Surface integral FdS where F(x,y,z) = xz i + x j + y k

[shakalandro]

FdS where F(x,y,z) = xz i + x j + y k and S is the hemisphere x[sup:3gy6rm93]2[/sup:3gy6rm93] + y[sup:3gy6rm93]2[/sup:3gy6rm93] + z[sup:3gy6rm93]2[/sup:3gy6rm93] = 25 and y > 0.

I'm not sure how to approach the problem, should I be finding a parametrization in spherical coordinates or what?

 

[Unco]

Re: Surface integral

If you are familiar with the divergence theorem, that's slightly neater. Otherwise, to calculate the surface integral directly, then yes, spherical coordinates are the way to go to parameterise the surface. Let's see how you go; show your work if you get stuck.

Edit: There's no curl here, so the divergence theorem is the one I meant!

 

[shakalandro]

I end up with a bunch of junk that then evaluates to zero.

I get the double integral of phi (which I'll denote x) from 0 to pi and theta (which I'll denote y) from o to pi of...
625sin^3xcosxcos^2y + 125sin^3xcosysiny + 125 sin^2xcosxsinO

all three terms seem to end up with a sin term somewhere whihc of course is evaluated to zero and then the whole thing is zero. Is that just the answer, or I am doing something wrong?

 

[Unco]

The final answer of 0 is correct. I meant to say the Divergence theorem (not Stokes', which involves the curl); using that agrees with your final answer.