**Top-dimensional Form**

In mathematics, a **volume form** on a differentiable manifold is a nowhere-vanishing top-dimensionial form (i.e., a differential form of top degree). Thus on a manifold *M* of dimension *n*, a volume form is an *n*-form, a section of the line bundle Ω*n*(*M*) = Λ*n*(*T*∗*M*), that is nowhere equal to zero. A manifold has a volume form if and only if it is orientable. An orientable manifold has infinitely many volume forms, since multiplying a volume form by a non-vanishing function yields another volume form. On non-orientable manifolds, one may instead define the weaker notion of a density.

A volume form provides a means to define the integral of a function on a differentiable manifold. In other words, a volume form gives rise to a measure with respect to which functions can be integrated by the appropriate Lebesgue integral. The absolute value of a volume form is a volume element, which is also known variously as a *twisted volume form* or *pseudo-volume form*. It also defines a measure, but exists on any differentiable manifold, orientable or not.

Kähler manifolds, being complex manifolds, are naturally oriented, and so possess a volume form. More generally, the *n*th exterior power of the symplectic form on a symplectic manifold is a volume form. Many classes of manifolds have canonical volume forms: they have extra structure which allows the choice of a preferred volume form. Oriented Riemannian manifolds and pseudo-Riemannian manifolds have an associated canonical volume form.

Read more about Top-dimensional Form: Orientation, Relation To Measures, Divergence, Invariants of A Volume Form, See Also

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