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 $f(x_{0})\eq 0$ 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.