# GL(s,R)

## 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

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

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.

I used projects last semester in an SBG linear algebra course, and I simply told the students

1. They are required to do a (satisfactorily-done) project in order to get an A, and
2. If they do not get an A but still do a project, that will be a very positive consideration when I compute final grades.

This worked very well for my class—the students (mostly) did excellent projects.
Bret

Comment by Bret Benesh — January 17, 2011 @ 4:00 pm

2. Thanks for the comments Bret. Linear algebra is the highest course I’ve heard SBG used so far (although looking at your blog, it seems abstract algebra will get its turn now). I’m enjoying reading your blog posts on the linear algebra projects. As to your specific suggestions, it sounds very similar to what I’ve done in the past with projects (with success). What I’m considering is breaking up the presentation itself into standards with the hope that it will get students to concentrate a bit more on some of the things I find important in a presentation, but which they usually ignore.

Comment by Adam Glesser — January 18, 2011 @ 3:54 pm

• I doubt that I will be trying it in abstract algebra for a while—I didn’t actually try it when I taught it last spring. For once, it seems like it might be too hard to come up with enough assessments (although I haven’t thought about it enough). By I think that it works really work for Linear Algebra.

I hadn’t thought about creating standards for presentations. That seems interesting; it is also great that you are giving them specific instruction on how to give a good presentation.
Bret

Comment by Bret Benesh — January 19, 2011 @ 8:22 am

• I see what you mean about assessments for an abstract algebra class. Here is a quick thought. In my system I have activities broken up into two categories: exercises and problems. The exercises are those questions that are solved immediately by some primary algorithm, e.g., find the derivative of $x^2$. Problems are questions for which the solution is more complicated, often requiring some creativity in choosing the algorithm. Most good word problems really are problems. Anyway, I give students a max score of 3 on exercises and a max score of 4 on problems. In an introductory abstract algebra class, one could give frequent level 3 assessments, mostly consisting of questions asking for definitions and perhaps results that follow in one step from the definitions. Level 4 assessments would be infrequent (a handful in addition to the midterm and final). Since, in my experience, it is usually pretty good if your students walk away from their first abstract algebra class knowing the definitions cold, I might set that as the bar for basic competency (a B- grade) and anything above that would merit an honor grade.

Hmmm…I don’t know. I need to think about it more.

Comment by Adam Glesser — January 21, 2011 @ 1:11 pm

I do like the idea of breaking activities down into “exercises” and “problems.” I did something similar for my linear algebra class last semester. The main difference is that you use points and I don’t (I only have two grades: “Acceptable” and “Incomplete”), but that is a matter of taste.

And I am a big believer in knowing definitions in proof-based courses.

Now that you encouraged me a little bit (I am not sure if that was your intent), I am going to think about figuring out a way to do something SBG-ish in my next abstract algebra course.

Thanks.
Bret

Comment by bretbenesh — January 24, 2011 @ 10:54 am

• For calculus level material and below, for some reason I feel compelled to assign numbers to everything. At the upper levels, though, I see why using acceptable vs. incomplete would be more appropriate. I am very glad—mostly for selfish reasons—that you will try to incorporate SBG into your next abstract algebra course. The typical department issues have kept me from teaching group theory or abstract algebra and so I haven’t had a chance to implement SBG at that level. I am dying to find out how it works.

Comment by Adam Glesser — January 26, 2011 @ 8:04 am

3. You don’t need to be an applied mathematician to have strong feelings on whether it is Stokes’ or Stokes’s: it depends on your manual of style, but Stokes’s is preferred, as in Frobenius’s theorem, and Jesus’s, the bus’s, etc.

As for multivariable calculus, can’t help you there as I’ve never taught it. I’m sticking with group theory next year, I hope…

Comment by DavCrav — January 24, 2011 @ 7:29 am

4. Could you republish your calendar and such? They aren’t available, and I would love to see this alternate approach to multivariable calc (specifically holding off on vectors!)

Comment by Zach — July 21, 2011 @ 11:27 pm

• Hi Zach,

I’m traveling through Denmark at the moment, so I won’t be able to put it up until I get back in a few days. Frankly, I don’t know why it disappeared in the first place! Perhaps I’ll find a different place to upload it, one that doesn’t randomly delete things.

As far as the alternate approach goes, my real preference would be to have a linear algebra class *before* multivariable, but that just isn’t so common it seems.