Du Bois singularity

2013-01-07T07:37:50Z

89.244.108.152: added <math> instead of ''foo''

{{about|an extension of the theory of the [[Lebesgue integral]] to [[manifold]]s|numerical method|geometric integrator}}
In the [[mathematics|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 [[manifold]]s. Rather than functions or [[differential form]]s, the integral is defined over [[current (mathematics)|currents]] on a manifold.

The theory is "homological" because currents themselves are defined by duality with differential forms. To wit, the space ''D''''k'' of ''k''-currents on a manifold ''M'' is defined as the [[dual space]], in the sense of [[distribution (mathematics)|distributions]], of the space of ''k''-forms Ω''k'' on ''M''. Thus there is a pairing between ''k''-currents ''T'' and ''k''-forms α, denoted here by
:<math>\langle T, \alpha\rangle.</math>
Under this duality pairing, the [[exterior derivative]]
:<math>d : \Omega^{k-1} \to \Omega^k</math>
goes over to a [[boundary operator]]
:<math>\partial : D^k \to D^{k-1} </math>
defined by
:<math>\langle\partial T,\alpha\rangle = \langle T, d\alpha\rangle</math>
for all α ∈ Ω''k''. This is a homological rather than [[cohomology theory|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
| id= {{MathSciNet|id=0257325}}
| series = series Die Grundlehren der mathematischen Wissenschaften
| volume = 153 }}
* {{citation
|first=H.
|last=Whitney
|author-link=Hassler Whitney
|title=Geometric Integration Theory
|series=Princeton Mathematical Series
|volume=21
|publisher=[[Princeton University Press]] and [[Oxford University Press]]
|place=Princeton, NJ and London
|year=1957
|pages= XV+387
|mr=0087148
|zbl=0083.28204
}}.

[[Category:Definitions of mathematical integration]]
[[Category:Measure theory]]

{{geometry-stub}}

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Du Bois singularity