[BUG] change of variables

[Tevian Dray]

As always, I saved change of variables for last -- not only after
discussing surface integrals, but after discussing both the Divergence
Theorem and Stokes' Theorem.  But this year, for the first time, I made
the students figure it out on their own.

I started the class by posing a fairly simple c.o.v. problem, but gave
no clues about how to solve it.  In the first instance, I asked for the
area bounded by 4 pairwise (and non-orthogonal) parallel lines, but I
made it clear at the beginning that I was also going to ask them to
integrate a function over this region.  I let the (very small) class
work on it for 10 minutes, then stopped them and asked what strategies
they were trying.

Most students found the "easy" solution to the area problem: take the
(magnitude of the) cross product of two adjacent sides.  At least one
found the "hard" solution: Use the area corollary to Green's Theorem
(which I do not teach as a separate topic, but which I had in fact
discussed privately with this particular student).  Several students set
up single integrals for the area; perhaps surprisingly, none used a
(traditional) double integral (in x and y).

But at least half the class (also) tried to chop up the region parallel
to the boundary, then use (the magnitude of) dr1 x dr2 to find the area
element -- which (eventually) leads to the standard formalism for (2-d)
change of variables.  Of course, this led to problems, since nobody
thought to actually change variables -- it's important to know how to
relate one's chosen parameters to one's choice of chopping.  But it set
the stage for me to guide the class through doing just that.

This activity is much like our last group activity (in the Instructor's
Guide), but done without any prior motivation or discussion.  In fact,
I'm now not sure whether I should use that activity next week -- it
would be good reinforcement, although perhaps too easy.  But yet again
I am struck by the lasting effect of motivating a new concept by having
the students themselves try a challenging problem.  First.

Tevian