Current (mathematics)

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In mathematics, more particularly in functional analysis and differential topology, a current in the sense of Georges de Rham is a functional on the space of compactly supported differential forms, on a smooth manifold M. Formally currents behave like Schwartz distributions on a space of differential forms. In a geometric sense they can represent quite singular versions of submanifolds: Dirac delta functions or even multipoles (directional derivatives of delta functions) spread out along subsets of M. A general form of Stokes theorem can be proved for currents.

Let \Lambda_c^m(\mathbb{R}^n) denote the space smooth m-forms with compact support in Rn. A continuous linear operator

T\colon \Lambda_c^m(\mathbb{R}^n)\to \mathbb{R}

is called an m-current. Let \mathcal D_m denote the space of m-currents in Rn. 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 represent. 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\,

whenever ω = 0 on C.

We denote with \mathcal E_m the vector subspace of \mathcal D_m of currents with compact support.

[edit] Topology

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

T_k(\omega) \to T(\omega),\qquad \forall \omega.\,

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

|\vert\omega|\vert:= \sup\{|\langle \omega,\xi\rangle|\colon\xi \mbox{ is a unit, simple, }m\mbox{-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

\mathbf M (T) := \sup\{ T(\omega)\colon \sup_x |\vert\omega(x)|\vert\le 1\}.

The mass of a current 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.

[edit] Examples

Recall that

\Lambda_c^0(\mathbb{R}^n)\equiv C^\infty_c(\mathbb{R}^n)\,

so that the following defines a 0-current:

T(f) = f(0).\,

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

T(f) = \int f(x)\, d\mu(x).

Let (x, y, z) be the coordinates in R3. 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.

This article incorporates material from Current on PlanetMath, which is licensed under the GFDL.