### 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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