• greens theorem intro
  • parametric surface drawing intro
  • surface area intro
  • surface integral applications (mass centroid center of mass)
  • Flux intro
  • Flux on multiple surfaces
  • stokes theorem intro
  • divergence theorem intro
  • planimeter?
  • proofs?
  • one problem focuses on verifying each
  • I would love to have a problem that gives them LOTS of parametric surfaces. Draw each with software, then pick a few to practice computing surface area. Or maybe just use a different surface with every problem (so they see lots of surfaces by the time they are done). Start with matching, end with "parametrize your own" (provide picture, they must give parametrization). Coming up with a parametrization is one of the toughest parts.
  • What if one problem, early on, just asked them to compute lots of integrals, using things they are completely familiar with (no need to employ a theorem, rather just set up and compute some integrals. Probably on the second day, after they have learned both surface area and flux. They could compute all the things involved in Stokes's and Divergence theorem, without having to know the theorem at all. I give them the surface, or the solid, or whatever, and they just set up and compute an integral.

As a side note, this unit is somewhat overwhelming at first. There are so many new ideas being thrown at the students. The second day was too much for the students. Not sure how to fix this. Maybe introduce surfaces earlier in the semester?

Task day.1

stuff

Task day.2

stuff

Task day.3

Mobius Strip - a non-orientable surface. 

r[u_, v_] := {(3 + u Cos[v/2]) Cos[v], (3 + u Cos[v/2]) Sin[v], u Sin[v/2]};
uBounds = {u, -1, 1};
vBounds = {v, 0, 2 Pi};

Put this code into the normal vector visualizer, and it's really easy to see why you cannot orient this surface. We will not compute flux on such a surface, but we can still talk about surface area without issue.

Physical properties. (this is my pet topic, but it's not required)

  1. Prove that $\iint_R x dA = \bar x A$.
  2. Prove that $\iiint y dV = \bar y V$.
  3. Compute a Green's theorem problem by referring to centroid and area facts.
  4. Compute a divergence theorem problem by referring to centroid and volume facts.

Green's theorem with multiple edges, including a hole inside. Use the winding number at some point.

Divergence theorem with multiple surfaces, including a hole inside one.

One problem needs to get out Gauss's law, and discuss the subtleties of the theorem (how the vector field must be differentiable EVERYWHERE inside the domain.