 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 309

Put $h_{ab}=\delta g_{ab}$. Varying $g^{ac}g_{cb}=\delta^a_b$ gives
$$
\boxed{\delta g^{ab}=-g^{ac}g^{bd}h_{cd}=-h^{ab}}.
$$
The determinant identity $\delta\log|\det g|=g^{ab}h_{ab}$ gives
$$
\boxed{\delta(d\operatorname{vol}_g)
=\frac12g^{ab}h_{ab}\,d\operatorname{vol}_g}.
$$
Finally, varying the formula for the <Christoffel symbols> and rewriting partial derivatives covariantly gives the tensor
$$
\boxed{\delta\Gamma^a{}_{bc}
=\frac12g^{ad}(\nabla_bh_{cd}+\nabla_ch_{bd}-\nabla_dh_{bc})}.
$$
These are the basic <metric variation> identities.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 309 / 3 / a / i / Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 309 / 3 / a / i / Solution

 Ancestors (12)

 Discussion (0)

 Articles by others on the same topic (0)

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 309 / 3 / a / i / Solution

 Ancestors (12)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)