Flux and Theorem Differential Equation This is my first post here!. So that, my question is, 
If $X$ is a field $C^{\infty}$ on the open set $V$ of the manifold $M$ and $p\in V$, then exists a open set $V_{0}\subset V$, $p\in V_{0}$, and number $\delta>0$, and application $C^{\infty}$, $\varphi:(-\delta,\delta)\times V_{0}\to V$ such that a curve $t\to\varphi(t,q), t\in(-\delta,\delta)$ is a unique path of $X$ in the time $t=0$ through point $q$, for each $q\in V_{0}$. So, what really means this? (a small drawing would ideally
) Any hint to show this, or actually a book of reference (this theorem maybe is very common in Differential equations) Thanks!

This is the "basic" Picard-Lindelöf theorem (some people would call it the Cauchy-Lipschitz theorem), look for example in Hale's ODE book (for the form that you need).
Although it is not there for manifolds, that is an easier part, since one can use charts to pass everything to $\mathbb R^n$ (after all, the result is local).
