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Let $M$ be a $C^{k}$ differentiable manifold of dimension $n$, and let $X$ be a $C^{k}$ vector field on $M$. Let $p$ be a point of $M$ such that $X(p) \neq 0$: is there a local parametrization $(\varphi, U)$ (where $U$ is a neighbourhood of $M$ containing $p$ and $\varphi : U \rightarrow \mathbb{R}^n$) such that the push forward of $X$ with respect to $\phi$ is a canonical field (that is, $\varphi_{*}X(u)=e_{i}(u)$ for every $u \in U$, where $e_{i}$ is the constant vector field parallel to the $i$-th versor) ?

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    $\begingroup$ This is often called the Flowbox Theorem. See this Wiki entry for a brief discussion. $\endgroup$ – Ted Shifrin Oct 30 '17 at 23:46

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