## Volume in differential geometry::Volume

### ::concepts

**Volume**::**volume** Center::style Radius::geometry Sphere::''h'' Height::cubic Manifold::integral**Volume in differential geometry**
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In differential geometry, a branch of mathematics, a **volume form** on a differentiable manifold is a differential form of top degree (i.e. whose degree is equal to the dimension of the manifold) 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. Integrating the volume form gives the volume of the manifold according to that form.

Any oriented Riemannian (or pseudo-Riemannian) manifold has a natural volume (or pseudo volume) form. In local coordinates, it can be expressed as

- <math>\omega = \sqrt{|g|} dx^1\wedge \dots \wedge dx^n</math>

where the <math>dx^i</math> are the 1-forms providing an oriented basis for the cotangent bundle of the *n*-dimensional manifold. Here, <math>|g|</math> is the absolute value of the determinant of the matrix representation of the metric tensor on the manifold.

**Volume sections**

Intro Units Related terms Volume in calculus Volume formulas Volume formula derivations Volume in differential geometry Volume in thermodynamics See also References External links

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