Evaluate the surface integral:
$$\iint_S \mathbf{\vec F} \cdot d\mathbf{\vec S}$$
for the vector field
$$ \mathbf{\vec F}(x,y,z) = xze^y \mathbf{ \hat i} - xze^y \mathbf{\hat j} + z\mathbf{\hat k}$$
where $S$ is part of the plane $x + y + z = 1$ in the first octant and has a downward orientation.