Thread: thoroughly Stuck!

alright, this bit of homework has really got to me, so I will post the original question here, along with the work I've done. Please help me out guys:

Calculate the surface integral Where and S is the entire sphere with outward orientation

SO, the tip I got from lectures is that we are basically only considering F on S. So:

I redifined S in terms of spherical coordinates such that because the radius is a constant, I don't need to consider it (as independent) when I make the parameterized surface, since it is always 3. I can rewrite the integrals now as:

and, because I switched to spherical coordinates, I also need a jacobian of which, in this case, is

Now, I take the derivative of this separetly with respect to each variable, and then cross them (still the tips from the lectures), which would, effectively, be

==

okay, that all said, here is where it's my own brain, and I feel like I've missed something obvious, but here goes anyway.

for my varibles,

the fun part:

= and I have no idea where to go from here. I only combined like terms and pulled out a ... any helpful hints appreciated.

Try using Stokes Theorem (or maybe you have heard it called the Divergence Theorem for this case).

Life is good. thank you Dr.

becomes
which, when we substitute back in our original problems we get:
which becomes:
and, as the last 2 terms are opposites, they cancel, and all I have is
which, when evaluated over spherical coordinates, becomes
if I'm not mistaken. Which, will evaluate to:
which is, more elegantly, , yes?

Originally Posted by Arcane_Mathamatition

Life is good. thank you Dr.

becomes
which, when we substitute back in our original problems we get:
which becomes:
and, as the last 2 terms are opposites, they cancel, and all I have is
which, when evaluated over spherical coordinates, becomes
if I'm not mistaken. Which, will evaluate to:
which is, more elegantly, , yes?

yep

It is even easier when you know the formula for the volume of a sphere -- -- and just substitute the radius of 3 and multiply by the constant that you are integrating which is 3 again. You can do the whole problem without evaluating a single integral directly.

ah, good to know, but I have a sneaking suspicion we wont have a sphere to integrate over on our final. good tip though, thanks!