About: Density on a manifold

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In mathematics, and specifically differential geometry, a density is a spatially varying quantity on a differentiable manifold that can be integrated in an intrinsic manner. Abstractly, a density is a section of a certain line bundle, called the density bundle. An element of the density bundle at x is a function that assigns a volume for the parallelotope spanned by the n given tangent vectors at x.

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rdfs:label	Dichtebündel (de) Density on a manifold (en) Densité sur une variété (fr) 밀도 다발 (ko)
rdfs:comment	Ein Dichtebündel ist ein Spezialfall eines Vektorbündels und wird im mathematischen Teilgebiet der Differentialgeometrie untersucht. Mit Hilfe dieser Bündel kann man einige aus der Analysis bekannte Objekte auf Mannigfaltigkeiten verallgemeinern. So kann man ähnlich wie mit Differentialformen einen Koordinaten-invarianten Integralbegriff auf Mannigfaltigkeiten definieren. Man findet mit Hilfe dieser Bündel Verallgemeinerungen der Lp-Räume und der Distributionenräume auf Mannigfaltigkeiten. (de) En géométrie différentielle, une densité est une notion qui sert à définir une intégrale indépendante de toute orientation. Ce faisant, elle sert d'abord à pouvoir intégrer sur une variété différentielle qui n'est pas orientable. Ensuite, la notion de densité sert aussi à définir une mesure positive sur une variété différentielle et, par conséquent, à pouvoir parler de densité de probabilité sur une variété différentielle. (fr) 미분기하학에서 밀도 다발(密度-, 영어: density bundle)은 적분을 정의할 수 있는 단면들을 갖는 선다발이다. (ko) In mathematics, and specifically differential geometry, a density is a spatially varying quantity on a differentiable manifold that can be integrated in an intrinsic manner. Abstractly, a density is a section of a certain line bundle, called the density bundle. An element of the density bundle at x is a function that assigns a volume for the parallelotope spanned by the n given tangent vectors at x. (en)
dct:subject	Differential geometry Manifolds
Wikipage page ID	3428935 (xsd:integer)
Wikipage revision ID	1053899369 (xsd:integer)
Link from a Wikipage to another Wikipage	Coordinate chart Riesz-Markov-Kakutani representation theorem Pseudotensor Differential geometry Compact support Mathematics General linear group Conformal geometry Lp space Partition of unity Lebesgue measure Line bundle Manifolds Parallelepiped Frame bundle Tensor density Radon measure Jacobian determinant Tensor product Atlas (topology) Differentiable manifold Differential form Differential geometry Distribution (mathematics) Associated bundle Group representation Integral Integration by substitution Section (fiber bundle) Vector bundle Dual vector bundle Springer-Verlag Oriented manifold Local trivialization
sameAs	Density on a manifold Density on a manifold Density on a manifold Density on a manifold Density on a manifold Density on a manifold Density on a manifold
dbp:wikiPageUsesTemplate	dbt:Citation dbt:Harv dbt:Short_description dbt:Sup dbt:Abs dbt:Manifolds
has abstract	Ein Dichtebündel ist ein Spezialfall eines Vektorbündels und wird im mathematischen Teilgebiet der Differentialgeometrie untersucht. Mit Hilfe dieser Bündel kann man einige aus der Analysis bekannte Objekte auf Mannigfaltigkeiten verallgemeinern. So kann man ähnlich wie mit Differentialformen einen Koordinaten-invarianten Integralbegriff auf Mannigfaltigkeiten definieren. Man findet mit Hilfe dieser Bündel Verallgemeinerungen der Lp-Räume und der Distributionenräume auf Mannigfaltigkeiten. (de) In mathematics, and specifically differential geometry, a density is a spatially varying quantity on a differentiable manifold that can be integrated in an intrinsic manner. Abstractly, a density is a section of a certain line bundle, called the density bundle. An element of the density bundle at x is a function that assigns a volume for the parallelotope spanned by the n given tangent vectors at x. From the operational point of view, a density is a collection of functions on coordinate charts which become multiplied by the absolute value of the Jacobian determinant in the change of coordinates. Densities can be generalized into s-densities, whose coordinate representations become multiplied by the s-th power of the absolute value of the jacobian determinant. On an oriented manifold, 1-densities can be canonically identified with the n-forms on M. On non-orientable manifolds this identification cannot be made, since the density bundle is the tensor product of the orientation bundle of M and the n-th exterior product bundle of T∗M (see pseudotensor). (en) En géométrie différentielle, une densité est une notion qui sert à définir une intégrale indépendante de toute orientation. Ce faisant, elle sert d'abord à pouvoir intégrer sur une variété différentielle qui n'est pas orientable. Ensuite, la notion de densité sert aussi à définir une mesure positive sur une variété différentielle et, par conséquent, à pouvoir parler de densité de probabilité sur une variété différentielle. (fr) 미분기하학에서 밀도 다발(密度-, 영어: density bundle)은 적분을 정의할 수 있는 단면들을 갖는 선다발이다. (ko)
prov:wasDerivedFrom	wikipedia-en:Density_on_a_manifold?oldid=1053899369&ns=0
page length (characters) of wiki page	9378 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Density_on_a_manifold
is rdfs:seeAlso of	Volume form
is Link from a Wikipage to another Wikipage of	Volume element Valuation (geometry) Volume Volume form Pseudotensor Lagrangian (field theory) Lp space Calculus on Euclidean space Density (disambiguation) GJMS operator Densities on a manifold Tensor field Tensor Paneitz operator Laplace operator Divergence Density bundle
is Wikipage redirect of	Densities on a manifold Density bundle
is Wikipage disambiguates of	Density (disambiguation)

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