Flow of a vector field

Summary

A flow is a family of maps $\varphi_t$ of a space to itself, indexed by real time $t$. The map $\varphi_0$ is the identity, and $\varphi_s \circ \varphi_t = \varphi_{s+t}$. The flow of a vector field sends each point to the position, after time $t$, of the solution of $x' = f(x)$ through it. See the definition.