# Easy flux surface integral

1. Jun 6, 2009

### cos(e)

1. The problem statement, all variables and given/known data
find the flux of the field F(vector) across the portion of the sphere x^2+y^2+z^2=a^2 in the first octant directed away from the origin

2. Relevant equations
F(x,y,z)=zk(hat)

3. The attempt at a solution
i used Flux=double integral over x-y plane F.n(unit normal)dsigma
i let g=x^2+y^2+z^2-a^2=0
p=n=(x,y,z)/sqrt(x^2+y^2+z^2)

and after some algebra i got:
double integral( (a^2-x^2-y^2)/a dA)
which i used polar co-ordinates
to get 1/a double integral( a^2-r^2)r dr dtheta
which gave me pi*a^3/8 yet the solutions in back of book is pi*a^3/6

Looks like iv made a simple error somewhere(iv looked like 3 times but cant find one), but i just need to check im using the right method.
sorry bout the way i worte out the maths but im unaware how to put it in nice maths form

cheers,
cos(e)

2. Jun 6, 2009

### cos(e)

ok i get the answer with p=k(hat)
now im sure why im letting p=k(hat), is it because p is the normal to the area of the double intergral i take, in this case the x-y plane with theta varying from 0 to pi/2 adn r varying from 0 to a?

3. Jun 6, 2009

### Count Iblis

If we call the integral I, then by symmetry, the integral over the entire surface of the sphere is 8 I. By the divergence theorem, we have:

8 I = Integral of div F over volume = 4/3 pi a^3 -------->

I = pi a^3/6