Evaluate the surface integral $\iint_S \mathbf{F} \cdot d\mathbf{S}$ for the given vector field \(\mathbf{F}\) and the oriented surface \(S\). In other words, find the flux of \(\mathbf{F}\) across \(S\). For closed surfaces, use the positive (outward) orientation.
$\mathbf{F}(x, y, z) = xy \mathbf{i} + 12x^2 \mathbf{j} + yz \mathbf{k}$
$\(S\) is the surface $z = xe^y$, $0 \le x \le 1$, $0 \le y \le 2$, with upward orientation
