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.
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 such that
The linear subspace of consisting of currents with support (in the sense above) that is a compact subset of is denoted .
To define a Principal current, we must first define a principal function.
Let and construct self-adjoint operators , with spectral resolutions and respectively.
Since it is hypothesised is trace class, then there must exist an , real-valued, compactly supported function called the principal function of so that
Since it has been shown for almost everywhere ,
If has a Lebesgue point at , then
Thus, If has finite dimensional range, then for almost all .
The principal function defined above is for a two-dimensional Lebesgue measure.
In cases where the essential spectrum is curve-like, the principal function can be defined on the curve as an average of the values on both of its sides, even when is weakly differentiable i.e, when and are measures.
For almost all in and a weakly-differentiable function 
whenever satisfies the conditions for a principal function.
It follows that if subspaces and are -comparable, then is equal to the average of the approximated upper and lower limits of principal function at .
This expression can be interpreted as a redefinition of the principal function on the set of points where the rotationally-symmetric regularisation of the principal function converge Hausdorff one-measure almost everywhere, which implies almost everywhere.
By definition, the principal current for the C* algebra which corresponds to a such that is in trace class is
The two-current defined by this relation has certain basic properties: with a suitable functional calculus
- when on
- where is a smooth function of and and is the current formed from the operator
- for in trace class.
In view of this formula we can define a boundary operator on arbitrary currents
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.
Topology and norms
It is possible to define several norms on subspaces of the space of all currents. One such norm is the mass norm. If ω is an m-form, then define its comass by
So if ω is a simple m-form, then its mass norm is the usual L∞-norm of its coefficient. The mass of a current T is then defined as
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.
An intermediate norm is Whitney's flat norm, defined by
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.
Let (x, y, z) be the coordinates in R3. Then the following defines a 2-current (one of many):
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