Currents, boundaries, and Stokes’ theorem

A k-current on an open Euclidean domain is a continuous linear functional on smooth compactly supported k-forms, with the usual test-function topology. Integration over a smooth oriented k-submanifold gives an example. The class of all currents is much larger: distributional objects need not have a rectifiable support or finite mass. Thus calling an arbitrary current an integer-weighted surface would build a false geometric conclusion into the definition.

Mass is the dual size obtained by testing against forms of comass at most one. Comass bounds evaluation on unit simple k-vectors, which is the appropriate area notion. A finite-mass current has a measure representation, but need not have integer multiplicity or a rectifiable carrier. A normal current has finite mass and finite boundary mass. These analytic bounds are useful, while integrality adds a separate geometric and arithmetic condition.

An integer-rectifiable current integrates over a countably k-rectifiable carrier with a measurable unit simple tangent orientation and positive integer multiplicity. The weighted area is locally finite, and its integral is the mass. A negative coefficient can be absorbed by reversing the orientation. Two presentations of overlapping sheets must combine their oriented coefficients before computing the resulting mass; otherwise cancellation is missed.

An integral current has both itself and its boundary integer rectifiable, with the corresponding finite mass conditions in the global class used here. Local integral currents impose these conditions only on compact subdomains. Normal currents with arbitrary real or diffuse coefficients need not be integral. The distinction is crucial in compactness: preserving integer-rectifiable structure in a limit is a deep theorem, not a general consequence of weak functional convergence.

Define the boundary by applying the current to the exterior derivative of a test form of one lower degree. For integration over an oriented smooth region, classical Stokes identifies this with its oriented geometric boundary. For every current, applying the boundary twice gives zero because the exterior derivative squares to zero. This identity survives weak limits even when the object is no longer a smooth region.

For polygons, the boundary orientation puts the interior on the left when the plane has its usual orientation. Shared edges of adjacent equally oriented regions occur in opposite directions and cancel. The boundary mass can therefore be smaller than the sum of the original boundary masses. This is not a violation of perimeter additivity, since perimeter is not additive under adjoining sets and current mass is a norm with cancellation, not an unsigned inventory of all initially drawn edges.