Riemannian volume form on an oriented manifold

On an oriented Riemannian $n$-manifold, the Riemannian volume form is ${\mathrm{vol}}_g=\sqrt{\det G_x}d{x}^1\wedge \cdots \wedge d{x}^n$ in positively oriented charts for $n\ge 1$. For $n=0$ it is the supplied orientation sign $\epsilon (p)\in \{1,-1\}$ at each point.

Oriented smooth manifolds and oriented charts supplies the orientation. On positive-chart overlaps the Jacobian determinant is positive, so the density calculation in The riemannian volume density is coordinate independent is also the top-form transformation law. Thus the formula glues, and $\mathrm{\mid }{\mathrm{vol}}_g\mathrm{\mid }={\mu }_g$. Reversing orientation negates the form but leaves the density unchanged, also in dimension zero.

Lee, Propositions 15.29–15.33 and Corollary 15.34, pp.389–391; density construction and integration pp.428–433.