Vector Calculus · Challenge problem

Let $S$ be the upper hemisphere $x^2 + y^2 + z^2 = 4$, $z\ge 0$, with outward (upward) normal, and $\mathbf F = (z - y,\; x + z^2,\; xyz)$. Evaluate $\displaystyle\iint_S (\nabla\times\mathbf F)\cdot d\mathbf S$. Give your answer in terms of $\pi$.