From 8322bcfd4f50955a50c142c5d2ce29255aa60c8d Mon Sep 17 00:00:00 2001 From: Tyler Clarke Date: Wed, 16 Apr 2025 09:15:17 -0400 Subject: [PATCH] 16.7 --- site/posts/multivariable-thomas-16.7.html | 18 +++++++++++------- 1 file changed, 11 insertions(+), 7 deletions(-) diff --git a/site/posts/multivariable-thomas-16.7.html b/site/posts/multivariable-thomas-16.7.html index 25c9485..4b6b101 100644 --- a/site/posts/multivariable-thomas-16.7.html +++ b/site/posts/multivariable-thomas-16.7.html @@ -8,16 +8,20 @@ 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 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)`. + 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!

[/] [=author "Tyler Clarke"] -[=date "2025-4-12"] +[=date "2025-4-16"] [=subject "Calculus"] [=title "Multivariable Exam 3 Review: Thomas 16.7"] -[=unpub] [#post.html] \ No newline at end of file