For one-forms, the defining pullback or the identity
$$
(\mathcal L_X\alpha)(Y)=X[\alpha(Y)]-\alpha([X,Y])
$$
gives
$$
(\mathcal L_X\alpha)_\mu=X^\rho\alpha_{\mu,\rho}
+\alpha_\rho X^\rho{}_{,\mu}.
$$
Applying the Leibniz rule to both covariant slots of a type-$(0,2)$ tensor yields
$$
\boxed{(\mathcal L_XT)_{\mu\nu}
=X^\rho T_{\mu\nu,\rho}
+T_{\rho\nu}X^\rho{}_{,\mu}
+T_{\mu\rho}X^\rho{}_{,\nu}}.
$$


Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-309/1/a/iv/solution