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.
Definition
Let
denote the space of smooth m-forms with compact support on a smooth manifold
. A current is a linear functional on
which is continuous in the sense of distributions. Thus a linear functional
is an m-current if it is continuous in the following sense: If a sequence
of smooth forms, all supported in the same compact set, is such that all derivatives of all their coefficients tend uniformly to 0 when
tends to infinity, then
tends to 0.