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I just had a student come to my office hours and ask me a ton of questions, the answer to all of which was "that's a slight variant to the implicit function theorem, which is proved by formal manipulation from the implicit function theorem". For example:

$\def\RR{\mathbb{R}}$ Let $f : \RR^n \to \RR$ be a smooth function and suppose that $\partial f/ \partial x_n \neq 0$. Then, locally, $(x_1, x_2, \ldots, x_{n-1}, f)$ are coordinates on $\RR^n$.

Let $f : \RR^n \to \RR$ be a smooth function and suppose that $\partial f/ \partial x_n \neq 0$. Then, locally, $(x_1, x_2, \ldots, x_{n-1})$ are coordinates on $\{ f=0 \}$.

Let $U$ be a small open set in $\RR^d$ and let $(f_1, \ldots, f_d): U \to \RR^n$ parameterize a patch on a manifold $M$ in $\RR^n$. Suppose that $\det (\partial f_i/\partial x_j)_{1 \leq i,j \leq d} \neq 0$. Then $x_1$, ..., $x_d$ are local coordinates on $M$.

Let $g_1$, ..., $g_{n-d}$ be smooth functions $\RR^n \to \RR$. Let $M= \{g_1=g_2=\ldots=g_{n-d} = 0 \}$. Suppose that $\det (\partial f_i/\partial x_j)_{1 \leq i,j \leq n-d} \neq 0$. Then $M$ is a smooth manifold of dimension $d$ and $x_{n-d+1}$, ..., $x_n$ are local coordinates near $0$.

Does anyone know a book which works through these sort of variants systematically?

I should mention that I actually need these facts for holomorphic functions. But I have a good reference for the holomorphic implicit function theorem: Gunning and Rossi, Chapter 1. The problem is that I want a reference which goes slowly through these variants, rather than assuming they are obvious corollaries.

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But isn't it reasonable for the student to struggle with figuring out how to prove these variants? Becoming facile with using the implicit function theorem is for me a rather basic and necessary skill for a differential geometer. –  Deane Yang Feb 7 '11 at 17:26
    
The proof of Sard's theorem is an excellent way to work through several of these reformulations. –  Ryan Budney Feb 7 '11 at 20:39
    
$g$ and $f$ are mixed up in one statement. –  S. Carnahan Feb 7 '11 at 21:29
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3 Answers

I have a good reference, which is even available online, but something tells me you will cry : the wikipedia page on the Théorème des fonctions implicites is more complete than its english counterpart!

Notice that they're refencing a Lang book... so perhaps that will do better.

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Broken link. Hopefully this works better: fr.wikipedia.org/wiki/… –  Hans Lundmark Feb 8 '11 at 15:49
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Perhaps The Implicit Function Theorem by Krantz and Parks. Not a textbook, but quite interesting.

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Well, not to all of them, but nevertheless a nice approach: in the differential topology book by Bröcker and Jänich, they discuss various applications of the implicit function theorem and the theorem of constant rank maps, using them to build coordinate systems etc. Maybe this is worth a look. I only have the german edition (there it is in Chap 5) but I think there is an english version around. They formulate it for the real/smooth setting, though :( But the ideas are the same of course.

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It's checked out of the MI library, but the part I can see on google books looks nice. –  David Speyer Feb 7 '11 at 16:59
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