GL(s,R)

September 12, 2012

Projecting onto Projections

Filed under: High Effort/Low Payoff Ideas — Adam Glesser @ 1:53 am
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The first time I saw the expression \int_C \mathbf{F} \cdot \mathbf{n}\ d\mathbf{r}, I thought, “Why should that dot product be in there. By the time I saw \iint_S \mathbf{F} \cdot\ d\mathbf{S}, I resigned myself to the fact that there was always a dot product in these seemingly random integrals. At some point, I decided that the dot products are in there to turn vectors (or vector fields) into scalar functions—which is something we know how to integrate. More recently, I’ve decided that the purpose of these dot products is to capture the projection of one vector on the other.

For example, if I apply a force \mathbf{F} to an object, then the work done by that force in moving the object a certain distance in a given direction (denote this shift by \mathbf{v}) is \mathbf{F} \cdot \mathbf{v}. If the force is not constant over some curve parametrized by \mathbf{r}(t) (a \leq t \leq b), then we compute the work by evaluating the integral \int_a^b \mathbf{F} \cdot \mathbf{r}'(t)\ dt since, at any given point, our \mathbf{v} from above is just the tangent vector to the curve at that point, i.e., \mathbf{r}'(t).

If you understand multivariable calculus, then you are probably laughing at me. “Duh. Why did it take you so long to figure that out?”

Here is my answer: We (or maybe just I) improperly motivate the dot product. This semester, I’m using Stewart for Multivariable Calculus*. He introduces vectors in a way that seems fairly standard for math texts.

Definition: The dot product of \langle x_1, \ldots, x_n \rangle and \langle y_1, \ldots, y_n \rangle is x_1y_1 + \cdots + x_ny_n.

The great thing about this definition is that it is bloody easy to compute and understand.

Theorem: If \mathbf{a} and \mathbf{b} be vectors with angle \theta between them, then \mathbf{a} \cdot \mathbf{b} = \mid\mid \mathbf{a} \mid\mid\ \mid \mid\mathbf{b}\mid\mid \cos(\theta).

The beauty here is that you can use the dot product to help compute angles and it is immediately obvious that the dot product of orthogonal vectors is 0.

*This wasn’t my choice, but rather the choice of my department. Oh, did I mention I got a new job? Indeed I finally gave up on east coast living and moved back to California. I am now in the mathematics department at California State University Fullerton.

I’ve heard that in physics textbooks, they switch the order of the above, i.e., they define the dot product via the cosine formula and then prove the above definition as a theorem. As a mathematician, I always went with the first definition. Now, I am not so sure. What follows is the introduction to the dot product I plan to give to my students (until I come up with something better, anyway*).

*In the comments, please do set me straight about the real purpose of the dot product or how you think it best to introduce it in this context.

Let’s start with two vectors, joined by their tales.

I am interested in how far \mathbf{b} extends along \mathbf{a}, so I drop a line perpendicular to \mathbf{a} from the end of \mathbf{b}.

At this point, I’m already confused by what would happen if I had tried to see how far \mathbf{a} goes along \mathbf{b}, but I decide that I could simply extend \mathbf{b} and at least draw the following picture:

Awesome, I have a couple of right triangles. And, heck, since they are right triangles that share the angle (let’s call it \theta) between \mathbf{a} and \mathbf{b}, they are similar triangles. Let’s give some names to the important sides.

The comment about similar triangles implies that \dfrac{h}{||\mathbf{b}||} = \dfrac{k}{||\mathbf{a}||}. Ugh, let’s clear denominators to get h||\mathbf{a}|| = k||\mathbf{b}||. On the other hand, \cos(\theta) = \dfrac{h}{||\mathbf{b}||}, and so if we multiply by ||\mathbf{a}||\ ||\mathbf{b}||, we get

||\mathbf{a}||\ ||\mathbf{b}||\cos(\theta) = h||\mathbf{a}||

The moral is that this important quantity—h||\mathbf{a}|| = k||\mathbf{b}||—coming from projecting the vectors onto each other, has a very simple reformulation as ||\mathbf{a}||\ ||\mathbf{b}||\cos(\theta) which only relies on knowing the original vectors and the angle between them. Since this projection property is so important to us physically, we give a short name to this expression: \mathbf{a} \cdot \mathbf{b}, and call it the dot product of \mathbf{a} and \mathbf{b}.

