Definition:Riemannian Volume Form/Definition 2

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

Let $T M$ be the tangent bundle of $M$.

Let $\tuple {E_1, \ldots, E_n}$ be a local oriented orthonormal frame of $T M$.

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

$\map {\rd V_g} {E_1, \ldots, E_n} = 1$