I've been watching Professor Ted Shifrin's brilliant lectures on Stokes's Theorem, and I had a few questions that I don't think were answered:
- Rectangles in the plane are not manifolds with boundary as the lectures define them, so I don't think Stokes's Theorem (as discussed in the lectures) would apply to them. But Professor Shifrin said that Stokes's Theorem was a generalization of Green's Theorem, which was grounded in rectangles in the lectures. How do these two ideas hold at once?
Professor Shifrin provided an example of a region on which the integral of a differential form on which wasn't parametrized "exactly," but the integral over the small segment canceled out. Then, he asserted that a similar cancellation would not occur when integrating over a Möbius strip. (The relevant portion of the lecture starts here: https://youtu.be/5k13cowATAw?t=644)
Why doesn't a similar cancellation occur when integrating over a Möbius strip? I would think that the segment would have opposite orientation when parametrized each time.
Why does the integral on that segment even matter? (I know this is probably a stupid question.) On first glance, my thought was that this boundary region had 2d volume zero, so it wouldn't matter in the grand scheme of integration.
Professor Shifrin said that the Möbius strip could be parametrized locally but not globally. Why don't we define the integral over the whole manifold to be the sum of the integrals along each coordinate patch, similar to what we do for piecewise smooth manifolds?
Thank you so much for your help!!
I think somewhere in there I did mention that Stokes's Theorem applies more generally to manifolds with corners but that we would not worry about proving those cases. There are certainly lots of examples that show up (like a hemisphere with a disk attached to its boundary).
The Möbius strip can certainly be parametrized globally (just as a circle can be). Indeed, one exercise in the text gives the parametrization. What I believe I said was that it could not be oriented globally, even though it can be oriented if you remove an appropriate line segment. Although we can easily define the integral of a $k$-form over a parametrized $k$-manifold by pulling back, to integrate a $k$-form over a general $k$-dimensional manifold requires orientation; if you check the definitions carefully, the different parametric patches (or charts, if you prefer) have to be compatibly oriented in order for the partition of unity argument to give you something well-defined.
A question worth thinking about (which I don't believe I raised in lecture) is this: Suppose you have a vector field defined on $\Bbb R^3$ (just for simplicity). Does its flux across a Möbius band make sense? What about finding its work done around the boundary of the Möbius strip (suitably oriented)? Then in what sense can you say that the flux of its curl across the Möbius strip gives that work? Indeed, what happens when you cut the Möbius strip and get an orientable surface? But, as you suggested in your second question, doesn't the cut show up twice (rather than cancelling out) when you do the line integral?