## Calculus 8th Edition

$\iint_S F \cdot dS=\iint_D [P \dfrac{\partial h}{\partial x}-Q+R\dfrac{\partial h}{\partial z}]dA$ where, $D$ is projection of $S$ onto xz-plane.
When the surface is orientable then the flux through a surface is defined as: $\iint_S F \cdot dS=\iint_S F \cdot n dS$ where, $n$ is the unit vector and $dS=n dA=\pm \dfrac{(r_u \times r_v)}{|r_u \times r_v|}|r_u \times r_v| dA=\pm (r_u \times r_v) dA$ Here, $r_x=i+(\dfrac{\partial h}{\partial x}) j ;r_z=(\dfrac{\partial h}{\partial z}) j+k$ Now, we have $r_x \times r_z=\dfrac{\partial h}{\partial x} i-j+\dfrac{\partial h}{\partial z}k$ Thus, we have $dS=(\dfrac{\partial h}{\partial x} i-j+\dfrac{\partial h}{\partial z}k) dA$ When $F=Pi+Qj+Rk$, then, we get $\iint_S F \cdot dS=\iint_D [P \dfrac{\partial h}{\partial x}-Q+R\dfrac{\partial h}{\partial z}]dA$ where, $D$ is projection of $S$ onto xz-plane.