Current (mathematics)
In mathematics, more particularly in functional analysis, differential topology, and geometric measure theory, a k-current in the sense of Georges de Rham is a functional on the space of compactly supported differential k-forms, on a smooth manifold M. Currents formally behave like Schwartz distributions on a space of differential forms, but in a geometric setting, they can represent integration over a submanifold, generalizing the Dirac delta function, or more generally even directional derivatives of delta functions (multipoles) spread out along subsets of M. Letdenote the space of smooth m-forms with compact support on a smooth manifoldA current is a linear functional onwhich is continuous in the sense of distributions.is an m-dimensional current if it is continuous in the following sense: If a sequenceof smooth forms, all supported in the same compact set, is such that all derivatives of all their coefficients tend uniformly to 0 whenis a real vector space with operations defined byMuch of the theory of distributions carries over to currents with minimal adjustments.For example, one may define the support of a currentas the complement of the biggest open setconsisting of currents with support (in the sense above) that is a compact subset ofIntegration over a compact rectifiable oriented submanifold M (with boundary) of dimension m defines an m-current, denoted byIf the boundary ∂M of M is rectifiable, then it too defines a current by integration, and by virtue of Stokes' theorem one has:This relates the exterior derivative d with the boundary operator ∂ on the homology of M. In view of this formula we can define a boundary operator on arbitrary currentsvia duality with the exterior derivative byfor all compactly supported m-formsCertain subclasses of currents which are closed undercan be used instead of all currents to create a homology theory, which can satisfy the Eilenberg–Steenrod axioms in certain cases.A classical example is the subclass of integral currents on Lipschitz neighborhood retracts.The space of currents is naturally endowed with the weak-* topology, which will be further simply called weak convergence.It is possible to define several norms on subspaces of the space of all currents.is an m-form, then define its comass byis a simple m-form, then its mass norm is the usual L∞-norm of its coefficient.The mass of a current represents the weighted area of the generalized surface.A current such that M(T) < ∞ is representable by integration of a regular Borel measure by a version of the Riesz representation theorem.This is the starting point of homological integration.Two currents are close in the mass norm if they coincide away from a small part.On the other hand, they are close in the flat norm if they coincide up to a small deformation.In particular every signed regular measureThis article incorporates material from Current on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.