Transforming vectors to coordinate vectors

Question

Let $X_1,\dots,X_n$ be n vector fields on an open subset $U$ of a manifold of dimension $n$ Suppose that at $p\in U$, the vectors $(X_1)_p,\dots,(X_n)_p$ are linearly indipendent. would any one say in detail how I will show that there is a chart $(V,x^1,\dots,x^n)$ about $p$ such that $(X)_p=(\partial/\partial x^i)_p$ for all $i=1,\dots,n$, here I use superscript for coordinates like $x^1,\dots,x^n$

Answer 1

By the Frobenius theorem, given a local basis $X_i$ of $TM$, there exists a chart $x_i$ with $X_i = \partial_i$ if and only if the for all $i,j$, $[X_i,X_j]=0$.

Basically, $[X,Y]$ measures the difference between infinitesmially flowing out $X$, then over on $Y$, versus infinitesimally flowing out $Y$, then flowing over on $X$. In order to have coordinates, the two must always agree, otherwise the "location" of a point is not well-defined - getting there from the origin by flowing out $X$, then $Y$, would result in a different position flowing out $Y$ from $X$.