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- Volume_form abstract "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 Ωn(M) = Λn(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 nth 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.".
- Volume_form wikiPageID "1855811".
- Volume_form wikiPageRevisionID "596750896".
- Volume_form hasPhotoCollection Volume_form.
- Volume_form subject Category:Determinants.
- Volume_form subject Category:Differential_forms.
- Volume_form subject Category:Integration_on_manifolds.
- Volume_form type Abstraction100002137.
- Volume_form type Cognition100023271.
- Volume_form type CognitiveFactor105686481.
- Volume_form type Determinant105692419.
- Volume_form type Determinants.
- Volume_form type DifferentialForms.
- Volume_form type Form106290637.
- Volume_form type LanguageUnit106284225.
- Volume_form type Part113809207.
- Volume_form type PsychologicalFeature100023100.
- Volume_form type Relation100031921.
- Volume_form type Word106286395.
- Volume_form comment "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 Ωn(M) = Λn(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.".
- Volume_form label "Forma di volume".
- Volume_form label "Forme volume".
- Volume_form label "Volume form".
- Volume_form label "Volumenform".
- Volume_form label "体积形式".
- Volume_form sameAs Volumenform.
- Volume_form sameAs Forme_volume.
- Volume_form sameAs Forma_di_volume.
- Volume_form sameAs m.061cg2.
- Volume_form sameAs Q781415.
- Volume_form sameAs Q781415.
- Volume_form sameAs Volume_form.
- Volume_form wasDerivedFrom Volume_form?oldid=596750896.
- Volume_form isPrimaryTopicOf Volume_form.