The traditional approach to curves and surfaces involves parameterization, which we have deliberately saved for last. Recall that a parametric curve can be written \begin{equation} \rr = \rr(u) = x(u)\,\xhat + y(u)\,\yhat + z(u)\,\zhat \end{equation} together with an appropriate domain for the parameter $u$. A parametric surface is similar, except there are now two parameters $u$,$v$ (and an appropriate domain): \begin{equation} \rr = \rr(u,v) = x(u,v)\,\xhat + y(u,v)\,\yhat + z(u,v)\,\zhat \end{equation}

Here are some examples of parametric surfaces (without domains): \begin{eqnarray*} \hbox{sphere}: \quad && \rr(\theta,\phi) = a\sin\theta\cos\phi\,\xhat + a\sin\theta\sin\phi\,\yhat + a\cos\theta\,\zhat \\ \hbox{cylinder}: \quad && \rr(\phi,z) = a\cos\phi\,\xhat + a\sin\phi\,\yhat + z\,\zhat \\ z=f(x,y): \quad && \rr(x,y) = x\,\xhat + y\,\yhat + f(x,y)\,\zhat\cr z=f(r,\phi): \quad && \rr(r,\phi) = r\cos\phi\,\xhat + r\sin\phi\,\yhat + f(r,\phi)\,\zhat\\ \hbox{surface of revolution}: \quad && \rr(x,\phi) = x\,\xhat + f(x)\cos\phi\,\yhat + f(x)\sin\phi\,\zhat \\ \hbox{change of variables}: \quad && \rr(u,v) = x(u,v)\,\xhat + y(u,v)\,\yhat \end{eqnarray*}

Recall further that for a parametric curve we have \begin{equation} d\rr = {d\rr\over du} \,du \end{equation} On a parametric surface it is natural to construct $d\SS$ using curves of constant parameter. But along a curve with $v=\hbox{constant}$, we have \begin{equation} d\rr_1 = \Partial{\rr}{u} \,du \end{equation} and similarly along a curve with $u=\hbox{constant}$ we have \begin{equation} d\rr_2 = \Partial{\rr}{v} \,dv \end{equation} Thus, \begin{equation} d\SS = d\rr_1\times d\rr_2 = \left(\Partial{\rr}{u} \times \Partial{\rr}{v}\right) \> du\,dv \end{equation} and \begin{equation} \dS = |d\SS| = \left| \Partial{\rr}{u} \times \Partial{\rr}{v} \right| \> du \, dv \end{equation} These formulas can be found in most calculus texts, and may help to relate our approach to the more traditional one.

However, expression (1) of § {Surface Elements} for $d\SS$ makes no reference to the parameters! In fact, $d\SS$ can be computed using any two (independent) infinitesimal displacements in the surface; no parameterization is needed! You will have to decide for yourself which approach you prefer. But real-world problems rarely come equipped with parameterizations, unlike those found in calculus texts. We reiterate that it is not always necessary to have an explicit parameterization in order to determine the surface element, as illustrated by our computation for the paraboloid in § {Less Symmetric Surfaces}.