I'm interested in the following result (chapter 5, theorem 7 in volume 1 of Spivak's Differential Geometry):
Let $X$ be a smooth vector field on an $n$-dimensional manifold M with $X(p)\neq0$ for some point $p\in M$. Then there exists a coordinate system $x^1,\ldots,x^n$ for $U$ (an open subset of $M$ containing $p$) in which $X=\frac{\partial}{\partial x^1}$.
Could someone please explain, in words, how to prove this (or, if you have the book, how Spivak proves this)?
I've read Spivak's proof, and have a few questions about it:
1) How is he using the assumption $X(p)\neq0$?
2) Why can we assume $X(0)=\frac{\partial}{\partial t^1}|_0$ (where $t^1,\ldots,t^n$ is the standard coordinate system for $\mathbb{R}^n$ and WLOG $p=0\in\mathbb{R}^n$)?
3) How do we know that in a neighborhood of the origin in $\mathbb{R}^n$, there's a unique integral curve through each point $(0,a^2,\ldots,a^n)$?