Riemannian volume form
= Riemannian volume form
{c}
{title2=$d\operatorname{vol}_g$}
{wiki=Volume_form#Riemannian_volume_form}
On an oriented Riemannian $n$-manifold, the volume form is the unique positive $n$-form taking value one on every positively oriented orthonormal frame. In positive local coordinates,
$$
d\operatorname{vol}_g=\sqrt{\det(g_{ij})}\,dx^1\wedge\cdots\wedge dx^n.
$$