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What textbook does a good job proving the multivariable implicit and inverse mapping theorems?

C. H. Edwards provides an adequate treatment of these theorems. I found his notation a bit hard to process, but I finally understood his proofs. I would like to compare his presentation to another treatment of the same material.

I would also like to know if there are proofs which do not involve contraction mappings.

Any suggestions?

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  • $\begingroup$ I think Tao's book Analysis II has particularly clean and clear proofs of these theorems. Hubbard and Hubbard has an approach based on Newton's method. They have tried very hard to explain this theorem clearly. $\endgroup$
    – littleO
    Dec 11, 2017 at 21:42

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Spivak (Calculus on Manifolds) and Munkres (Analysis on Manifolds) have a (very finite-dimensional) proof that avoids the Contraction Mapping Principle. I think the Contracting Mapping proof is superior, inasmuch as it works verbatim in Banach spaces. It also seems more natural to me to avoid a compactness argument to prove that the map is open.

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  • $\begingroup$ Not to mention, the contraction mapping principle is something very useful to know for its own sake. $\endgroup$
    – littleO
    Dec 11, 2017 at 21:36
  • $\begingroup$ @littleO I actually used contraction mapping in Mathematica to generate illustrations to support my version of Edwards's proof. I agree that it's a wonderful thing to know. It converges very fast, too. $\endgroup$
    – Steven Thomas Hatton
    Dec 12, 2017 at 2:34
  • $\begingroup$ @Ted Shifrin this is a completely fair assasment of Edwards's proof of the inverse mapping theorem: con·vo·lut·ed/ˈkänvəˌlo͞odəd/ adjective -- (especially of an argument, story, or sentence) extremely complex and difficult to follow. $\endgroup$
    – Steven Thomas Hatton
    Dec 12, 2017 at 2:38
  • $\begingroup$ @StevenHatton: I definitely disagree. You can find my treatment of the proof in my own text and in my lectures on YouTube (see my profile if you care). I should remark this is one of the deepest theorems of undergraduate mathematics and the proof — any proof — is involved. $\endgroup$
    – Ted Shifrin
    Dec 12, 2017 at 3:05
  • $\begingroup$ Read the definition of "convoluted" again. It says nothing about the reason for the convolution. I am absolutely sure there is room for improvement in Edwards's presentation. But that doesn't mean I don't respect and value it. I will look into the referenced sources. $\endgroup$
    – Steven Thomas Hatton
    Dec 12, 2017 at 4:07
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I really like the discussion and details in Zorich's text. In the first volume there is a long discussion, motivation, and careful derivation without invoking the contraction mapping principle, but everything is done for $\mathbb R^n$. In the second volume, the same theorem now is proved for arbitrary Banach space, and now the contraction mapping principle is used. What I most like about this analysis book that it does not try to skip any "obvious" details.

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  • $\begingroup$ I finally got around to looking at Zorich's Vol I. His approach to math is frighteningly similar to mine. It may prove to be a relatively easy read since he covers in a similar order the same topics which I have treated in my self-inflicted study notes. I will not be so presumptuous as to compare the quality of my scribblings to his obviously world-class textbook. One missing feature which I look for in a textbook is the use of graphical illustration. But that would have expanded the thousand or so pages to twice that. Note: Yes Dr. Shifrin, I know... It's expensive. $\endgroup$
    – Steven Thomas Hatton
    Sep 16, 2022 at 13:07

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