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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

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. Thus a linear functional
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 tends to infinity, then tends to 0.

The space of *m*-dimensional currents on 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 such that
whenever

The linear subspace of consisting of currents with support (in the sense above) that is a compact subset of is denoted

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*. A sequence of currents, converges to a current if

It is possible to define several norms on subspaces of the space of all currents. One such norm is the *mass norm*. If is an *m*-form, then define its **comass** by

So if is a simple *m*-form, then its mass norm is the usual L^{∞}-norm of its coefficient. The **mass** of a current 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.

Recall that so that the following defines a 0-current:

In particular every signed regular measure is a 0-current:

Let (*x*, *y*, *z*) be the coordinates in Then the following defines a 2-current (one of many):

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