 Home

Codex Wiki

List of universities

University of Cambridge

Course of the University of Cambridge

Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2024

Paper 115

The metric induces an inner product on decomposable $r$-forms by
$$
\langle\alpha_1\wedge\cdots\wedge\alpha_r,
\beta_1\wedge\cdots\wedge\beta_r\rangle
=\det(\langle\alpha_i,\beta_j\rangle),
$$
extended bilinearly. On an oriented Riemannian manifold, the <Hodge star operator> is characterized by
$$
\alpha\wedge *\beta=\langle\alpha,\beta\rangle\,d\operatorname{vol}_g.
$$
On $r$-forms in dimension $n$,
$$
\boxed{*^2=(-1)^{r(n-r)}.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 115 / 4 / d / Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 115 / 4 / d / Solution

 Ancestors (11)

 Discussion (0)

 Articles by others on the same topic (0)

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 115 / 4 / d / Solution

 Ancestors (11)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)