Current (mathematics)

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In mathematics, more particularly in functional analysis,differential topology, and geometric measure theory, 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.

Let $\Lambda_c^m(\mathbb{R}^n)$ denote the space of smooth m-forms with compact support on 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

$\partial\colon \mathcal D_{m+1}\to \mathcal D_m$

$\partial T(\omega) := T(d\omega).\,$

giving a general form of Stokes' theorem by definition. 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 on manifolds with boundary:

$\int_{\partial M} \omega = \int_M d\omega.\,$

The space $\mathcal D_m$ of m-dimensional currents is a real vector space with operations defined by

$(T+S)(\omega):= T(\omega)+S(\omega),\qquad (\lambda T)(\omega):=\lambda T(\omega).$

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

$\mathrm{spt}(T),\,$

is the complementary of the biggest open set U such that

$T(\omega) = 0\,$

whenever

$\mathrm{spt}(\omega) \subset U \,$ .

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 T_k 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 weighted 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.