About: Vector-valued differential form

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In mathematics, a vector-valued differential form on a manifold M is a differential form on M with values in a vector space V. More generally, it is a differential form with values in some vector bundle E over M. Ordinary differential forms can be viewed as R-valued differential forms. An important case of vector-valued differential forms are Lie algebra-valued forms. (A connection form is an example of such a form.)

Attributes	Values
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rdfs:label	Vektorwertige Differentialformen (de) 벡터 값 미분 형식 (ko) Vector-valued differential form (en) 向量值微分形式 (zh)
rdfs:comment	Als Vektorwertige Differentialformen bezeichnet man in der Mathematik eine Verallgemeinerung des Begriffs der Differentialformen auf Funktionen, die jedem Punkt einer Mannigfaltigkeit eine vektorwertige multilineare und alternierende Abbildungen zuordnen. Ein wichtiger Spezialfall bilden sogenannte Lie-Algebra-wertige Differentialformen, die zum Beispiel eine wichtige Anwendung in der Theorie der Zusammenhänge und Krümmung eines Hauptfaserbündels finden. Eine Verallgemeinerung des Konzepts der vektorwertigen Differentialformen sind . (de) In mathematics, a vector-valued differential form on a manifold M is a differential form on M with values in a vector space V. More generally, it is a differential form with values in some vector bundle E over M. Ordinary differential forms can be viewed as R-valued differential forms. An important case of vector-valued differential forms are Lie algebra-valued forms. (A connection form is an example of such a form.) (en) 미분기하학에서 벡터 값 미분 형식(vector값微分形式, 영어: vector-valued differential form)의 개념은 미분 형식의 개념의 일종의 일반화이다. 벡터 값 미분 형식은 미분 형식 다발과 임의의 벡터 다발과의 텐서곱 다발의 단면이며, 일종의 "뒤틀린 미분 형식"으로 여겨질 수 있다. 그 위에는 당김과 쐐기곱이 정의되지만, 일반적으로 외미분은 정의되지 않는다. (ko) 数学中，流形 M 上一个向量值微分形式（vector-valued differential form）是 M 上取值于一个向量空间 V 的微分形式。更一般地，它是取值于 M 上某个向量丛 E 的微分形式。通常的微分形式可以视为 R-值微分形式。向量值微分形式是微分几何中的自然对象并有广泛的应用。 (zh)
dcterms:subject	Differential forms Vector bundles
Wikipage page ID	3772116 (xsd:integer)
Wikipage revision ID	1045692679 (xsd:integer)
Link from a Wikipage to another Wikipage	Module (mathematics) Basis (linear algebra) Algebra bundle Algebra over a field Curvature form Vacuously true Vector space Connection form Mathematics Chern–Weil homomorphism Pullback (differential geometry) Pullback bundle Graded-commutative Connection (vector bundle) Differential forms Shoshichi Kobayashi Commutative algebra Vector bundles Wedge product Linearity Adjoint representation Exterior derivative Exterior power Differential operator Foundations of Differential Geometry Frame bundle Equivariant Ring (mathematics) Group action (mathematics) Cotangent bundle Covariant derivative Tensor product Tensor product of modules Katsumi Nomizu Tautological one-form Differential form Associated bundle Associative algebra Manifold Group representation Section (fiber bundle) Siegel modular form Skew-symmetric matrix Smooth manifold Vector bundle Principal bundle Vertical bundle Flat vector bundle Siegel modular variety Tensor product bundle Vector bundle morphism Associated vector bundle Lie algebra-valued forms Covariant exterior derivative Naturally isomorphic Wiley Interscience Smooth map Exterior covariant differentiation Local trivialization Trivial bundle Strong monoidal functor
sameAs	Vector-valued differential form Vector-valued differential form Vector-valued differential form Vector-valued differential form Vector-valued differential form Vector-valued differential form Vector-valued differential form
dbp:wikiPageUsesTemplate	dbt:Reflist
has abstract	Als Vektorwertige Differentialformen bezeichnet man in der Mathematik eine Verallgemeinerung des Begriffs der Differentialformen auf Funktionen, die jedem Punkt einer Mannigfaltigkeit eine vektorwertige multilineare und alternierende Abbildungen zuordnen. Ein wichtiger Spezialfall bilden sogenannte Lie-Algebra-wertige Differentialformen, die zum Beispiel eine wichtige Anwendung in der Theorie der Zusammenhänge und Krümmung eines Hauptfaserbündels finden. Eine Verallgemeinerung des Konzepts der vektorwertigen Differentialformen sind . (de) In mathematics, a vector-valued differential form on a manifold M is a differential form on M with values in a vector space V. More generally, it is a differential form with values in some vector bundle E over M. Ordinary differential forms can be viewed as R-valued differential forms. An important case of vector-valued differential forms are Lie algebra-valued forms. (A connection form is an example of such a form.) (en) 미분기하학에서 벡터 값 미분 형식(vector값微分形式, 영어: vector-valued differential form)의 개념은 미분 형식의 개념의 일종의 일반화이다. 벡터 값 미분 형식은 미분 형식 다발과 임의의 벡터 다발과의 텐서곱 다발의 단면이며, 일종의 "뒤틀린 미분 형식"으로 여겨질 수 있다. 그 위에는 당김과 쐐기곱이 정의되지만, 일반적으로 외미분은 정의되지 않는다. (ko) 数学中，流形 M 上一个向量值微分形式（vector-valued differential form）是 M 上取值于一个向量空间 V 的微分形式。更一般地，它是取值于 M 上某个向量丛 E 的微分形式。通常的微分形式可以视为 R-值微分形式。向量值微分形式是微分几何中的自然对象并有广泛的应用。 (zh)
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