In mathematics, particularly in differential geometry and multivariable calculus, a volume form is a differential form that provides a way to define volume on a manifold. It is a useful concept in areas such as integration on manifolds and the study of geometric structures. ### Definition 1. **Differential Forms**: In the context of manifolds, a differential form of degree \( n \) on an \( n \)-dimensional manifold represents an infinitesimal volume element.
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A volume form is a nowhere-vanishing differential form of top degree on an oriented manifold. A metric supplies the canonical local density and hence a metric volume form after choosing orientation.