rectification theorem

Let $U$ be an open subset of $\mathbb{R}^{n}$ and let $f\in C^{1}(U)$ be a continuous differentiable vector field

$f\colon U\to\mathbb{R}^{n}.$

If there exists $x_{0}\in U$ such that $f(x_{0})\neq 0$ then there exists $U_{0}\subset U$ an open neighborhood of $x_{0}$ such that there exists a diffeomorphism of class $C^{1}$

$F\colon U_{0}\to V$

where $V$ is an open subset of $\mathbb{R}^{n}$ such that

$[DF(x)]f(x)=e_{1}$

for all $x\in U_{0}$ where $[DF(x)]$ is the Jacobian of the diffeomorphism $F$ evaluated at $x$ and $e_{1}=(1,0,\ldots,0)$ is the first vector of the canonical basis of $\mathbb{R}^{n}$ . More generally if the vector field $f$ is of class $C^{r}$ then so is the diffeomorphism $F$ . [AVI]

References

AVI Arnold, V.I.: Ordinary Differential Equations (translated by R.A. Silverman). The MIT Press, Cambridge, 1973.

Title	rectification theorem
Canonical name	RectificationTheorem
Date of creation	2013-03-22 14:57:15
Last modified on	2013-03-22 14:57:15
Owner	Daume (40)
Last modified by	Daume (40)
Numerical id	12
Author	Daume (40)
Entry type	Theorem
Classification	msc 34-00
Classification	msc 34A12
Related topic	ImplicitFunctionTheorem
Related topic	TangentMap