[BUG] The Valley

[Tevian Dray]

Because of the bad weather earlier in the month, we're still about a lecture
behind where we would normally be at this point in the term.  Rather than
postpone the Valley lab, I went ahead and did it last week anyway, although I
had spent barely 5 minutes in class talking about line integrals -- just
enough time to remind students that integrals are sums, mention work and dot
products, and pretend to push the overhead projector one way while it moved
"on rails" another way.

The lab went very smoothly, although the students were busy the entire time.
And the punch line -- that the answer was just the height difference -- was
well-received.

Even better was what happened in class the next day.  I had finally decided to
try Corinne's suggestion of doing vector line integrals (e.g. work) before
scalar line integrals (e.g. charge).  We do surface integrals in this order --
and the Calculus Consortium books don't even mention scalar line or surface
integrals.  But I had always resisted doing this for line integrals, feeling
that the length or mass of a wire makes a better starting point than the
somewhat obscure (for many students) notion of work.  On the other hand, we
really want to tie things to a single idea (drvector), and starting with
scalar line integrals makes this tough -- ds=|drvector| seems artificial when
there are no vectors in sight.

But having done the lab, I didn't really have much more to say about vector
line integrals, so I simply skipped that part of my lecture, and spent most of
the period with our "chocolate factory" -- a (pretend!) tub full of melted
chocolate into which variously shaped wafers are dipped, then slowly withdrawn
in such a way that the density of chocolate on the wafer is directly
proportional to the time spent in the tub, and hence to the depth each part of
the wafer starts at.

After the lab, the use of drvector to get ds seemed natural, and so the same
skills are being used for both kinds of line integrals: Start with drvector
and use what you know.  I will now return to vector line integrals in order to
discuss conservative vector fields, but I feel that I have stumbled on the
ideal combination: Use vector line integrals to get started, but do the lab
without much preparation.  Then do some scalar integrals with an obvious
physical context.  Then go on to discuss why not all vector line integrals are
like the one in the Valley lab (which is of course path independent).  This
engages the students from the beginning, making all the ideas seem both simple
and obvious.

In sum, the Valley lab does a nice job of setting things up for line
integrals, and I encourage its use early -- all that's really needed is an
understanding of topographic maps and the gradient.  Perhaps that's the beauty
of it -- the only vector field involved is a gradient, so it's probably not
even necessary to mention work beforehand.  On the other hand, I've been
emphasizing in class that not all vector fields correspond to topo maps,
motivating a later discussion of (non)conservative vector fields.

Tevian