= Variation of metric volume density
{title2=$\delta\sqrt{|\det g|}=\frac12\sqrt{|\det g|}\,g^{ab}\delta g_{ab}$}
For a <nondegenerate> <metric tensor> whose signature is fixed during the <variation>, the <Jacobi formula> gives $\delta\det g=(\det g)g^{ab}\delta g_{ab}$. Differentiating the square root of the absolute <determinant> proves the displayed identity. Expressed through the <variation of inverse metric>, the formula is $-\sqrt{|\det g|}\,g_{ab}\delta g^{ab}/2$. Absolute values accommodate either Riemannian or Lorentzian signature.
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