Looking For a Flow-Free Proof of the Rectification Theorem.

[caffeinemachine]

I am looking for a proof of the following theorem:Rectification Theorem: Let $\mathbf V:\mathbf R^n\to\mathbf R^n$ be a smooth function and $\mathbf p\in \mathbf R^n$ be such that $V(\mathbf p)\neq \mathbf 0$.
Then there exists a neighborhood $U$ of $\mathbf p$ in $\mathbf R^n$, and a diffeomorphism $f:U\to W\subseteq \mathbf R^n$ such that $D_{\mathbf V(\mathbf x)}f(\mathbf x)=\mathbf e_1$ for all $\mathbf x\in U$, where $\mathbf e_1=(1,0,\ldots,0)$.I googled 'rectification theorem proof' and I found many proofs all using the concept of 'flow of a vector field' and other concepts from differential equations literature.I am not acquainted with these concepts.Does anybody know of a proof which does not use any concepts beyond what are usually found in an Advanced Calculus textbook (like Spivak's Calculus on Manifolds or Munkres' Analysis on Manifolds.)Thanks.

 

[jvicens]

Thank you for your inquiry about the Rectification Theorem. I am happy to provide you with a proof of this theorem that does not require any advanced concepts beyond those typically found in an Advanced Calculus textbook.

First, let us define some terms that will be used in the proof. A diffeomorphism is a smooth and invertible function between two manifolds. A manifold is a space that locally resembles Euclidean space, such as a curve, surface, or higher-dimensional space. A neighborhood is a set that contains an open set around a particular point. Finally, a vector field is a function that assigns a vector to each point in space.

Now, let us proceed with the proof of the Rectification Theorem.

Proof:

Let $\mathbf{V}:\mathbf{R}^n\to\mathbf{R}^n$ be a smooth function and $\mathbf{p}\in\mathbf{R}^n$ such that $\mathbf{V}(\mathbf{p})\neq\mathbf{0}$. Since $\mathbf{V}$ is a smooth function, it has a well-defined Jacobian matrix at every point in its domain. Let $\mathbf{J}(\mathbf{x})$ be the Jacobian matrix of $\mathbf{V}$ at the point $\mathbf{x}\in\mathbf{R}^n$.

By the Inverse Function Theorem, if $\mathbf{J}(\mathbf{p})$ is invertible, then there exists a neighborhood $U$ of $\mathbf{p}$ and a diffeomorphism $f:U\to W\subseteq\mathbf{R}^n$ such that $D_{\mathbf{V}(\mathbf{x})}f(\mathbf{x})=\mathbf{J}^{-1}(\mathbf{p})$ for all $\mathbf{x}\in U$. Since $\mathbf{V}(\mathbf{p})\neq\mathbf{0}$, $\mathbf{J}(\mathbf{p})$ is invertible, and the conditions of the Inverse Function Theorem are satisfied.

Now, let us consider the function $\mathbf{F}(\mathbf{x})=\mathbf{V}(\mathbf{x})-\mathbf{