Let $\Phi_s$ be the local flow generated by the vector field $X$. For a covariant tensor, its <Lie derivative> is
$$
\boxed{\mathcal L_XT=\frac{d}{ds}\bigg|_{s=0}\Phi_s^*T}.
$$
For 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 $X$.

Solution

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

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