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

I used this activity on day 4 of the Paradigm, just after deriving/motivating the div operator (first, divergance as “outward flux per unit volume”, and then getting to the Cartesian formula).

I stepped the class through the first MAPLE examples myself, modeling how I wanted the activity to go: First, look at a formula of a vector field. PREDICT what it will look like. Then, check with Maple. Next, PREDICT what the divergence will look like by thinking visually about “flux/volume” at several *different* points. Then, again, checking with Maple. I did this with the class as a whole (1-2 students per terminal) through the first three examples (uniform, then “r” field, then the first field with a Gaussian in it), at which point I left them to work through the rest of the sheet on their own. Activity took about 20 minutes, and was very productive - lots of good questions, many groups were successfully predicting both fields and divergences. They got stumped on the “comic relief” one (it's pretty nasty), so I had to reassure them that sometimes it's just too hard to “visualize” the divergence in advance.

I did find that when talking with groups, they were so focused on the “conceptual” way of solving that they weren't ALSO checking by just taking the divergence mathematically, so it was good to remind them to just look at “partial Ex/dx + partial Ey/dy” and see that it ALSO jived with their outcome.

The last example (Coulomb field) is subtle and hard. We discussed this collectively on the board, but for this class (which is not yet familiar with the delta function) it seemed a little challenging. Working out why the divergence of the Coulomb field is zero (almost) everywhere is important, and I got lots of feedback during this activity as the students argued about what it should be, and what it meant that it works out to be zero.

Note: I preceded this activity with two quick clicker questions about divergence, which helped get them started. ( I showed 2 simple fields and asked if they zero or non-zero divergence. (One was effectively E=(cy, 0,0) the other was effectively E=(cx, 0, 0), but formulas were not shown, just sketches of the vector field in 2-D) They argued and struggled, and again we had a productive discussion, followed by this MAPLE activity.

-S

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