In this lecture we make contact with the theory of ordinary differential equations. A vector field defines an ordinary differential equation on a manifold, and just as in the Euclidean case, solutions to this ordinary differential equation exist (at least for a short time) and are unique.

This gives rise to the notion of an integral curve for a vector field on a manifold, and hence also the flow of a vector field. In the case where the manifold is compact, we prove that the flow of a vector field $X$ on $M$ is a one-parameter family of diffeomorphisms $t \mapsto \Phi_t \in \mathrm{Diff}(M)$.