I've worked through a computation-heavy, "standard" but quite nonrigorous treatment of multivariable calculus in the past. What book would do well as a rigorous (but not overly) "second course"? In particular, I'm looking for a book that

  • treats differential forms

  • treats the inverse and implicit function theorems

and leads well into an intro manifolds book like Tu's one. (Bonus points if it actually talks a bit about manifolds itself or works with differential forms defined on manifolds instead of only $\Bbb R^n$!)

  • $\begingroup$ Zorich's Mathematical Analys (vol. I & II) perhaps? $\endgroup$ – Hans Lundmark Nov 12 '15 at 5:47
  • $\begingroup$ I do not think I want an analysis book. I'd like something with plenty of pictures and one that does not spend a lot of time with epsilon-delta arguments. (My idea of analysis books is that they don't do that kind of thing; I may be wrong.) $\endgroup$ – Soham Chowdhury Nov 12 '15 at 5:51
  • 1
    $\begingroup$ Fine, it was just a suggestion. But what do you mean by "treating" the inverse and implicit function theorems then? I got the impression that you already know how to use them, and were looking for a more advanced book where you could see how they actually are proved. $\endgroup$ – Hans Lundmark Nov 12 '15 at 7:06
  • $\begingroup$ No, I do not know those two theorems. I only know that they are something that I should know before studying manifolds. $\endgroup$ – Soham Chowdhury Nov 12 '15 at 10:52
  • $\begingroup$ I see. Yes, in that case, some other book is probably more suitable. $\endgroup$ – Hans Lundmark Nov 12 '15 at 11:05

Possible recommendations - I'm not 100% sure if what you are looking for exists but these are all well-written and worth investigating.

  • Edwards, Advanced calculus: a differential forms approach
  • Bloch, A First Course in Geometric Topology and Differential Geometry
  • Bachmann, A Geometric Approach to Differential Forms

The last one is particularly nice, in my opinion.

  • $\begingroup$ Bachmann doesn't do the implicit function theorem, I suppose. :( $\endgroup$ – Soham Chowdhury Nov 12 '15 at 5:25
  • $\begingroup$ Yeah, I wasn't sure - it might be in the "Prerequisites" section. $\endgroup$ – kcrisman Nov 12 '15 at 13:31

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