Question

define current, give tis mathematical expression and also give its unit of measurement

Answer 1

Let  denote the space of smooth m-forms with compact support on a smooth manifold . A current is a linear functionalon  which is continuous in the sense of distributions. Thus a linear functional



is an m-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 .