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}


\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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