxy

    x(1 x y)dxdy = 1x=0

    ( 1xy=0

    x(1 x y)dy)dx

    =

    1x=0

    x(1

    2 x+ x

    2

    2)dx =

    1

    24

    H sunistsa tou kntrou mzac ston xona twn x ja enai

    xG =1

    m

    V

    xdV =1

    m

    V

    x2dxdydz

    =1

    m

    Dxy

    ( z=1xyz=0

    x2dz

    )dxdy =

    Dxy

    x2(1 x y)dxdy

    =

    1x=0

    ( 1xy=0

    x2(1 x y)dy)dx =

    1x=0

    x2(1

    2 x+ x

    2

    2)dx =

    2

    5

    H sunistsa tou kntrou mzac ston xona twn y ja enai

    yG =1

    m

    V

    ydV =1

    m

    V

    yxdxdydz

    =1

    m

    Dxy

    ( z=1xyz=0

    yxdz

    )dxdy =

    1

    m

    Dxy

    yx(1 x y)dxdy

    =

    1x=0

    ( 1xy=0

    yx(1 x y)dy)dx

    =1

    m

    1x=0

    x(1

    6 x

    2+x2

    2 x

    3

    6)dx =

    1

    5

    H sunistsa tou kntrou mzac ston xona twn z ja enai

    zG =1

    m

    V

    zdV =1

    m

    V

    zxdxdydz

    =1

    m

    Dxy

    ( z=1xyz=0

    zxdz

    )dxdy

    =1

    m

    Dxy

    x1

    2(1 x y)2dxdy

    =

    1x=0

    ( 1xy=0

    x1

    2(1 x y)2dy

    )dx

    =1

    m

    1x=0

    1

    2x(1

    3)(x 1)3dx = 1

    5

    43

  • Sunepc to kntro mzac qei suntetagmnec G( 615, 15, 15)

    'Askhsh 8. Na breje to kntro mzac tmmatoc omogenoc sfarac ak-

    tnac a pou apokptetai ap kno gwnac me koruf to kntro thcsfarac.

    Lsh:

    Lgw summetrac to kntro mzac ja brsketai pnw ston xona twn z kai ja enai

    zG =1

    m

    V

    zdxdydz

    Ja upologsoume arqik thn mza tou tmmatoc qrhsimopointac sfairikc suntetagmnec

    pou

    x = r cos sin , y = r sin sin , z = r cos , dxdydz = r2 sin drdd

    me ria

    0 r a, 0 , 0 2pi'Etsi ja enai

    m =

    V

    dxdydz =

    2pi=0

    d ar=0

    r2dr =0

    sin d

    = 2pi 13a3 (1 cos) = 2pia

    3

    3(1 cos)

    Gia to kntro mzac ja qoume

    zG =1

    m

    V

    zdxdydz =

    2pi=0

    d ar=0

    r3dr =0

    cos sin d

    =1

    m 2pi 1

    4a4 sin

    2

    2=

    3a

    8

    sin2

    1 cos

    'Askhsh 9. Na brejon h mza kai h rop adraneac omogenoc elleiy-

    oeidoc me exswsh

    x2

    4+y2

    9+z2

    16= 1

    wc proc thn arq twn axnwn

    Lsh: Gia na upologsoume thn mza kai thn rop adraneac sto parapnw elleiyoeidc

    mporome na efarmsoume nan katllhlo metasqhmatism se sfairikc suntetamnec pou

    ja enai

    x = 2r cos sin , y = 3r sin sin , z = 4r cos , dV = 24r2 sin drdd

    me ria

    0 r 1, 0 2pi, 0 pi

    44

  • H mza tou elleiyoeidoc dnetai ap to paraktw tripl oloklrwma

    m =

    V

    dxdydz = 24

    G

    r2 sin drdd

    = 24

    1r=0

    r2dr

    2pi=0

    d

    pi=0

    d = 24 13 2 2pi = 32pi

    H rop adrneiac dnetai ap to paraktw oloklrwma

    I =

    V

    (x2 + y2 + z2)dxdydz

    = 24

    G

    (4r2 sin2 cos2 + 9r2 sin2 sin2

    + 16r2 cos2 )r2 sin drdd

    = 24

    1r=0

    r4dr

    ( pi=0

    2pi=0

    (4 sin2 cos2 + 9 sin2 sin2

    + 16 cos2 ) sin dd)

    = 241

    5

    ( pi=0

    2pi=0

    4 sin3 cos2 dd+

    pi=0

    2pi=0

    9 sin3 sin2 dd

    +

    pi=0

    2pi=0

    16 cos2 sin dd

    )= 24

    1

    5

    (16pi

    3+ 12pi +

    64pi

    3

    )=

    928pi

    5

    'Askhsh 10. Leptc dskoc omogenc stajero pqouc k puknthtac kalptei thn perioq tou epipdou Oxy pou orzetai ap tic sqseic y = x2,x = y2. Na brejon to kntro mzac kai h rop adrneiac tou dskou wcproc thn arq twn axnwn.

