In verifying the divergence theorem for a unit sphere and a vector field F I have no problem evaulating the triple integral using spherical coordinates for the RHS. This can be done by hand. Looking at the surface integral on the LHS I have used rectangular coordinates and switched the limits of integration at the end to polar. In both of the cases the integral turns nasty. I get the same answer if I leave it in rect or polar when I evaluate using Maple. I need to do everything by hand so I am assuming I should start with sperical coordinates.

r^2=1 (for top r = 1)
R()=rsin(phi)cos(theta) i + rsin(phi)cos(theta) j + r cos(phi) k
unit normal * dA = (r*sin(phi) i + r*sin(phi) j + r*sin(phi) k)*|dphi|*|dtheta|

After dotting F with the unit normal*dA what would my limits of integration be for the top half? I cannot seem to make it work out. Am I missing something.