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
$$f:U\to {\mathbb{R}}^{n}.$$ |
If there exists ${x}_{0}\in U$ such that $f({x}_{0})\ne 0$ then there exists ${U}_{0}\subset U$ an open neighborhood of ${x}_{0}$ such that there exists a diffeomorphism of class ${C}^{1}$
$$F:{U}_{0}\to V$$ |
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 ${e}_{1}=(1,0,\mathrm{\dots},0)$ 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 |