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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