Warner page-37: Theorem: Let $X$ be a $C^{\infty}$ vector field on a differentiable manifold $M$ For each $m\in M$ there exist $a(m)$ and $b(m)$ in extended real line, and smooth curve $\gamma_m:(a(m),b(m))\rightarrow M$ such that
a) $0\in (a(m),b(m))$ and $\gamma_m(0)=m$
b)$\gamma_m$ is an integral curve of $X$
c) If $\mu:(c,d)\rightarrow M$ is a smooth curve satisfying conditions $a$ and $b$ then $(c,d)\subseteq (a(m),b(m))$ and $\mu=\gamma_m|(c,d)$
my question is I confused about the dependency of $a$ , $b$ on $m$ and not getting any feeling of that definition, could any one give one simple example satisfying above conditions?
Next is based on above:
Definition: $\forall t\in \mathbb{R}$ we define a transformation $X_t$ with domain $D_t=\{m\in M:t\in (a(m),b(m))\}$ by setting $X_t(m)=\gamma_m(t),$ I understand that this a map from $M$ to $M$, but not able to visualize what is exactly going on, I need some example on $\mathbb{R}$ or $\mathbb{R}^n$, I will be really happy if some one help me to understand these.