---
tag: 06738GGWWK
kind: concept
name: Flow of a vector field
aliases: [flow]
msc: [37C10]
sources:
  - url: https://en.wikipedia.org/wiki/Flow_(mathematics)
    locator: lead section
same_as:
  wikidata: Q1434290
  wikipedia: Flow (mathematics)
---

## 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|02903SSA48]].