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I plan on working through "Geometry of Physics" by Frankel. I keep on running into little snags here and there, and I am wondering if that's just part of the process, or whether I am ill-prepared.

For example, in order to understand what differentiability in higher dimensions means (one of the things Frankel assumes you know already), I have been trying to make my way through Spivak's "Calculus on Manifolds". Progress is there, but it's slow.

Is it just me, or is there something incredibly ugly about how Spivak proves the closed interval $[a,b]$ is compact(proof of Heine-Borel theorem)? Am I just missing a lot of "back-story" (i.e. stuff I should be pretty comfortable with already)? Is each page supposed to take me an hour to work through?

Just thought I'd ask others who might have something to share from their experience.

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My first question to you is "Why did you choose Frankel's book ?" Well, Frankel is a nice book, and I would recommend it to a physics major. But if you are ready to digest a bit more of abstraction with plenty of details, I would suggest you to go for John Lee's "Introduction to smooth manifolds" 2nd ed. Another advantage of following Lee is that he also talks about basic Topology and multivariable analysis in appendices making the book rather complete.


Secondly, don't get demotivated just because you had to spend an hour or so understanding one page. Mathematics is more like an art. You may have to struggle to learn the techniques and language. But once you are done with it, you can understand and even create masterful pieces of art with it. All the best.

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