-
SHMMUMajhmatik Anlush
(Sunartseic Polln Metablhtn - Dianusmatik Anlush)Lseic tou 10ou
fulladou asksewn
Askhsh 1. Na upologsete to epifaneiak oloklrwma tou dianusmatiko
pe-dou F = (2x; 3y; z) pnw sthn epifneia pou frssetai ap to
paraboloeidcz = 4 x2 y2 kai ap ton kuklik dsko, kntrou (0; 0; 0)
tou xy epipdou,aktnac 2.
Lsh. Prkeitai gia thn epifneia S h opoa enai to snoro tou
kanonikoqwrou W to opoo apoteletai ap la ta (x; y; z) 2 R3, gia ta
opoa isqei
x2 + y2 4 kai 0 z 4 x2 y2 :Efarmzontac to Jerhma Gauss qoume tiZ
Z
S
F dS =Z Z Z
W
r F dV :
Qrhsimopointac kulindrikc suntetagmnec kai afo r F = 6 brskoume
tiZ ZS
F dS = 6Z 20
Z 20
Z 4r20
r dzdrd = 48 :
Askhsh 2. Na epalhjesete to Jerhma Stokes gia to dianusmatik
pedo
F (x; y; z) = (y + x; x+ z; z2)
kai thn epifneia tou knou z2 = x2+y2 pou brsketai anmesa sta
eppeda z = 0kai z = 1.
Lsh. Upologzoume prta to epifaneiak oloklrwma. An S h
dosmnhepifneia ma parametrikopohsh thc enai h
(r; ) = (r cos ; r sin ; r) ; 0 2 ; 0 r 1 :Upologzoume ta
diansmata Tr, T kai brskoume ti
Tr T = (r cos ;r sin ; r) :Epshc
r F = (1; 0; 0) :Sunepc Z Z
S
r F dS =Z 20
Z 10
r cos drd = 0 :
1
-
Gia to epikamplio oloklrwma qoume ti ma parametrikopohsh tou
sunrouthc S, me jetik for, enai h
r(t) = (cos t; sin t; 1); 0 t 2 :Opte I
@S
F ds =Z 20
F (r(t)) r0(t) dt = 0 :
Askhsh 3 (H fusik shmasa thc apklishc). 'Estw Br h kleist
sfarakntrou (x0; y0; z0) kai aktnac r > 0. Epshc stw
F : Br ! R3
na C1 dianusmatik pedo kai n to monadiao kjeto dinusma sthn
epifneia thcsfarac, me katejunsh proc ta xw. Na dexete ti
r F (x0; y0; z0) = limr!0
3
4r3
Z Z@Br
F ndS :
Ermhneste to parapnw apotlesma,(Updeixh: afo to F enai C1, ja
isqei r F (x; y; z) = r F (x0; y0; z0) +G(x; y; z), me G(x; y; z)!
0, kajc (x; y; z)! (x0; y0; z0).)
Lsh. To dianusmatik pedo enai C1, opte gia kje (x; y; z) sthn
Brqoume ti
r F (x; y; z) = r F (x0; y0; z0) +G(x; y; z) ;me G(x; y; z)! 0,
kajc (x; y; z)! (x0; y0; z0).
Efarmzoume to Jerhma Gauss kai qoume ti
3
4r3
Z Z@Br
F ndS = 34r3
Z Z ZBr
r F (x; y; z) dV
=3
4r3
Z Z ZBr
(r F (x0; y0; z0) +G(x; y; z)) dV
= F (x0; y0; z0) +3
4r3
Z Z ZBr
r G(x; y; z) dV :
'Omwc
lim(x;y;z)!(x0;y0;z0)
3
4r3
Z Z ZBr
G(x; y; z) dV
ap pou prokptei to sumprasma.
Askhsh 4. 'Estw h prosanatolismnh epifneia S kai
f; g : S ! RC1 sunartseic. Na dexete ti
2
-
(i) Z@S
(frg) ds =Z Z
S
(rf rg) ndS ;
pou n enai to monadiao kjeto dinusma sthn S me katejunsh proc
taxw kai ti
(ii) Z@S
(frg + grf) ds = 0 :
Lsh.
(i) Ap to Jerhma Stokes qoume tiZ@S
(frg) ds =Z Z
S
(r (frg)) ndS =Z Z
S
(rf rg) ndS :
(ii) Ap to (i) qoume tiZ@S
(frg + grf) ds =Z Z
S
[(rf rg) + (rg rf)] ndS = 0 :
Askhsh 5. 'EstwW qwro tou R3 gia to opoo efarmzetai to Jerhma
Gausskai
f; g : W ! RC2 sunartseic. An me
@f
@n
sumbolzoume thn kateujunmenh pargwgo thc f , sthn katejunsh tou
mona-diaou diansmatoc n me katejunsh proc ta xw, na dexete tiZ
Z
@W
g@f
@ndS =
Z Z ZW
(gr2f +rf rg)dV :
Lsh. JtoumeF = grf
kai efarmzoume to Jerhma Gauss. To sumprasma prokptei mesa
afo
@f
@n= rf n
kair (grf) = gr2f +rf rg :
3
