We can apply these techniques to surfaces which are the graphs of functions. Suppose $z=f(x,y)$. 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 \end{equation} we obtain \begin{eqnarray} d\rr_1 &=& dx\,\xhat + dz\,\zhat = \left(\xhat + \Partial{f}{x}\,\zhat\right) dx \\ d\rr_2 &=& dy\,\yhat + dz\,\zhat = \left(\yhat + \Partial{f}{y}\,\zhat\right) dy \end{eqnarray} so that \begin{equation} d\SS = d\rr_1\times d\rr_2 = \left( -\Partial{f}{x}\,\xhat - \Partial{f}{y}\,\yhat + \zhat \right) \,dx\,dy \end{equation}

Similarly, if a scalar surface integral is needed, we can compute \begin{equation} \dS = |d\SS| = \sqrt{1+\left(\Partial{f}{x}\right)^2+\left(\Partial{f}{y}\right)^2}\>dx\,dy \end{equation}

You can find these formulas for $d\SS$ and $\dS$ in most textbooks on vector calculus.