Evaluating Divergence Thm: V, S & Verify for x^2+y^2+z^2=1

[jlmac2001]

I need help evaluating both sides of the divergence theorem if V=xi+yj+zk and the surface S is the sphere x^2+y^2+z^2=1, and so verify the divergence theorem for this case.

Is the divergence theorem the triple integral over V (div V) dxdydz= the double integral over S (V dot normal)dS? If so I would I evaluate it for the above problem?

 

[matt grime]

Try looking these things up, (Wolfram/mathworld). And yes, verify means evaluate both sides of the equality.

 

[HallsofIvy]

Note that, in this problem, div V is very simple so the integration over the volume is trivial. Integrating V.n dS on the surface is a bit more challenging but if you "project" the surface into the xy-plane and then use polar coordinates, it should be easy. (Don't forget to do both the part of the sphere above the xy-plane and the part below!)