Suppose V is a smooth vector field of a differentiable manifold $\mathcal{M}$, and let $p\in \mathcal{M}$, we know that with some coordinate chart $(U,\phi)$, V can be represented by $V=\sum_{i=1}^{n}a^{i}\frac{\partial}{\partial x_i}$ for some smooth function $a^{i}$ on $\phi (U)$. My question is, can we choose a special coordinate chart such that $V=\frac{\partial}{\partial x_1}$?
Thanks!
This is true if and only if the vector field does not vanish at $p$. This is sometimes called the flow box theorem or the straightening theorem, or the rectification theorem. This is proven in most textbooks on ordinary differential equations, such as Arnol'd, Ordinary Differential Equations.
Some hints can be found in this question. Particular function in proof of flow box theorem
