Choosing a Surface in Stokes Theorem

In this class 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 M 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, I've included an animation below. This shows you 40 different surfaces, all of which have the same boundary, namely the unit circle in the xy-plane. You could use any of these with Stokes Theorem.

(Remember, though, I said there's usually a "best" choice. In this case, I think the best choice would be the unit disk in the xy-plane. I definitely wouldn't use 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. You could discuss that with your instructor.)

Move the mouse over the picture to start.
  • Double click the picture to start/stop the animation.
  • Click and drag the picture to rotate it
  • Press the shift key, click and drag up/down to zoom in/out.

  • Created using Live Graphics 3D.

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