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. Definition 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 Homological theory 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. Topology and norms 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