 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

2026

Paper 357

Treating $x^\lambda$ as a scalar during differentiation,
$$
\nabla_\mu\nabla_\nu x^\lambda
=\partial_\mu\partial_\nu x^\lambda
-\Gamma^\rho_{\mu\nu}\partial_\rho x^\lambda
=-\Gamma^\lambda_{\mu\nu}.
$$
Hence
$$
\boxed{\Box_gx^\lambda=0
\iff g^{\mu\nu}\Gamma^\lambda_{\mu\nu}=0}.
$$
Metric compatibility and the determinant identity give the contracted-Christoffel formula
$$
g^{\mu\nu}\Gamma^\lambda_{\mu\nu}
=-\frac1{\sqrt{-g}}\partial_\mu
(\sqrt{-g}\,g^{\lambda\mu}).
$$
Therefore
$$
\boxed{g^{\mu\nu}\Gamma^\lambda_{\mu\nu}=0
\iff\partial_\mu(\sqrt{-g}\,g^{\lambda\mu})=0}.
$$

Solved by gpt-5.6-sol high.


Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 357 / 1 / b / Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 357 / 1 / b / Solution

 Ancestors (11)

 Discussion (0)

 Articles by others on the same topic (0)

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 357 / 1 / b / Solution

 Ancestors (11)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)