1. ## Surface flux integral

EDIT: I've put this in the wrong forum, haven't I? can this be moved/deleted?

Hi, just need a quick hand on an assignment question:

Find the flux of $\displaystyle F$ across the surface $\displaystyle S$, where $\displaystyle F = xzi+xj+yk$, and $\displaystyle S$ is the hemisphere $\displaystyle x^2 + y^2 + z^2 = 25, y >= 0$, oriented in the direction of the positive y-axis.
What I've done so far is (convert to spherical) parameterise $\displaystyle S$ by $\displaystyle r(\theta,\phi)=5sin{\phi}cos{\theta}i+5sin{\phi}si n{\theta}j+5cos{\phi}k$

And taken the vectors $\displaystyle r_\theta$ and $\displaystyle r_\phi$, found their cross product as a normal vector, then taken the dot product with $\displaystyle F$.

What results is something horrible (seeing as the integral must be taken), and it makes me think that something has gone very wrong:
$\displaystyle F \cdot (r_\theta \times r_\phi) =625sin^3{\phi}cos{\phi}cos^2{\theta}+125sin^3{\ph i}sin{\theta}cos{\theta}+125sin^2{\phi}cos{\phi}si n{\theta}$

Obviously I'm not looking for an answer to the problem, but I'm pretty sure that I've made an error somewhere, and I would appreciate it if someone could point it out. More details of the in-between steps can be supplied, if needed.

2. Originally Posted by backflip
Hi, just need a quick hand on an assignment question:

What I've done so far is (convert to spherical) parameterise $\displaystyle S$ by $\displaystyle r(\theta,\phi)=5sin{\phi}cos{\theta}i+5sin{\phi}si n{\theta}j+5cos{\phi}k$

And taken the vectors $\displaystyle r_\theta$ and $\displaystyle r_\phi$, found their cross product as a normal vector, then taken the dot product with $\displaystyle F$.

What results is something horrible (seeing as the integral must be taken), and it makes me think that something has gone very wrong:
$\displaystyle F \cdot (r_\theta \times r_\phi) =625sin^3{\phi}cos{\phi}cos^2{\theta}+125sin^3{\ph i}sin{\theta}cos{\theta}+125sin^2{\phi}cos{\phi}si n{\theta}$

Obviously I'm not looking for an answer to the problem, but I'm pretty sure that I've made an error somewhere, and I would appreciate it if someone could point it out. More details of the in-between steps can be supplied, if needed.
Why does integrating $\displaystyle 625sin^3{\phi}cos{\phi}cos^2{\theta}+125sin^3{\phi }sin{\theta}cos{\theta}+125sin^2{\phi}cos{\phi}sin {\theta}$ worry you?
The integration can be broken up into three bits. In each bit the integration wrt $\displaystyle \theta$ and the integration wrt $\displaystyle \phi$ can be done independent to each other. The trig integrals are (relative to the standard of the problem) routine.
All you have to do is get the integral terminals for the $\displaystyle \theta$ and $\displaystyle \phi$ integrals correct.