Proof Review. A Theorem in Spivak's Book.

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Discussion Overview

The discussion revolves around the proof of a theorem from Spivak's "Calculus on Manifolds," specifically Theorem 2-13, and its application to a variation of Theorem 5.1. Participants are reviewing the proof's correctness and addressing potential oversights or stylistic issues in the argumentation.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • One participant seeks confirmation of their proof regarding the theorem's implications for the manifold structure of the zero set of a function.
  • Another participant notes that Spivak's original text does not specify "continuously differentiable," suggesting it may have been an oversight, while a third participant clarifies that Spivak defined "differentiable" as meaning $C^\infty$ in chapter 5.
  • Concerns are raised about the clarity and conciseness of the proof, with some participants finding certain claims obvious and suggesting that the proof could be more succinct.
  • A participant expresses skepticism about the validity of Theorem 2.13's proof, questioning the existence of the composite function involved in the argument.
  • Another participant questions whether the proof correctly establishes that $h(M)$ is open and whether it should intersect with $h(A}$.

Areas of Agreement / Disagreement

There is no consensus on the correctness of the proof. Some participants support the proof's validity, while others express doubts about specific aspects, particularly regarding the assumptions and the clarity of the argumentation.

Contextual Notes

Participants highlight potential limitations in the proof, including the need for clarity on the continuity of functions involved and the requirement for certain sets to be open. There are also unresolved questions about the assumptions made in the original theorem and its implications.

caffeinemachine
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Hello MHB.
I need your help to confirm that I have got the proof right of a very important theorem.

Theorem 2-13 in Spivak's Calculus on Manifolds.

Let $p\leq n$ and $f:\mathbb R^n\to\mathbb R^p$ be a continuously differentiable function in an open set $O$ of $\mathbb R^n$.
Let $a$ be a point in $O$ such that $f(a)=0$ and assume that the $p\times n$ matrix $M$ with the $i,j$-th entry $[M]_{i,j}=D_jf_i(a)$ has rank $p$.
Then there exists an open set $A$ of $\mathbb R^n$ which contains $a$, and a diffeomorphism $h:A\to\mathbb R^n$ such that $f\circ h(x_1,\ldots,x_n)=(x_{p-n+1},\ldots,x_n)$.

In Spivak's book, it says that Theorem 5.1 is immediate using the above theorem. I have tried to prove a slight variation of Theorem 5.1 below.

Notation:
Let $x\in \mathbb R^n$. We write $x_k^+$ as a shorthand for $(x_{k+1},\ldots,x_n)$ and $x_k^-$ as a shorthand for $(x_1,\ldots,x_{k-1})$.

To Prove:
Let $p\leq n$ and $f:\mathbb R^n\to\mathbb R^p$ be a continuously differentiable function such that $Df(x)$ has rank $p$ whenever $f(x)=0$.
Then $f^{-1}(0)$ is a $(n-p)$-dimensional manifold in $\mathbb R^n$.

Proposed Proof:
Let $a=(a_1,\ldots,a_n)$ be in $f^{-1}(0)$.
Then $f(a)=0$ and thus by Spivak's Theorem 2.13 there exists an open set $A$ of $\mathbb R^n$ which contains $a$, and a diffeomorphism $h:A\to\mathbb R^n$ such that $f\circ h(x)=x_{n-p}^+$.
Write $M=\{x\in A:x_{n-p}^+=0\}$.
We now show that $h(M)=f^{-1}(0)$.

Claim 1: $h(M)\subseteq f^{-1}(0)$.
Proof:
Let $y\in M$.
Then $f\circ h(y)=y_{n-p}^+$ and since $y_{n-p}^+=0$, we have $f(h(y))=0$.
Therefore $f(h(M))=\{0\}$.
This gives $h(M)\subseteq f^{-1}(0)$ and the claim is settled.

Claim 2: $f^{-1}(0)\subseteq h(M)$.
Proof:
Let $x\in f^{-1}(0)$.
Since $h$ is a diffeomorphism, it is a bijection and thus there is $y\in A$ such that $h(y)=x$.
Thus $f(h(y))=f(x)=0$.
This means $f\circ h(y)=0$, that is $y_{n-p}^+=0$, meaning $y\in M$.
Hence $x\in h(M)$ and since $x$ was arbitrarily chosen in $f^{-1}(0)$, we conclude that $f^{-1}(0)\subseteq h(M)$ and the claim is settled.

From the above two claims we have shown that $f^{-1}(0)=h(M)$.
Note that $M=A\cap \{x\in \mathbb R^n:x_{n-p}^+=0\}$.
Since $A$ is an open set in $\mathbb R^n$ and $\{x\in \mathbb R^n:x_{n-p}^+=0 \}$ is a $(n-p)$-dimensional manifold in $\mathbb R^n$, we infer that $M$ is a $(n-p)$-dimensional manifold in $\mathbb R^n$.
Now since $h$ was a diffeomorphism, and since diffeomorphisms take manifolds to manifolds and preserve the dimension, we know that $h(M)$ is a $(n-p)$-dimensional manifold in $\mathbb R^n$.
Having already shown that $h(M)=f^{-1}(0)$, our lemma is proved.
___

Can anybody please check the proof and confirm that it's correct or else point point out the errors?

Thanks in advance for taking the time out.
 
