[!]
[=post-]
<p>
    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
    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.
</p>
<p>
    Let's do an example. Take the hemisphere with radius 2 above the plane `x - z = 0`, centered on origin: what is the circulation of the vector field
    `F(x, y, z) = \[-y, x, z]` about this surface? In 2 dimensions this is a somewhat complicated integral to perform, but Stokes' theorem allows us to reduce
    it to a single dimension, if we can parameterize the boundary curve. The plane parameterization in 2 variables is obviously `p(u, v) = \[u, v, u]`,
    and the sphere is `s(u, v) = \[2cos(u)sin(v), 2sin(u)sin(v), 2cos(v)]`. This gives us three equations for the intersection: `u = 2cos(u)sin(v)`, `v = 2sin(u)sin(v)`,
    and `u = 2cos(v)`.
</p>
[/]
[=author "Tyler Clarke"]
[=date "2025-4-12"]
[=subject "Calculus"]
[=title "Multivariable Exam 3 Review: Thomas 16.7"]
[=unpub]
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