From 97a06432a70fba3bbdd4bbe8f9975dfd4ba3778d Mon Sep 17 00:00:00 2001 From: Tyler Clarke Date: Thu, 3 Apr 2025 22:27:55 -0400 Subject: [PATCH] 16.2 --- site/posts/multivariable-thomas-16.2.html | 39 +++++++++++++++++++++++ 1 file changed, 39 insertions(+) create mode 100644 site/posts/multivariable-thomas-16.2.html diff --git a/site/posts/multivariable-thomas-16.2.html b/site/posts/multivariable-thomas-16.2.html new file mode 100644 index 0000000..1e1a31e --- /dev/null +++ b/site/posts/multivariable-thomas-16.2.html @@ -0,0 +1,39 @@ +[!] +[=post-] +

+ Welcome back! We continue chapter 16 with a discussion of vector fields. Essentially, a vector field is a function with the same input dimensions as output dimensions. In `R^2` and `R^3`, + we visualize this as a bunch of vectors permeating all of space. If you've taken physics, you've probably encountered vector fields - the electric and magnetic fields are in fact + vector fields! A way to think about vector fields is as the velocity of a given infinitesimal region of a fluid: it's at a position, and the velocity vector is pointing in some direction + with some magnitude. This is actually a very good way to represent a flowing fluid, for our purposes. +

+

+ An example of a vector field we've used extensively thus far is the gradient function of a surface! At every point along the surface, + it outputs a vector representing the slope at that point. +

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+ One of the most useful operations we can perform on a vector field is a line integral (remember 16.1?). The integral of a vector field over a line is a fairly abstract concept, + but it has real-world uses: imagine we have a function for the force at any given point, and we want to know the total force over a line. Vector fields! The form for a vector + field integral is simply the dot product of the vector field and the tangent vector to the point on our line: given a line `r(t)` in a vector field `f(x, y) = [u, v]`, we have + `int f(x, y) * dr`. This is one of those very elegant things that has interesting consequences. +

+

+ Let's do an example. Straight from the textbook, we have `f(x, y, z) = \[z, xy, -y^2]` and `r(t) = \[t^2, t, sqrt(t)]`, where `a <= t <= b`. Getting `f(x, y, z)` in terms of `t` is simple enough: + just substitute the components of `r`: `f(t) = \[sqrt(t), t^3, -t^2]`. `dr` is going to be `frac {dr} {dt} * dt`, so let's find `frac {dr} {dt}`: + `frac d {dt} \[t^2, t, sqrt(t)] = \[2t, 1, frac 1 {2sqrt(t)}]`. + So our integral is `int_a^b \[sqrt(t), t^3, -t^2] * \[2t, 1, frac 1 {2sqrt(t)}] dt`. Evaluating the dot product gives us `int_a^b 2t^{3/2} + t^3 - frac {t^{3/2}} {2} dt`. + I'll leave further simplification as an exercise to the reader. +

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+ This section is cut rather short, as the introduced idea is simple and it's too late at night to try making it more complex. I might expand it + with more examples and applications later. Section 16.3 will go into quite a bit more detail, as we'll be dealing with the problem of conservative vs + nonconservative paths. For now, da svidanya and bonne nuit! +

+

+ P.S: If you notice broken or missing math, it's probably due to my template engine freaking out about the square brackets. Will fix in a bit! +

+[/] +[=title "Multivariable Exam 3 Review: Thomas 16.2"] +[=author "Tyler Clarke"] +[=date "2025-4-3"] +[=subject "Calculus"] +[#post.html]