For any differential form $\omega$, <Cartan's magic formula> gives
$$
\mathcal L_X\omega=d(\iota_X\omega)+\iota_X(d\omega).
$$
Using $d^2=0$,
$$
d\mathcal L_X\omega=d\iota_Xd\omega
=\mathcal L_Xd\omega.
$$
Thus the <Lie derivative> commutes with the <exterior derivative> on every differential form.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 309 / 4 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-309/4/a/solution