Answer
$\iint_S F \cdot dS=\iint_D [P \dfrac{\partial h}{\partial x}-Q+R\dfrac{\partial h}{\partial z}]dA $
where, $D$ is the projection of $S$ onto the xz-plane.
Work Step by Step
The flux through a surface can be defined only when the surface is orientable.
We know that $\iint_S F \cdot dS=\iint_S F \cdot n dS$
Here, $n$ denotes the unit vector.
Since, $dS=n dA=\pm \dfrac{(r_u \times r_v)}{|r_u \times r_v|}|r_u \times r_v| dA=\pm (r_u \times r_v) dA$
Here, $r_x=i+\dfrac{\partial h}{\partial x} j$ and $r_z=\dfrac{\partial h}{\partial z} j+k$
and $r_x \times r_z=\dfrac{\partial h}{\partial x} i-j+\dfrac{\partial h}{\partial z}k$
Thus, $dS=(\dfrac{\partial h}{\partial x} i-j+\dfrac{\partial h}{\partial z}k) dA$
When $F=Pi+Qj+Rk$
Then, we have, $\iint_S F \cdot dS=\iint_D [P \dfrac{\partial h}{\partial x}-Q+R\dfrac{\partial h}{\partial z}]dA $
where, $D$ is the projection of $S$ onto the xz-plane.
