Past exam of the mathematical methods II course of the University of Oxford / 2016 / 10 / a

Parametrize the graph by $\mathbf{r}(x,y)=(x,y,f(x,y))$ over its projection $D$ onto the $xy$ plane. The tangent vectors are $(1,0,f_x)$ and $(0,1,f_y)$, so their cross product is $(-f_x,-f_y,1)$. Hence the scalar surface element isandThe printed subscript $S$ is shorthand for integrating over the projected parameter region: $dxdy$ belongs to $D$, while the square-root factor converts projected area into actual surface area. The corresponding upward vector surface element of a graph is $(-f_x,-f_y,1)dxdy$.