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In the mathematical fields of differential geometry and geometric measure theory, homological integration or geometric integration is a method for extending the notion of the integral to manifolds. Rather than functions or differential forms, the integral is defined over currents on a manifold. Under this duality pairing, the exterior derivative goes over to a boundary operator defined by for all α ∈ Ωk. This is a homological rather than cohomological construction.

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rdfs:label	Homological integration (en)
rdfs:comment	In the mathematical fields of differential geometry and geometric measure theory, homological integration or geometric integration is a method for extending the notion of the integral to manifolds. Rather than functions or differential forms, the integral is defined over currents on a manifold. Under this duality pairing, the exterior derivative goes over to a boundary operator defined by for all α ∈ Ωk. This is a homological rather than cohomological construction. (en)
dcterms:subject	Definitions of mathematical integration Measure theory
Wikipage page ID	23029956 (xsd:integer)
Wikipage revision ID	764394117 (xsd:integer)
Link from a Wikipage to another Wikipage	Princeton University Press Boundary operator Current (mathematics) Mathematics Geometric measure theory Definitions of mathematical integration Dual space Exterior derivative Oxford University Press Measure theory Differential form Differential geometry Distribution (mathematics) Manifold Integral Lebesgue integral Cohomology theory
sameAs	Homological integration Homological integration Homological integration Homological integration
dbp:wikiPageUsesTemplate	dbt:About dbt:Citation dbt:Geometry-stub dbt:Math dbt:Mvar
has abstract	In the mathematical fields of differential geometry and geometric measure theory, homological integration or geometric integration is a method for extending the notion of the integral to manifolds. Rather than functions or differential forms, the integral is defined over currents on a manifold. The theory is "homological" because currents themselves are defined by duality with differential forms. To wit, the space Dk of k-currents on a manifold M is defined as the dual space, in the sense of distributions, of the space of k-forms Ωk on M. Thus there is a pairing between k-currents T and k-forms α, denoted here by Under this duality pairing, the exterior derivative goes over to a boundary operator defined by for all α ∈ Ωk. This is a homological rather than cohomological construction. (en)
prov:wasDerivedFrom	wikipedia-en:Homological_integration?oldid=764394117&ns=0
page length (characters) of wiki page	2189 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Homological_integration
is Link from a Wikipage to another Wikipage of	Current (mathematics) Geometric integration Geometric integration theory
is Wikipage redirect of	Geometric integration theory
is foaf:primaryTopic of	wikipedia-en:Homological_integration

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