Calculate the flux of the vector field $\vec{F}(x,y,z) = (e^{xy} + 3 z + 4) \vec{i} + (e^{xy} + 4 z + 3) \vec{j} + (3 z + e^{xy}) \vec{k}$ through the square of side length $3$ with one vertex at the origin, one edge along the positive y-axis, one edge in the xz-plane with $x \geq 0$ and $z \geq 0$, oriented downward with normal $\vec{n} = \vec{i} - \vec{k}$.

Flux = $\displaystyle \iint\limits_S \vec{F} \cdot d\vec{A}$ =