Definition:Riemannian Volume Form/Definition 3

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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 $g_{i j}$ be a local form of metric $g$.

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

$\d V_g = \sqrt {\map \det {g_{i j} } } \rd x^1 \wedge \ldots \wedge \rd x^n$