Abstract
The notion of a distribution, which we studied on open subsets of ℝn in [P2], can be defined in the same way on a locally compact manifold B that is countable at infinity. The notion of a current, due to de Rham and further specified by L. Schwartz, generalized the notion of a distribution: while the Dirac distribution represents the electric charge of a sphere as the radius becomes infinitely small, a current represents the electric current density in a conductive wire as the cross-section becomes infinitely small. Point distributions (such as the Dirac distribution δb : φ ↦ φ (b) and its derivatives, where φ∈ℰB) are defined on Banach manifolds and can be identified with scalar differential operators.
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