It does not do much justice to give Lie derivative just one day, but it is nice to first meet it then slowly get famliar with it.

\gdef\cD{\mathcal D}

Let $M$ be a smooth manifold, $X$ be a vector field, $\Phi: \cD \to M$ is the flow, where $\cD \subset M \times \R$ is the open subset of the flow domain. For simplicity, assume $M$ is compact, and then $\cD = M \times \R$.