Vector Calculus · Superchallenge problem

Let $S$ be the side of the cylinder $x^2 + y^2 = 1$, $0 \le z \le 2$ (without its top or bottom), with normal pointing away from the $z$-axis, and let $\mathbf F = \left(-yz,\; xz,\; e^{xy}\right)$. Evaluate $\displaystyle\iint_S (\nabla\times\mathbf F)\cdot d\mathbf S$. Give your answer in terms of $\pi$.