If \mathbf{b} is orthogonal to \mathbf{a}, then the projection should be 0, which of course it is since the cosine of 90^\circ is 0.

At this point one can go about proving that the dot product is obtained directly from the components, i.e., without knowing the angle between them. Of course,  there is still the issue of when \theta is obtuse, and it will probably be helpful to cover that case as well. Geometrically it will look a bit different, but the algebra and trig will be almost the same*.

*You do get to use the fact that the cosine of an angle equals the cosine of the supplementary angle!

There is nothing really new here, but I think the ordering is important. Their first impression of the dot product should convey the purpose of the dot product, not just the easiest algorithm for computing it. As it stands, the projection of a vector onto another vector gets a a somewhat token reference at the end of the dot product chapter. As ubiquitous as the idea is throughout the end of the class, it deserves its time in the sun.

January 17, 2011

A multivariable calculus list

In addition to my calculus course this semester, I also get to teach a multivariable calculus course with only six students. I’ll start with the standard list for those interested in that sort of thing.

Spring 2011 Multivariable Calculus Standards List

Let me admit something, here, in between two documents—less likely to read in here—about teaching this course, now for the third time: I’m a fraud. That’s right, I’m a fake, a charlatan, an impostor. I’ve created a counterfeit course and hustle the students with a dash of hocus-pocus and a sprinkle of hoodwinking. It is only through mathematical guile that my misrepresentations, chicanery and flim-flam go unnoticed. In short, and in the passing Christmas spirit, I am a humbug. This is a physics course. It should be taught be someone proficient in physics, someone with honed intuition about the geometry of abstract mathematical notions like div, grad, curl and all that, someone who sees everything as an application of Stokes’ theorem and has strong feelings about whether it should be written Stokes’ theorem or Stokes’s theorem. About the only thing I bring to the table is that I can teach students to remember that:

\mathrm{curl}(\mathbf{F}) = \nabla \times \mathbf{F}

and

\mathrm{div}(\mathbf{F}) = \nabla \cdot \mathbf{F}

because curl and cross both start with c, while div and dot both start with d.

Here is the calendar for the course. After it, I’ll explain a little bit of what I’m trying.

Spring 2011 Multivariable Calculus Calendar

There are several big differences here from how I’ve taught this course in the past. First, I am going to try with all my might to get to Stokes’ theorem before the last week. Part of the way I plan to do this is, similar to my calculus class, to cut out most of the stuff on limits and continuity that I usually get bogged down on in the first couple of weeks—am I the only person who finds interesting the pathological examples that make Clairaut’s theorem necessary? I get to teach an extra hour a week to a subset of the class and that stuff will fit perfectly in there. For the science majors, I’m more interested in helping them figure out how to use this stuff and how to develop intuition. Second, I’m skipping Green’s theorem until the end. Yes, it changes the story I normally tell, one that progresses so nicely up the dimension chart, but the trade-off is that I get more time to show them Stokes’ theorem and more time to focus on the physical interpretation.

Speaking of interpretation, you will notice in the calendar  eleven or so ‘Group Activities’. These are stolen from an excellent guide produced by Dray and Manogue at Oregon State as part of their Bridge Project. To work within their framework, I’ve made another structural change that I’d never considered given how I think about the subject. Immediately after finishing triple integration (which, essentially, finishes the first half of the course), we start with vectors (I never start with vectors as most calculus books do) and then I want to get to line integrals and surface integrals as fast as possible. Normally, I mess around with div and curl before getting to integration of vector fields. Instead, I’m going to push out the Divergence theorem—the theorem I always cover in the last 45 minutes of the course—and use this to motivate the definition of div. Then I’ll push out Stokes’ theorem and use this to help motivate the definition of curl. This ought to give me two solid weeks to explore the physical meaning of these theorems as well as to use them to prove some of the standard cool corollaries (like Green’s theorem).

This class will also be the first of my SBG courses to incorporate a final project. If anyone has good suggestions based on experience about how best to incorporate projects into the SBGrading scheme, I would live to hear them. My current system is quite simplistic. The standards for the course are given a 90% weighting for the overall grade—did I mention that midterms and finals now are simply extended assessments whose grades are treated like an arbitrary quiz, just with a lot more standards tested?—and 10% weighting for the project.

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