Multivariable Exam 3 Review: Thomas 16.7

By Tyler Clarke in Calculus on 2025-4-16

Hello again! We're on the penultimate section in the chapter and in the entire book. In this section, we're covering Stokes' Theorem. Stokes' theorem is an application of Green's theorem in 3 dimensions. This is much harder to visualize, but essentially, with Stokes' theorem, we can find the circulation of a vector field on a curved surface. The critical idea is that, for a smooth open surface `S` with a smooth boundary curve `C`, the circulation about `C` is the surface integral of the curl over `S`: `int_C F cdot dr = int int_k (grad times F) cdot k dA`. This has an interesting consequence: the circulation about any two open surfaces with the same boundary curve is the same.

Let's do an example. Take the hemisphere with radius 2 above the xy plane: what is the circulation of the vector field `F(x, y, z) = [-y, x, z]` about this surface? This is a somewhat difficult problem in two dimensions, but Stokes' theorem means we can reduce it to a single dimension integral. The parametrization of our boundary curve is the radius 2 circle on the xy plane: `r(u) = [2cos(u), 2sin(u), 0]` for `0 <= u <= 2pi`. Taking the derivative gives us `dr = [-2sin(u), 2cos(u), 0] du`, and substituting our parametrization gives us `F(u) = [-2sin(u), 2cos(u), 0]`. We know the circulation is going to be `int_C F cdot dr`; `F cdot dr` is just `4sin^2(u) + 4cos^2(u)`. Substituting in gives us `int_0^{2pi} 4(sin^2(u) + cos^2(u)) du` - which simplifies to `int_0^{2pi} 4 du = 8pi`. Easy!

This post was much shorter than I was hoping, largely due to time constraints. I recommend practicing all the complexities of Stokes' theorem in depth- for instance, Stokes' can be applied to surfaces with holes, unlike Green's. See you in 16.8!