Flow of a vector field ($\frac{d}{dt}{\Phi }_t(p)=V({\Phi }_t(p))$)

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.