# Definition:Riemannian Volume Form/Definition 3

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## 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:

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

## Sources

- 2018: John M. Lee:
*Introduction to Riemannian Manifolds*(2nd ed.): $\S 2$: Riemannian Metrics. Basic Constructions on Riemannian Manifolds