Volume in differential geometry::Volume


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Volume in differential geometry {{#invoke:main|main}}

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  

Volume in differential geometry