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. 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. 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. Thus a linear functional is an m-dimensional current if it is continuous in the following sense: If a sequence ω k {displaystyle omega _{k}} of smooth forms, all supported in the same compact set, is such that all derivatives of all their coefficients tend uniformly to 0 when k {displaystyle k} tends to infinity, then T ( ω k ) {displaystyle T(omega _{k})} tends to 0. The space D m ( M ) {displaystyle {mathcal {D}}_{m}(M)} of m-dimensional currents on M {displaystyle M} is a real vector space with operations defined by Much of the theory of distributions carries over to currents with minimal adjustments. For example, one may define the support of a current T ∈ D m ( M ) {displaystyle Tin {mathcal {D}}_{m}(M)} as the complement of the biggest open set U ⊂ M {displaystyle Usubset M} such that The linear subspace of D m ( M ) {displaystyle {mathcal {D}}_{m}(M)} consisting of currents with support (in the sense above) that is a compact subset of M {displaystyle M} is denoted E m ( M ) {displaystyle {mathcal {E}}_{m}(M)} . Integration over a compact rectifiable oriented submanifold M (with boundary) of dimension m defines an m-current, denoted by [ [ M ] ] {displaystyle ]} : If the boundary ∂M of M is rectifiable, then it too defines a current by integration, and by virtue of Stokes' theorem one has: This relates the exterior derivative d with the boundary operator ∂ on the homology of M.