Flow of a vector field
= Flow of a vector field
{title2=$\frac{d}{dt}\Phi_t(p)=V(\Phi_t(p))$}
The local flow of a smooth <vector field> is the family of solutions of its <ordinary differential equation>, with $\Phi_0(p)=p$. Uniqueness gives $\Phi_{s+t}=\Phi_s\circ\Phi_t$ wherever defined. A compactly supported smooth <vector field> on a manifold without boundary is complete, so its flow exists for all real times.