The determinant identity $\delta\log|\det g|=g^{ab}\delta g_{ab}$ implies
$$
\delta\sqrt{|g|}=\frac12\sqrt{|g|}\,g^{ab}\delta g_{ab}.
$$
Hence the metric <volume form> varies as
$$
\boxed{\delta(d\operatorname{vol}_g)
=\frac12g^{ab}\delta g_{ab}\,d\operatorname{vol}_g}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 309 / 2 / a / ii / Solution

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A volume form is a nowhere-vanishing differential form of top degree on an oriented manifold. A metric supplies the canonical local density $\sqrt{|\det g|}$ and hence a metric volume form after choosing orientation.


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