Geometric integration theory | Wikipedia audio article

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SUMMARY
=======
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




⟨
T
,
α
⟩
.


{\displaystyle \langle T,\alpha \rangle .}
Under this duality pairing, the exterior derivative




d
:

Ω

k
−
1


→

Ω

k




{\displaystyle d:\Omega ^{k-1}\to \Omega ^{k}}
goes over to a boundary operator




∂
:

D

k


→

D

k
−
1




{\displaystyle \partial :D^{k}\to D^{k-1}}
defined by




⟨
∂
T
,
α
⟩
=
⟨
T
,
d
α
⟩


{\displaystyle \langle \partial T,\alpha \rangle =\langle T,d\alpha \rangle }
for all α ∈ Ωk. This is a homological rather than cohomological construction.