Current (mathematics)
In mathematics, more particularly in functional analysis, differential topology, and geometric measure theory, a k-current in the sense of Georges de Rham is a functional on the space of compactly supported differential k-forms, on a smooth manifold M. Currents formally behave like Schwartz distributions on a space of differential forms, but in a geometric setting, they can represent integration over a submanifold, generalizing the Dirac delta function, or more generally even directional derivatives of delta functions (multipoles) spread out along subsets of M. == Definition == Let Ω c m ( M ) {\displaystyle \Omega _{c}^{m}(M)} denote the space of smooth m-forms with compact support on a smooth manifold M . {\displaystyle M.} A current is a linear functional on Ω c m ( M ) {\displaystyle \Omega _{c}^{m}(M)} which is continuous in the sense of distributions.