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. Formally currents behave like Schwartz distributions on a space of differential forms. 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.

Let ${\displaystyle \Omega _{c}^{m}(M)}$ denote the space of smooth m-forms with compact support on a smooth manifold ${\displaystyle M}$. A current is a linear functional on ${\displaystyle \Omega _{c}^{m}(M)}$ which is continuous in the sense of distributions. Thus a linear functional

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