One flux example two ways: using Stokes' and the Divergence Theorem
In this exercise, we compute a vector surface integral using two different methods: Stokes' theorem and the Divergence theorem. First, we use Stokes' theorem to relate our surface integral to a circulation integral around the boundary of the paraboloid, ultimately getting a result of 64pi.
Next, we introduce a "lid" to enclose the region of interest, allowing us to apply the Divergence theorem. We give the lid an outward orientation and consider the combined surface S (paraboloid) and L (lid) as enclosing a solid region E. The Divergence theorem relates the flux of the curl of F across the surfaces S and L to a triple integral of the divergence of the curl of F over the region E. However, since the divergence of the curl of F is 0, the right-hand side of the Divergence theorem is zero. We end up computing the surface integral across L only, which matches our result of 64pi.
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