[BUG] The Net

[Smith, Alexander J.]

I gave "The Fishing Net" out in class about a week before we will get to
the divergence theorem, and right after the section on computing flux
(when the surface is given parametrically).


It works great! After working on the activity in class, students are
supposed to read  ahead to the section on the divergence theorem, and
use the divergence theorem as part (b) asks.

I can't tell you how many emails I get from students asking what to do
about the "0". (Most think they are computing the divergence wrong.)
Then I give them the hint about the word >closed< in the div. theorem,
and suggest they look at the imaginary top to the net. The next
conversation is usually about the semantics of "net outward flux=0"
versus "what comes out is balanced by what goes in".


Here is an example of such an email:

%%%
Dr. Smith, I can't for the life of me figure out this group project. For
the part where we're supposed to figure out the flux using the
divergence theorem it doesn't make sense. The divergence of the vector
field that we're given is 0 every which way I do it. Can you share any
insight that I'm not comprehending?
%%%

I really do recommend that you give this activity out before formally
covering the divergence theorem in class. My students had lots of build
up to the concept of divergence, however, by using Vector Field Analyzer
intensively when we did the unspeakable Green's Theorem. 

As far as whether or not to cover Green's theorem, I think that given
the existence of Vector Field Analyzer, it is a no brainer. Do it. It is
worth it.