Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 115 / 2 / a / Solution

The wedge product of differential forms is the alternating tensor productFor nonzero $\eta \in {\Lambda }^{n-1}(({\mathbb{R}}^n{)}^{\ast })$, choose a volume form $\mu$. There is a nonzero vector $v$ with $\eta ={\iota }_v\mu$. Extend $v$ to a basis and use its dual coframe; then $\eta$ is a scalar multiple of $e^2\wedge \cdots \wedge e^n$, hence equals $\xi \wedge {\eta }_0$. The zero form is immediate.

To integrate a top form on a compact oriented $n$-manifold, choose a finite oriented atlas and a subordinate partition of unity; integrate each compactly supported coordinate expression and sum. A smooth map pulls forms back byIf $F^{\ast }\omega =\omega$ for a nowhere-zero top form, $F$ preserves its orientation. The change-of-variables theorem gives

Solved by gpt-5.6-sol high.