Currents

Abstract

In an n-dimensional manifold V, a current¹ is a functional T [ø], defined on the vector space of all C^∞ forms ø with compact support in V, which is linear, that is, such that

$$T[{k_1}{\phi _1} + {k_2}{\phi _2}] = {k_1}T[{\phi _1}] + {k_2}T[{\phi _2}]$$

for all forms ø and ø of this space and for all constants k₁and k₂, and which is continuous in the following sense:

If ø _h (h= 1, 2,…) is a sequence of C^∞ forms with supports all contained in a single compact set which is in the interior of the domain of a local coordinate system x¹,…, xⁿ such that each derivative of each coefficient of the form ø _h (represented using x¹,…, xⁿ ) tends uniformly to zero as h→∞, then T [ø _h ]→0².