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Swapping surfaces in Stokes' Theorem

Example 6 in Section 16.5 of Thomas Calculus computes the circulation of a vector field over the curve shown below two ways.
  1. By directly evaluating the flow integral along the blue curve
  2. By parameterizing the red surface and evaluating the curl integral over the surface.
It is probably easier to use the gradient method to set up and evaluate the surface integral. Doing so results in the following double integral, where R is the region inside the unit circle:

Evaluating this integral using polar coordinates is much simpler than the integral at the bottom of page 1019. However, there is an even easier way to find this circulation. Stokes' Theorem implies that the curl integral over any surface whose boundary is the blue curve must equal the value of the flow integral. So we can change the surface to one that makes this double integral easier to evaluate. Drag the point P up or down the z-Axis to change the surface and it's equation in a way that might lead to an easier curl integral.
When , the surface becomes , and as a result, we get a gradient vector of . Since the curl is , that makes the curl integral

where R is the unit circle. Hence the curl over the surface and also the circulation is .