“Homological integration”的意思、由来-开放百科全书

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Homological integration

释义

References

{{about|an extension of the theory of the Lebesgue integral to manifolds|numerical method|geometric integrator}}

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 {{math|D^k}} of {{mvar|k}}-currents on a manifold {{mvar|M}} is defined as the dual space, in the sense of distributions, of the space of {{mvar|k}}-forms {{math|Ω^k}} on {{mvar|M}}. Thus there is a pairing between {{mvar|k}}-currents {{mvar|T}} and {{mvar|k}}-forms {{mvar|α}}, denoted here by

Under this duality pairing, the exterior derivative

goes over to a boundary operator

defined by

for all {{math|α ∈ Ω^k}}. This is a homological rather than cohomological construction.

References

{{citation

|last = Federer
|first = Herbert
|authorlink = Herbert Federer
|title = Geometric measure theory
|publisher = Springer-Verlag New York Inc.
|location = New York
|year = 1969
|pages = xiv+676
|isbn = 978-3-540-60656-7
|series = Die Grundlehren der mathematischen Wissenschaften
|volume = 153
|mr=0257325
|zbl=0176.00801}}.

{{citation

}}.

2 : Definitions of mathematical integration|Measure theory

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