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.
Read more about this topic: Current (mathematics)
Famous quotes containing the word theory:
“every subjective phenomenon is essentially connected with a single point of view, and it seems inevitable that an objective, physical theory will abandon that point of view.”
—Thomas Nagel (b. 1938)