Prove that if \(\mathcal{S}\) is the part of a graph \(z=g(x, y)\) lying over a domain \(\mathcal{D}\) in the \(x y\)-plane, then

\[
\iint_{\mathcal{S}} \mathbf{F} \cdot d \mathbf{S}=\iint_{\mathcal{D}}\left(-F_{1} \frac{\partial g}{\partial x}-F_{2} \frac{\partial g}{\partial y}+F_{3}ight) d x d y
\]