Let Λcm(n) denote the space of C differentiableMathworldPlanetmathPlanetmath m-forms with compact support in n. A continuous linear operator T:Λcm(n) is called an m-current. Let 𝒟m denote the space of m-currents in n. We define a boundary operatorMathworldPlanetmath :𝒟m+1𝒟m by


We will see that currents represent a generalizationPlanetmathPlanetmath of m-surfaces. In fact if M is a compactPlanetmathPlanetmath m-dimensional oriented manifold with boundary, we can associate to M the current [[M]] defined by


So the definition of boundary T of a current, is justified by Stokes Theorem:


The space 𝒟m of m-dimensional currents is a real vector space with operationsMathworldPlanetmath defined by


The sum of two currents represents the union of the surfaces they represents. Multiplication by a scalar represents a change in the multiplicity of the surface. In particular multiplication by -1 represents the change of orientation of the surface.

We define the supportMathworldPlanetmath of a current T, denoted by spt(T), the smallest closed setPlanetmathPlanetmath C such that

T(ω)=0whenever ω=0 on C.

We denote with m the vector subspace of 𝒟m of currents with compact support.


The space of currents is naturally endowed with the weak-star topologyMathworldPlanetmath, which will be further simply called weak convergence. We say that a sequencePlanetmathPlanetmath Tk of currents, weakly convergesPlanetmathPlanetmath to a current T if


A stronger norm on the space of currents is the mass norm. First of all we define the mass norm of a m-form ω as

||ω||:=sup{|ω,ξ|:ξ is a unit, simple, m-vector}.

So if ω is a simple m-form, then its mass norm is the usual norm of its coefficient. We hence define the mass of a current T as


The mass of a currents represents the area of the generalized surface.

An intermediate norm, is the flat norm defined by


Notice that two currents are close in the mass norm if they coincide apart from a small part. On the other hand the are close in the flat norm if they coincide up to a small deformation.


Recall that Λc0(n)Cc(n) so that the following defines a 0-current:


In particuar every signed measure μ with finite mass is a 0-current:


Let (x,y,z) be the coordinates in 3. Then the following defines a 2-current:

Title current
Canonical name Current
Date of creation 2013-03-22 14:27:39
Last modified on 2013-03-22 14:27:39
Owner paolini (1187)
Last modified by paolini (1187)
Numerical id 7
Author paolini (1187)
Entry type Definition
Classification msc 58A25
Defines mass
Defines support