Definition:Riemannian Volume Form/Definition 3

Definition
Let $\struct {M, g}$ be an oriented $n$-dimensional Riemannian manifold.

Let $\tuple {x_1, \ldots, x_n}$ be a set of local oriented coordinates.

Let $g_{ij}$ be a local form of metric $g$.

The Riemannian volume form, denoted by $\rd V_g$, is an $n$-form such that:


 * $\rd V_g = \sqrt {\det \paren {g_{ij}}} \rd x^1 \wedge \ldots \wedge \rd x^n$