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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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    $\begingroup$ 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. $\endgroup$ – Deane Yang Feb 7 '11 at 17:26
  • $\begingroup$ The proof of Sard's theorem is an excellent way to work through several of these reformulations. $\endgroup$ – Ryan Budney Feb 7 '11 at 20:39
  • $\begingroup$ $g$ and $f$ are mixed up in one statement. $\endgroup$ – S. Carnahan Feb 7 '11 at 21:29
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Shameless plug: In my thesis, I introduced the following notion of an (abstract) normal form. It consists of a tuple $(X, Y, \hat{f}, f_s)$ where:

  • $X$ and $Y$ finite-dimensional vector spaces with compositions $X = Ker \oplus Coim$ and $Y = Coker \oplus Im$,
  • $\hat{f}: Coim \to Im$ is a linear isomorphism,
  • $f_s: X \to Coker$ is a smooth map with vanishing derivative at $0$ which satisfies $f_s(0, x) = 0$ for all $x \in Coim$.

The inverse function theorem can then be used to show that every smooth map $f: M \to N$ can be brought into such a normal form, i.e. there exist charts such that $f$ locally coincides with $\hat{f} + f_s$. The abstract spaces $Ker$, $Coimg$, $Coker$ and $Im$ in the normal form are then of course identified with the corresponding kernel, coimage, ... of the derivative of $f$. This normal form theorem has the immersion, the submersion and the constant rank theorem as direct corollaries, and it also serves as the natural "parent" for all your statements.

For example consider the second statement.

Let $f : \mathbb R^n \to \mathbb R$ 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 $\{ f=0 \}$.

Proof: Since $\partial f/ \partial x_n \neq 0$, we have $Ker = span (x_1, \dots, x_{n_1})$, $Coimg = span (x_n)$, $Img = \mathbb R$ and trivial $Coker$. Moreover, $f$ is locally given by a linear isomorphism $\hat{f}: Coimg \to Img$ so that $\{f = 0\}$ is identified with $Ker = span (x_1, \dots, x_{n_1})$.

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  • $\begingroup$ You might just use the usual rank theorem to say that $(x_1,\dots,x_{n-1},f)$ are local coordinates, so can just write them as $(x_1,\dots,x_{n-1},x_n)$, and then $(x_n=0)$ has coordinates $x_1,\dots,x_{n-1}$. I don't see why you need to worry about $f_s$, and without $f_s$, this is just the usual rank theorem. $\endgroup$ – Ben McKay Nov 12 '19 at 15:51
  • $\begingroup$ Sure, this particular statement follows from the rank theorem. If I understood @David E Speyer correctly, he wanted to have a general theorem/treatment of various applications of the inverse function theorem, and this is what this abstract normal theorem provides. The singular part $f_s$ is of course only important if you work with submersions/constant rank maps. $\endgroup$ – Tobias Diez Nov 12 '19 at 16:10
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    $\begingroup$ Thanks! This looks nice. I may try this out next time I teach our honors calculus on manifolds course. $\endgroup$ – David E Speyer Nov 12 '19 at 16:50
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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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  • $\begingroup$ It's checked out of the MI library, but the part I can see on google books looks nice. $\endgroup$ – David E Speyer Feb 7 '11 at 16:59
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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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Perhaps The Implicit Function Theorem by Krantz and Parks. Not a textbook, but quite interesting.

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