1. ## Divergence Theorem

I have recently done a proof (verification) of the divergence theorem of a sphere centered at the origin with no problem using polar coordinates to complete the LHS and spherical coordinates on the RHS. I now have a problem that gives a sphere at x^2 + y^2 + z^2 + 2*x - 2*y =7. Before the rhs volume integral was very nice in spherical coordinates when the spere was at 0,0,0. I need to determine either side of the divergence theorem. del dotted with F gives me a constant but not 0. Where should I start (is the surface integral easier to deal with than the volume integarl)? The obivous choice of spherical coordinates from before does not look so great now? Could I take advantage of the y = 0 surface in the surface integral case?

2. Originally Posted by MarionButler
I have recently done a proof (verification) of the divergence theorem of a sphere centered at the origin with no problem using polar coordinates to complete the LHS and spherical coordinates on the RHS. I now have a problem that gives a sphere at x^2 + y^2 + z^2 + 2*x - 2*y =7. Before the rhs volume integral was very nice in spherical coordinates when the spere was at 0,0,0. I need to determine either side of the divergence theorem. del dotted with F gives me a constant but not 0. Where should I start (is the surface integral easier to deal with than the volume integarl)? The obivous choice of spherical coordinates from before does not look so great now? Could I take advantage of the y = 0 surface in the surface integral case?

If $\nabla \cdot F = C$ where $C$ is constand then the divergence theorem gives

$\iiint_VCdV=C\iiint_VdV=C\cdot (\text{ Volume of the sphere})$

This is becuase the integral $\iiint_VdV$ is the just the volume of the region of integration!

So just complete the squares in the equation of the sphere to find the radius and use the volume formula for the sphere!