Learning Differential Forms and Questions


After years of putting it off, I finally prepaid a tutor to force me to actually read about differential forms.

The tiny bit of algebraic topology I remember came in handy, since I once gave a lecture on singular homology, and it taught me about the boundary operator, that it squares to 0, and how to define orientation over a simplex.

Differential forms carry over from that well, thank god.

The wedge product, Hodge star, and exterior derivative make remembering Green’s Theorem and the non-general Stokes Theorem easy. Those never stuck in my head before.

Stokes Theorem now seems super obvious and using it as a tattoo now feels tacky (sorry people who have this).

Questions

  • How to unite measure theory and forms?
  • How does the ordinary derivative relate to the exterior derivative beyond the total derivative of a smooth function (0 form)?
  • Is there a meaningful way to integrate n-forms over non n-dimensional regions?
  • Can you do all this on Banach manifolds or at least Hilbert ones, or does any of this fundamentally rely on some property of \(\mathbb{R}^n\)?

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