start on 16.7: will finish later

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+[!]
+[=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]
+[#post.html]