A <diffeomorphism> $\phi:M\to M$ is a smooth bijection with smooth inverse. It acts covariantly on a differential form by the <pullback of a differential form>:
$$
\omega\longmapsto\phi^*\omega.
$$
For vector fields $X_0,\ldots,X_p$,
$$
(\phi^*\omega)_x(X_1,\ldots,X_p)
=\omega_{\phi(x)}(d\phi_xX_1,\ldots,d\phi_xX_p).
$$
The coordinate definition of the <exterior derivative>, or its characterization as the unique natural graded derivation extending the differential of functions, gives
$$
d(\phi^*\omega)=\phi^*(d\omega).
$$

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 309 / 4 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-309/4/a/solution