\begin{definition}[Definition:Riemannian Volume Form/Definition 1]
Let $\struct {M, g}$ be an oriented $n$-dimensional Riemannian manifold.
Let $T^* M$ be the cotangent bundle of $M$.
Let $\tuple {\epsilon^1, \ldots, \epsilon^n}$ be a local oriented orthonormal coframe of $T^* M$.
The '''Riemannian volume form''', denoted by $\rd V_g$, is an $n$-form such that:
:$\rd V_g = \epsilon^1 \wedge \ldots \wedge \epsilon^n$
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{{Research|orientation, form, coframes}}
\end{definition}