Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 309 / 1 / a / i / Solution

Let ${\Phi }_s$ be the local flow generated by the vector field $X$. For a covariant tensor, its Lie derivative isFor contravariant indices one equivalently uses the differential of the inverse flow, and the construction extends to every tensor product by the Leibniz rule. It measures the infinitesimal change of $T$ under transport by the flow of $X$.