Graphs of Functions

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Section 5.11 Graphs of Functions

We 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{5.11.1} \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{5.11.2} \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{5.11.3} \end{equation}

You can find these formulas for \(d\AA\) and \(\dA\) in most textbooks on vector calculus.