Use the Divergence Theorem to evaluate
$\iint_S x^2y^2+y^2z^2+z^2x^2 dS$
where $S$ is the surface of the sphere $x^2+y^2+z^2=1$.
So the divergence of F is $2y^2x+2z^2y+2x^2z$. Then I tried to integrate using spherical coordinates, but the integral seemed to be very exhausting. Is there a better way to do it?