Section 7.12 Graphs of Functions
ΒΆWe can apply these techniques to surfaces which are the graphs of functions. Suppose \(z=f(x,y)\text{.}\) We slice the surface using curves along which either \(y\) is constant or \(x\) is constant. Since
\begin{equation}
dz = df = \Partial{f}{x}\,dx + \Partial{f}{y}\,dy\tag{7.12.1}
\end{equation}
we obtain
\begin{align*}
d\rr_1
\amp= dx\,\xhat + dz\,\zhat
= \left(\xhat + \Partial{f}{x}\,\zhat\right) dx\\
d\rr_2
\amp= dy\,\yhat + dz\,\zhat
= \left(\yhat + \Partial{f}{y}\,\zhat\right) dy
\end{align*}
so that
\begin{equation}
d\AA
= d\rr_1\times d\rr_2
= \left(
-\Partial{f}{x}\,\xhat - \Partial{f}{y}\,\yhat + \zhat
\right) \,dx\,dy .\tag{7.12.2}
\end{equation}
Similarly, if a scalar surface integral is needed, we can compute
\begin{equation}
\dA
= |d\AA|
= \sqrt{1+\left(\Partial{f}{x}\right)^2+\left(\Partial{f}{y}\right)^2}
\> dx\,dy .\tag{7.12.3}
\end{equation}
You can find these formulas for \(d\AA\) and \(\dA\) in most textbooks on vector calculus.