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Section 10.5 Vector Surface Elements

In this section, you will find the vector surface elements for common shapes with high symmetry.

Activity 10.1. Surface Elements for Planes, Cylinders, and Spheres.

Using the general formula for the scalar surface elements from Section (10.5.1) or the vector surface element

\begin{equation} d\AA = d\rr_1 \times d\rr_2\tag{10.5.1} \end{equation}

find explicit formulas for the vector surface element in each of the following cases:

  • a plane in both rectangular and polar coordinates;

  • the three surfaces (top, bottom, and curved side) of a cylinder with finite length;

  • the surface of a sphere.

Answer.

In this activity, you should have obtained the following common surface elements:

\begin{align*} \hbox{(plane)}\quad z\amp=\hbox{ const}: \amp d\AA\amp=dx\,dy \> \zhat ;\\ \hbox{(plane)}\quad z\amp=\hbox{ const}: \amp d\AA\amp=s\,ds\,d\phi \> \zhat ;\\ \hbox{(top of cylinder)}\quad z\amp=\hbox{ const}: \amp d\AA\amp=s\,ds\,d\phi \> \zhat ;\\ \hbox{(bottom of cylinder)}\quad z\amp=\hbox{ const}: \amp d\AA\amp=-s\,ds\,d\phi \> \zhat ;\\ \hbox{(side of cylinder)}\quad s\amp=\hbox{ const}: \amp d\AA\amp=s\,d\phi\,dz \> \shat ;\\ \hbox{(sphere)}\quad r\amp=\hbox{ const}: \amp d\AA\amp=r^2\sin\theta\,d\theta\,d\phi \> \rhat . \end{align*}
Figure 10.3. A disk oriented upward (left) and downward (right).

In each of these examples, the direction of the vector surface element is given by the basis vector associated with the coordinate that you held constant when describing the surface — up to sign. This sign corresponds to a choice about the orientation of the surface. The standard conventions, adopted above, are to orient the surface element outward for a closed surface such as a sphere, and upward otherwise, unless otherwise stated. In the absence of such a convention, each of the surface elements above would have to be preceded by a factor of \(\pm1\text{.}\) Compare the two drawings in Figure 10.3.

When using formula (10.5.1), the choice of orientation is determined by the order in which \(d\rr_1\) and \(d\rr_2\) are multiplied. By convention, one usually computes \(d\AA=d\rr_1\times d\rr_2\text{,}\) so the orientation is determined by the choice of which vector is \(d\rr_1\text{,}\) and which is \(d\rr_2\text{.}\) It is up to you to choose this labeling to correctly match the desired orientation. Which orientation is determined by the choices shown in Figures 10.2 and Figure 11.2?

Formula (10.5.1) will work for all kinds of complicated surfaces, so we wanted you to get practice in learning how to use it. However, when a surface can be described as a “coordinate equals constant” surface in an orthogonal coordinate system, then the cross product is trivial. You can think of the scalar area element as an infinitesimal “rectangle” whose area is just the product of the infinitesimal lengths of the two sides.