General Relativity: An Introduction for Physicists · Problem 3.7

Vector calculus on manifolds — Hobson, Efstathiou & Lasenby

Physics · metric-determinant-derivative

Problem

Let

g

be the determinant of the matrix

[g_{ab}]

. By considering the cofactor of the element

g_{ab}

in this determinant, or otherwise, show that

\partial_c g = g g^{ab}(\partial_c g_{ab})