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This fall I'm teaching the "second half" of the standard entry-level undergraduate multivariable calculus course: the focus is on double and triple integrals, path integrals, Green's theorem, Stokes' theorem, and the divergence theorem. This is apparently not usually enough to fill the entire semester, and I'm told that other instructors often use the last two weeks or so to briefly introduce another topic, often a brief discussion of functions of a complex variable.

I seem to have free reign to do what I want with those last two weeks (within reason), so I thought I would solicit suggestions from MO. I would like to pick a topic that is at least somewhat related to the rest of the course; aside from the complex analysis idea I am also considering a brief introduction to Maxwell's equations. It should of course be suitable for undergraduate math, physics, engineering, and economics majors (mostly in their second year I think). Any ideas would be appreciated!

The question doesn't really have one correct answer, so I made it a community wiki. I realize that this is not really a research question, but I believe that it is likely to be relevant to many MO users and that the MO community is uniquely suited to answering it. I hope my record shows that I do not post questions on here lightly.

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    $\begingroup$ Nitpick: you give a horse free rein $\endgroup$
    – Yemon Choi
    Sep 2, 2012 at 23:37
  • $\begingroup$ relevant: mathoverflow.net/questions/83965/… $\endgroup$
    – Steven Gubkin
    Sep 2, 2012 at 23:40
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    $\begingroup$ might I suggest that if you are finishing those topics two weeks early, you might just be going too fast? $\endgroup$
    – Steven Gubkin
    Sep 2, 2012 at 23:48
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    $\begingroup$ How do you want to serve your students? By giving them a peek at other things they may or may not find useful? If you do an excellent job and they master everything you teach them, then you might pick something based on class interest. Otherwise, a review with applications may result in them remembering the material after the final exam. Gerhard "That Would Be Really Something" Paseman, 2012.09.02 $\endgroup$
    – Gerhard Paseman
    Sep 3, 2012 at 0:19
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    $\begingroup$ @Yemon: I'm glad someone pointed this out! (That said, I imagine that the OP and his students aren't horses, so I suppose that he has more reign than rein here.) $\endgroup$
    – Ramsey
    Sep 3, 2012 at 0:25

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Time spent on the heat equation would be well-spent: $$\frac{\partial u}{\partial t} = \alpha \nabla^2 u$$
           Wikipedia image on heat equation
           (Wikipedia image due to Oleg Alexandrov)

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    $\begingroup$ This is a cool video, but it has a certain dermatological aspect. $\endgroup$
    – Igor Rivin
    Sep 3, 2012 at 1:49
  • $\begingroup$ Others in the past have done a unit on Fourier series. My concern with that was that it doesn't really tie into the rest of the course, but if I could manage to solve the heat equation... $\endgroup$
    – Paul Siegel
    Sep 6, 2012 at 2:32
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Probability and Statistics have a wide array of applications for an entry-level undergraduate multivariable calculus course. All the integration concepts previously taught could be illustrated by, say, deriving the multivariate normal distribution, its properties, or properties of its transforms... There is a nice book with plenty of illustrations for just that purpose:

http://www.amazon.com/Advanced-Calculus-Applications-Statistics-Khuri/dp/0471534595

Furthermore, most of the majors mentioned (math, physics, engineering, and economics) are exposed to probability or statistics at some time or another.

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    $\begingroup$ I hadn't thought of this, and it's a really nice idea. I was concerned because all of my other ideas kind of left out the economics majors - this one definitely doesn't. $\endgroup$
    – Paul Siegel
    Sep 6, 2012 at 2:31

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