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Paul
Posts:
719
Registered:
7/12/10


Learning the basic theory and language of differential forms
Posted:
Jul 16, 2013 6:20 PM


I would be interested to know some very basic theory of differential forms like Stoke's theorem etc. I wonder whether anyone has any words of wisdom about what some good material would be to consult? For me, it's extremely important to be able to prove every detail of what I read. I really don't like informal handwaving explanations. I was reading baby Rudin on this topic. However, without really getting stuck, I seem to have found it a bit of a slog, and haven't read it in over a week.
I'll give some flavour of stuff I've done recently, although it has nothing to do with differential forms.
Basically, I finally achieved my goal of mastering a proof of Jordan Curve Theorem.
The route I took was Gale's proof that a full board of Hex stones always contains a winning path, Gale's proof that this implies Brouwer's fixed point theorem, standard proofs of Urysohn's lemma and Tietze extension and finally Maehara's proof that Brouwer + Tietze Extension Theorem implies JCT.
I found all this quite straightforward (I mean I didn't get seriously stuck, not that everything was obvious).
The only slight stumbling block was the fact that Hex can apparently be played on a square board. Rather than puzzle about why the boards were equivalent, I stayed with the hexagonal board for the Hex game, and identified the square board with the hexagonal board by means of the obvious linear homeomorphism. (Modifying Gale's proof in that way was not my idea. I saw it in a blog.)
So, having said a bit about my interests and stuff I've been doing, I wonder if any readers have any thoughts about a recommended guide to differential forms.
Perhaps Spivak's Calculus on Manifolds?
Many thanks for your advice.
Paul Epstein



