Tyler Clarke
3faf1ab200
All checks were successful
Build / Build-Docker-Image (push) Successful in 3m8s
40 lines
3.3 KiB
HTML
40 lines
3.3 KiB
HTML
[!]
|
|
[=post-]
|
|
<p>
|
|
Hello, everybody! This is the very last section in the Thomas textbook, and it's (fortunately) not a very hard one. This section is on something called <i>divergence theorem</i>.
|
|
</p>
|
|
<p>
|
|
The critical idea behind divergence theorem can be summed up with this formula, where `S` is a closed smooth surface, `C` is the volume contained by it, and `F` is a differentiable vector field:
|
|
`int int_S F cdot n dA = int int_C int grad cdot F dV`. The flux of a vector field through a closed smooth surface is equal to the volume integral of the divergence of the vector field in
|
|
the volume contained by the surface.
|
|
</p>
|
|
<p>
|
|
This has obvious applications. In many cases, it's <i>much</i> easier to compute `grad cdot F` than `F cdot n`: finding a normal vector can be extremely difficult for complex surfaces.
|
|
It also means you don't have to worry about orientation, and in some cases you can greatly simplify the bounds of integration. Let's do an example!
|
|
</p>
|
|
<p>
|
|
Straight from the textbook, we have nice spherical one: find the flux of the vector field `F = \[x, y, z]` over the sphere `x^2 + y^2 + z^2 = a^2`. Finding this the
|
|
conventional way is pretty easy, but I'll leave that as an exercise to the reader. This is very obviously a sphere with radius `a` centered at origin, so we set up
|
|
our substitution with `0 <= theta <= 2 pi`, `0 <= phi <= pi`, `0 <= rho <= a`, and `dV = rho^2 sin(phi)`. It's easy enough to find that `grad cdot F = 3`. Substituting
|
|
everything in gives us the integral `int_0^{2pi} int_0^{pi} int_0^a 3rho^2 sin(phi) d rho d phi d theta`. The inner integral is trivial - it works out to
|
|
`int_0^a 3 rho^2 sin(phi) d rho = a^3 sin(phi)`. This gives us an integral in `phi` of `int_0^{pi} a^3 sin(phi) d phi = |_0^{pi} -a^3 cos(phi) = 2a^3`.
|
|
Finally, because there is no `theta` term here, we just multiply by `2pi` to get `4pi a^3`.
|
|
</p>
|
|
<p>
|
|
This technique can also be used to make finding the flux in a nonsmooth surface much less terrible, if it can be reduced to smooth bounds. For instance:
|
|
to find the flux of `F(x, y, z) = \[cos^3(z), cos^3(z), z - e^{cos(x)sin(y)}]` through a cube described by the points `(0, 0, 0)` and `(1, 4, 9)`, we would normally have to calculate
|
|
the flux through each individual face, which would be absolutely terrible; but with divergence theorem, we can turn this into a much more manageable
|
|
triple integral over the volume `int_0^1 int_0^4 int_0^9 grad cdot F dz dy dx`. We're in the incredibly lucky position of having `grad cdot F = 1` - meaning
|
|
the flux is just the volume, `1 * 4 * 9 = 36`. If you don't believe that that was easier, I invite you to calculate `int e^{cos(x)sin(y)} dx`. It's not pleasant.
|
|
</p>
|
|
<p>
|
|
That's the end of the whole textbook - my copy, at least. See you in the summer with differential equations! Also keep a look-out in the physics section -
|
|
after finals, I'll be building railguns with the folks at <a href="https://scrappysgarage.org/">Scrappy's Garage</a>, and I'll definitely be posting the results
|
|
of our experimentation here. For the very last time (well, not really), <i>au revoir</i>, and good luck!
|
|
</p>
|
|
[/]
|
|
[=title "Multivariable Exam 3 Review: Thomas 16.8"]
|
|
[=author "Tyler Clarke"]
|
|
[=date "2025-4-23"]
|
|
[=subject "Calculus"]
|
|
[#post.html] |