    Lsh:

    Ja upologsoume arqik thn mza tou dskou pou enai

    m =

    Dxy

    dxdy =

    1x=0

    ( xy=x2

    dy

    )dx =

    1x=0

    (x x2) dx

    =

    (2

    3 13

    )=

    3

    a) Oi suntetagmnec tou kntrou mzac xG kai yG upologzontai ap ta paraktwoloklhrmata

    xG =1

    m

    Dxy

    xdxdy =

    m

    1x=0

    ( xy=x2

    xdy

    )dx

    =

    m

    1x=0

    x(

    x x2) dx = m

    3

    20=

    9

    20

    45

  • yG =1

    m

    Dxy

    ydxdy =

    m

    1x=0

    ( xy=x2

    ydy

    )dx

    =

    m

    1

    2

    1x=0

    (x x4) dx =

    2m

    3

    10=

    9

    20

    Sunepc to kntro mzac qei suntetagmnec G( 920, 920)b) H rop adrneiac, wc proc to shmeo O(0, 0) ja enai

    I =

    Dxy

    (x2 + y2)dxdy =

    1x=0

    ( xy=x2

    (x2 + y2)dy

    )dx

    =

    1x=0

    (x3/2

    3+ x5/2 x4 x

    6

    3

    )dx =

    6

    35

    'Askhsh 11. Na breje h z sunistsa tou kntrou mzac tou omogenocstereo pou perilambnetai metax twn epifanein z = x2 + 3y2 kai z =8 x2 y2Lsh:

    H probol tou koino gkou twn do epifanein sto eppedo Oxy prokptei apale-fontac thn metablht z kai aut pou prokptei enai mia lleiyhDxy me exswsh x

    2+2y2 = 4.Upologzoume arqik thn mza tou koino gkou pou prokptei ap to paraktw tripl

    oloklrwma

    m =

    V

    dxdydz =

    Dxy

    ( 8x2y2z=x2+3y2

    dz

    )dxdy

    =

    Dxy

    (8 2x2 4y2) dxdyTo parapnw oloklrwma mpore na upologiste ekola qrhsimopointac polikc sunte-

    tagmnec tic opoec orzoume wc exec

    x =2 cos , y = sin , dxdy =

    2dd

    kai me ria

    0 2, 0 2piTo oloklrwma grfetai plen

    m =

    Dxy

    (8 2x2 4y2) dxdy =

    G

    4(2 2)2dd

    = 42

    2pi=0

    d 2=0

    (2 2)d = 42 2pi 1 = 8pi

    2

    46

  • H z sunistsa tou kntrou mzac upologzetai ap to oloklrwma

    zG =1

    m

    V

    zdxdydz =

    m

    Dxy

    ( 8x2y2z=x2+3y2

    zdz

    )dxdy

    =

    2m

    Dxy

    4(4 + y2)(4 x2 2y2)dxdy

    =

    2m

    G

    4(4 + 2 sin2 )(4 22)2dd

    =42

    m

    2pi=0

    ( 2=0

    (4 + 2 sin2 )(2 2)d)d

    =42

    m

    2pi=0

    82

    15(10 + sin2 )d

    =42

    m

    82

    1521pi =

    448pi

    5m=

    282

    5

    'Askhsh 12. Jewreste to omogenc stere pou orzetai ap tic epifneiec

    x + y = 2, 2y + x = 4, z2 + y2 = 4, x, y, z 0. Na breje h z sunistsa toukntrou mzac

    Lsh: H probol tou koino gkou sto eppedo Oxy enai h epifneia Dxy h opoaorzetai ap tic kamplec x+ y = 2, 2y + x = 4 kai y = 0. Upologzoume arqik thn mzatou smatoc h opoa dnetai ap to paraktw tripl oloklrwma

    m =

    V

    dxdydz =

    Dxy

    ( 4y2z=0

    dz

    )dxdy

    =

    Dxy

    (4 y2)1/2dxdy = 2y=0

    ( 42yx=2y

    (4 y2)1/2dx)dy

    =

    2y=0

    (2 y)(4 y2)1/2dy = (2pi 8

    3

    )H z sunistsa tou kntrou mzac upologzetai ap to oloklrwma

    zG =1

    m

    V

    zdxdydz =

    m

    Dxy

    ( 4y2z=0

    zdz

    )dxdy

    =

    m

    Dxy

    1

    2(4 y2)dxdy =

    2m

    2y=0

    ( 42yx=2y

    (4 y2)dx)dy

    =

    2m

    2y=0

    (4 y2)(2 y)dy = 2m

    203=

    10

    3m=

    10

    3(2pi 83)

    47