Let $\Lambda_c^m(\R^n)$ denote the space of $C^\infty$ differentiable $m$ -forms with compact support in $\R^n$ . A continuous linear operator $T\colon \Lambda_c^m(\R^n)\to \R$ is called an $m$ -current. Let $\mathcal D_m$ denote the space of $m$ -currents in $\R^n$ . We define a boundary operator $\partial\colon \mathcal D_{m+1}\to \mathcal D_m$ by $$ \partial T(\omega) := T(d\omega). $$

We will see that currents represent a generalization of $m$ -surfaces. In fact if $M$ is a compact $m$ -dimensional oriented manifold with boundary, we can associate to $M$ the current $[[M]]$ defined by $$ [[M]](\omega)=\int_M \omega. $$ So the definition of boundary $\partial T$ of a current, is justified by Stokes Theorem: $$ \int_{\partial M} \omega = \int_M d\omega. $$

The space $\mathcal D_m$ of $m$ -dimensional currents is a real vector space with operations defined by $$ (T+S)(\omega):= T(\omega)+S(\omega),\qquad (\lambda T)(\omega):=\lambda T(\omega). $$ 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 support of a current $T$ , denoted by $\mathrm{spt}(T)$ , the smallest closed set $C$ such that $$ T(\omega)=0\ \text{whenever $\omega=0$ on $C$}. $$ We denote with $\mathcal E_m$ the vector subspace of $\mathcal D_m$ of currents with compact support.

Topology

A stronger norm on the space of currents is the mass norm. First of all we define the mass norm of a $m$ -form $\omega$ as $$ ||\omega||:= \sup\{|\langle \omega,\xi\rangle|\colon \text{$\xi$ is a unit, simple, $m$-vector}\}. $$ So if $\omega$ 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 $$ \mathbf M (T) := \sup\{ T(\omega)\colon \sup_x ||\omega(x)||\le 1\}. $$ The mass of a currents represents the area of the generalized surface.

An intermediate norm, is the flat norm defined by $$ \mathbf F (T) := \inf \{\mathbf M(A) + \mathbf M(B) \colon T= A + \partial B,\ A\in\mathcal E_m,\ B\in\mathcal E_{m+1}\}. $$

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.

Examples

Let $(x,y,z)$ be the coordinates in $\R^3$ . Then the following defines a $2$ -current: $$ T(a\,dx\wedge dy + b\,dy\wedge dz + c\,dx\wedge dz) = \int_0^1 \int_0^1 b(x,y,0)\, dx \, dy. $$