From a20581b57eb5a5b6c6f2809a216d5b7dcf133af0 Mon Sep 17 00:00:00 2001
From: Tyler Clarke <plupy44@gmail.com>
Date: Wed, 16 Apr 2025 09:03:02 -0400
Subject: [PATCH] start on 16.7: will finish later

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 site/default.html                         | 12 +++++++-----
 site/posts/multivariable-thomas-16.7.html | 23 +++++++++++++++++++++++
 2 files changed, 30 insertions(+), 5 deletions(-)
 create mode 100644 site/posts/multivariable-thomas-16.7.html

diff --git a/site/default.html b/site/default.html
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@@ -36,11 +36,13 @@
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                     <div>
                         [f posts post]
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-                                    <b>[^post.title]</b><br>
-                                    <i>by [^post.author] on [^post.date]</i>
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+                            [i post.unpub not]
+                                [i post.subject subj equals]
+                                    <a [i post.title title equals] class="current" [e] href="[^baseurl]/[^post.filename]" [/] >
+                                        <b>[^post.title]</b><br>
+                                        <i>by [^post.author] on [^post.date]</i>
+                                    </a>
+                                [/]
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diff --git a/site/posts/multivariable-thomas-16.7.html b/site/posts/multivariable-thomas-16.7.html
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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]
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