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I don't see anything wrong with this. One thing I noticed in the original text was that Spivak did not include the condition "continuously differentiable" but merely said $f$ is differentiable, which may have been an oversight on his part.

As far as your proof goes, the proof of the 2 claims is a bit wordy, but this is just a stylistic objection. I find it obvious that:

$h(M) \subseteq f^{-1}(0)$ as soon as you show that $f(h(y)) = 0$, and likewise that:

$f^{-1}(0) \subseteq h(M)$ as soon as you show $y \in M$, which is also obvious from:

$f(h(y)) = 0$.

Having shown the existence of $h$, you're done, because that is all that is required to show $f^{-1}(0)$ is an $(n-p)$-dimensional manifold according to the definition at the beginning of chapter 5 (with $M$ playing the role of the set $V \cap (\Bbb R^k \times \{0\}) \subseteq \Bbb R^n$).
 
Deveno said:
I don't see anything wrong with this. One thing I noticed in the original text was that Spivak did not include the condition "continuously differentiable" but merely said $f$ is differentiable, which may have been an oversight on his part.
Actually at the beginning of chapter 5 Spivak had said that 'differentiable' henceforth will mean $C^\infty$ so he didn't make a mistake. I wanted to prove the theorem using the much weaker hypothesis of continuous differentiability only and that's why I changed the statement.

Deveno said:
As far as your proof goes, the proof of the 2 claims is a bit wordy, but this is just a stylistic objection. I find it obvious that:

$h(M) \subseteq f^{-1}(0)$ as soon as you show that $f(h(y)) = 0$, and likewise that:

$f^{-1}(0) \subseteq h(M)$ as soon as you show $y \in M$, which is also obvious from:

$f(h(y)) = 0$.
I am new to these things so I am writing the proofs in full. :o

Deveno said:
Having shown the existence of $h$, you're done, because that is all that is required to show $f^{-1}(0)$ is an $(n-p)$-dimensional manifold according to the definition at the beginning of chapter 5 (with $M$ playing the role of the set $V \cap (\Bbb R^k \times \{0\}) \subseteq \Bbb R^n$).

Yes. Thanks. :)
 
Hi there all,

My first post, so apologies if this turns out to be cr*p. I think I am probably at about the stage caffeinemachine was when he posted - i.e. struggling to wade through Spivak chapter 5. I am not at an educational institution, so I'm on my own, which can be a lonely place in Calculus on Manifolds!

Anyway, I also struggle with Theorem 5.1, not least because I don't think the proof of Theorem 2.13 referenced from chapter 2 is valid. In that proof a claim is made about a cont. diff. function defined on an open set A containing a point a. For the general case in paragraph 2, the author first permutes the coordinates of a via a function g, then proceeds to apply the inverse function theorem to the composite f o g. My issue with this is that surely we don't even know this composite exists at a, let alone in an open set containing a!

That aside, with respect caffeinemachine, I have some trouble following your proof of Thm 5.1:
In showing that h(M)=f-1(0), should this not be h(M) = f-1(0) intersected with h(A)? This notwithstanding, do we not require h(M) to be open? You have not shown this. I know these are details, but I'm trying to be rigorous. No offence is meant.
 
Last edited:
Semillon said:
Hi there all,

My first post, so apologies if this turns out to be cr*p. I think I am probably at about the stage caffeinemachine was when he posted - i.e. struggling to wade through Spivak chapter 5. I am not at an educational institution, so I'm on my own, which can be a lonely place in Calculus on Manifolds!

Anyway, I also struggle with Theorem 5.1, not least because I don't think the proof of Theorem 2.13 referenced from chapter 2 is valid. In that proof a claim is made about a cont. diff. function defined on an open set A containing a point a. For the general case in paragraph 2, the author first permutes the coordinates of a via a function g, then proceeds to apply the inverse function theorem to the composite f o g. My issue with this is that surely we don't even know this composite exists at a, let alone in an open set containing a!

That aside, with respect caffeinemachine, I have some trouble following your proof of Thm 5.1:
In showing that h(M)=f-1(0), should this not be h(M) = f-1(0) intersected with h(A)? This notwithstanding, do we not require h(M) to be open? You have not shown this. I know these are details, but I'm trying to be rigorous. No offence is meant.

Hey Semillon!

I wrote this post quite some time ago.

I will review my proof keeping in mind the possible mistakes you have mentioned and reply by the end of the day.

EDIT: I am sorry that I could not reply because I have been quite busy today. I will try to study this post tomorrow and post my response.
 
Last edited:
@ Semillon... You don't need to worry about the "auto-save" function on here. It just stores what you've written as a handy back-up. If you then delete, change, etc your text, the next 'save' will keep a record of your changes. In short, the auto-save is your friend; you don't need to rush. Really... :cool:
 
So I finally reviewed my proof. Sorry for the very late reply.

Semillon said:
That aside, with respect caffeinemachine, I have some trouble following your proof of Thm 5.1:
In showing that h(M)=f-1(0), should this not be h(M) = f-1(0) intersected with h(A)?
Isn't $h(A)=\mathbb R^n$? In this light, $f^{-1}(0)\cap h(A)=f^{-1}(0)\cap\mathbb R^n=f^{-1}(0)$.

Semillon said:
I know these are details, but I'm trying to be rigorous. No offence is meant.
None taken. :)
 

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