Let \(\omega_g\) be the Riemannian volume form on a Riemannian manifold \((M,g)\). Then it has the local coordinate expression
\begin{equation*} \omega_g=\sqrt{\det(g_{ij})}\, dx^1\wedge \cdots \wedge dx^n. \end{equation*}Let \(\omega_g\) be the Riemannian volume form on a Riemannian manifold \((M,g)\). Then it has the local coordinate expression
\begin{equation*} \omega_g=\sqrt{\det(g_{ij})}\, dx^1\wedge \cdots \wedge dx^n. \end{equation*}