Homological Theory
Integration over a compact rectifiable oriented submanifold M (with boundary) of dimension m defines an m-current, denoted by :
If the boundary ∂M of M is rectifiable, then it too defines a current by integration, and one has Stokes' theorem:
This relates the exterior derivative d with the boundary operator ∂ on the homology of M.
More generally, a boundary operator can be defined on arbitrary currents
by dualizing the exterior derivative:
for all compactly supported (m−1)-forms ω.
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