I'm asked to find the flux (going out) of the upper-unitary-semisphere of the following vector field: $$F=(xy, -xy, z\sin^2ze^{x^2+y^2})$$ Every component is $C^{\infty}$ so we can apply the divergence theorem. We obtain: $$\int_S F\times n d\Sigma=\int_S div(F)dV$$ so we have to compute: $$\int_S y-x+e^{x^2+y^2}(\sin^2z+2z\sin z\cos z)dxdydz$$ or at least I need to say it's positive. I'd like to see a fast computation of this integral, any suggestion?
Thanks in advance.