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I am studying for finals and I'm having trouble calculating flux over sections of spheres. I can do it using the divergence theorem, but I need to know how to do it without divergence thm also.

The problem is, when calculating a vector field such as F(x, y, z) = <z, y, x>, say over the unit sphere (x^2 + y^2 + z^2 = 1), I always end up with weird terms like sin^3(phi) and cos^2(phi)sin(theta) that must be integrated

So, is this normal? Should I memorize integrals for sin^3(phi) and such, or is there an easier method?

thanks