Prerequisites

Simple Surface Elements

The simplest surfaces are those given by holding one of the coordinates constant. Thus, the $xy$-plane is given by $z=0$. Its (surface) area element is $dA=(dx)(dy)=(dr)(r\,d\phi)$, as can easily be seen by drawing the appropriate small rectangle. The surface of a cylinder is nearly as easy, as it is given by $r=a$ in cylindrical coordinates, and drawing a small “rectangle” yields for the surface element \begin{eqnarray} \hbox{cylinder:} \qquad && \dS = (a\,d\phi)(dz) = a\, d\phi \, dz \quad\qquad\qquad\nonumber\end{eqnarray} while a similar construction for the sphere given by $r=a$ in spherical coordinates yields \begin{eqnarray}\nonumber \hbox{sphere:} \qquad && \dS = (a\,d\theta)(a\sin\theta\,d\phi) = a^2 \sin\theta \, d\theta \, d\phi \end{eqnarray} The last expression can of course be used to compute the surface area of a sphere, which is  1) \begin{equation} \Int_{\rm sphere} \!\! \dS = \int_0^{2\pi} \int_0^\pi a^2 \sin\theta \, d\theta \, d\phi = 4 \pi a^2 \end{equation}

What about more complicated surfaces?

The basic building block for surface integrals is the infinitesimal area $\dS$, obtained by chopping up the surface into small pieces. If the pieces are small parallelograms, then the area can be determined by taking the cross product of the sides!

1) We write a single integral sign when talking about adding up “bits of area” (or “bits of volume”), reserving multiple integral signs for iterated single integrals. The notation $\DS \DInt{} \dS$ is also common.