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. Let
denote the space of smooth m-forms with compact support on a smooth manifold
A current is a linear functional on
which is continuous in the sense of distributions.
is an m-dimensional current if it is continuous in the following sense: If a sequence
of smooth forms, all supported in the same compact set, is such that all derivatives of all their coefficients tend uniformly to 0 when
is a real vector space with operations defined by
Much of the theory of distributions carries over to currents with minimal adjustments.
For example, one may define the support of a current
as the complement of the biggest open set
consisting of currents with support (in the sense above) that is a compact subset of
Integration over a compact rectifiable oriented submanifold M (with boundary) of dimension m defines an m-current, denoted by
If 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 currents
via duality with the exterior derivative by
for all compactly supported m-forms
Certain subclasses of currents which are closed under
can 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 by
is 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 measure
This article incorporates material from Current on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.