# Math Insight

### Applet: Choosing a surface in Stokes' theorem

Sometimes, you might be given an integral of a vector field over some given curve, and then be asked to compute it using Stokes' Theorem.

You can only use Stokes' Theorem do to this if you have a surface integral whose boundary is your given curve. So the first step of any such problem is to choose a surface $\dls$ whose boundary is your curve.

This is actually much tougher than it sounds, because there's no system we can give you to choose such a surface. In fact, there are always infinitely many choices, although usually one choice stands out as being the best. To illustrate this point, this applet shows you 40 different surfaces, all of which have the same boundary, namely the unit circle in the $xy$-plane. (Move the mouse over the applet to start the animation.) You could use any of these with Stokes Theorem.

Remember, though, there's often a “best” choice. In this case, the best choice is probably the unit disk in the $xy$-plane. We definitely wouldn't recommend any of these surfaces, which were created using trig functions. The surfaces shown here aren't even smooth, although — as with a cone — it turns out not to matter.

Applet file: stokes_theorem_choose_surface.m

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#### General information about LiveGraphics3D applets

This applet was created using LiveGraphics3D. To manipulate it, you can

• click and drag to rotate in any direction,
• shift + click and drag up/down to zoom out/in,
• shift + click and drag left/right to rotate around an axis coming out of the screen, and
• press Home to reset to the original view.

Most of the LiveGraphics3D applets have parameters that you can change. In general, you can click and drag points or balls around to change the parameters.

Since these applets use Java, you must have Java installed and properly configured in your browser for the them to display. You can get Java here. If you have trouble getting them to display in your browser, you can visit the LiveGraphics3D troubleshooting page for information on how to get the Java applets to work.

On occasion, even when Java is configured correctly, the Java applets don't load the first time and one instead sees an error message. You can wait a few moments to see if the applet will eventually load itself anyway. You can also reload the page to see if the applet will properly load. Sometimes, one needs to do this multiple times.