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Comment from S. Pollock (CU Boulder, visiting OSU and teaching Paradigm “Vector Fields), Nov 2009:

I did this activity around lecture 7, and we spent about 10 minutes on it, though I timed my break to happen as it finished, and most of the class just sat around for the 5 minute break continuing to work through it. This one worked very well, just like the divergence visualization - I didn't need to “walk them through” more than the first one (since they know the drill by now), and I found my students were trying to first visualize the field before letting MAPLE show it to them, AND then try to predict the curl, again before looking at the answer. They were doing pretty well on this (I had derived the curl in Cartesian coordinates on the board before doing this activity, and had done one simple example, as well as giving them the “pinwheel” visualization idea too, all just before this worksheet). Some groups were working out the curl in a couple of cases to see if it agreed.

One thing I did with almost all groups was to have them walk me through why the x and y components of the “v4” field (which looks like $[0, e^{-x^2},0])$ vanished. In one case, it's because all legs in the “circulation around the axis” are zero, and in the other, it's because of cancellation of 2 legs, I wanted them to be very clear about this, so they weren't just focusing on the one “obvious” non-zero component.

Afterwards, I asked if they had any questions or puzzles, and there was considerable discussion about the last one (which is basically $(1/r) \hat\phi$). MAPLE says the curl is (0,0,0), which is correct (but surprising for students!) everywhere except the origin. So I used this to lead to a chalkboard/group discussion about this case, working through both origin and non-origin separately. It's subtle (with the Delta function curl!) and the vanishing of a curl that “looks curly” nearly everywhere is worth talking/working through.

I finished this with a “Jeopardy question”: “The ANSWER is, “a nontrivial field that looks like the one on the screen which has zero curl everywhere but the origin”. And the QUESTION is….? This was fun. One student said “What is a phihat field that drops off like 1/r” (or something like that) so I said, good, that's a MATH question, but can you come up with a PHYSICS question? Someone else said “What is the magnetic field of a long wire”. Ahh, so this was the segue to the next topic, namely why are we computing A?…