Solution
= Solution
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 $X$.