Simple Surface Elements

Section 7.2 Simple Surface Elements

The simplest surfaces are those given by holding one of the coordinates constant. Thus, the \(xy\)-plane is given by \(z=0\text{.}\) Its (surface) area element is \(dA=(dx)(dy)=(dr)(r\,d\phi)\text{,}\) 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{align*} \textrm{cylinder:} \qquad \amp \amp \dA = (a\,d\phi)(dz) = a\, d\phi \, dz \end{align*}

while a similar construction for the sphere given by \(r=a\) in spherical coordinates yields

\begin{align*} \textrm{sphere:} \qquad \amp \amp \dA = (a\,d\theta)(a\sin\theta\,d\phi) = a^2 \sin\theta \, d\theta \, d\phi . \end{align*}

The last expression can of course be used to compute the surface area of a sphere, which is ¹

\begin{equation} \Int_{\textrm{sphere}} \!\! \dA = \int_0^{2\pi} \int_0^\pi a^2 \sin\theta \, d\theta \, d\phi = 4 \pi a^2 .\tag{7.2.1} \end{equation}

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 \(\DInt{} \dA\) is also common.

What about more complicated surfaces?

The basic building block for surface integrals is the infinitesimal area \(\dA\text{,}\) 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!