We recall standard results from ordinary differential equations and use this to define integral curves for a vector field on a manifold, and also the flow of a vector field. In the case where the manifold is compact, the flow of a vector field $X$ on $M$ is a one-parameter family of diffeomorphisms $ t \mapsto \theta_t^X \in \mathrm{Diff}(M)$.

We then use the flow of a vector field to define the Lie derivative $ \mathcal{L}_X$ of a vector field, first as an operator on functions, and then as an operator on vector fields. We conclude by showing that the Lie derivative is equal to the Lie bracket: $ \mathcal{L}_X Y = [X ,Y]$, which thus explains the name.

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