Current (mathematics)

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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and For functional analysis as used in psychology see the Functional analysis (psychology article In Mathematics, differential topology is the field dealing with differentiable functions on Differentiable manifolds It is closely related to Differential In Mathematics, geometric measure theory ( GMT) is the study of the geometric properties of the measures of sets (typically in Georges de Rham ( 10 September 1903 &ndash 9 October 1990) was a Swiss Mathematician, known for his contributions to In Mathematics, the support of a function is the set of points where the function is not zero or the closure of that set In the mathematical fields of Differential geometry and Tensor calculus, differential forms are an approach to Multivariable calculus which is A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. Formally currents behave like Schwartz distributions on a space of differential forms. In Mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and Probability distributions 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. The Dirac delta or Dirac's delta is a mathematical construct introduced by the British theoretical physicist Paul Dirac. Multipole moments are the Coefficients of a Series expansion of a Potential due to continuous or discrete sources (e In Mathematics, the directional derivative of a multivariate Differentiable function along a given vector V at a given point P intuitively represents the

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. In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that 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. In Mathematics, an operator is a function which operates on (or modifies another function In Differential geometry, Stokes' theorem is a statement about the integration of Differential forms which generalizes several Theorems from 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. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be In Mathematics, the real numbers may be described informally in several different ways In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Linear algebra, Real numbers are called Scalars and relate to vectors in a Vector space through the operation of Scalar multiplication 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

$\mathrm{spt}(\omega) \subset U \,$ . In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in

We denote with $\mathcal E_m$ the vector subspace of $\mathcal D_m$ of currents with compact support. The concept of a linear subspace (or vector subspace) is important in Linear algebra and related fields of Mathematics.

Topology

The space of currents is naturally endowed with the weak-star topology, which will be further simply called weak convergence. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Mathematics, weak convergence may refer to The weak Convergence of random variables of a Probability distribution. 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. In Mathematics, a sequence is an ordered list of objects (or events In Mathematics, informally speaking a limit point of a set S in a Topological space X is a point x in X that can be "approximated" In Linear algebra, Functional analysis and related areas of Mathematics, a norm is a function that assigns a strictly positive length 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. In Mathematics, specifically in Ring theory, an algebra is simple if it contains no non-trivial two-sided ideals and the set { 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.

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 regular measure $μ$ is a 0-current:

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

Let (x, y, z) be the coordinates in R³. In Mathematics, signed measure is a generalization of the concept of measure by allowing it to have negative values In Mathematics, a regular measure on a Topological space is a measure for which every Measurable set is "approximately open" and "approximately Then the following defines a 2-current (one of many):

$T(a\,dx\wedge dy + b\,dy\wedge dz + c\,dx\wedge dz) = \int_0^1 \int_0^1 b(x,y,0)\, dx \, dy.$

This article incorporates material from Current on PlanetMath, which is licensed under the GFDL. PlanetMath is a free, collaborative online Mathematics Encyclopedia.