Compute \(\iint_{\mathcal{S}} \mathbf{F} \cdot d \mathbf{S}\) for the given oriented surface or parametrized surface.
\(\mathbf{F}(x, y, z)=\left\langle 0,0, x z e^{x y}ightangle, \quad z=x y, \quad 0 \leq x \leq 1,0 \leq y \leq 1\), upward-pointing normal