Tyler Clarke
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28 lines
1.9 KiB
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[!]
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[=post-]
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<p>
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Hello again! We're on the penultimate section in the chapter and in the entire book. In this section, we're covering <i>Stokes' Theorem</i>. Stokes' theorem
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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
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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
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`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:
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the circulation about any two open surfaces with the same boundary curve is the same.
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</p>
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<p>
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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]`
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about this surface? This is a somewhat difficult problem in two dimensions, but Stokes' theorem means we can reduce it to a single dimension
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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
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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
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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
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simplifies to `int_0^{2pi} 4 du = 8pi`. Easy!
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</p>
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<p>
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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-
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for instance, Stokes' can be applied to surfaces with holes, unlike Green's. See you in 16.8!
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</p>
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[/]
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[=author "Tyler Clarke"]
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[=date "2025-4-16"]
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[=subject "Calculus"]
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[=title "Multivariable Exam 3 Review: Thomas 16.7"]
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[=unpub]
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[#post.html] |