So the definition of boundary of a current, is justified by Stokes Theorem:
The space of -dimensional currents is a real vector space with operations defined by
The sum of two currents represents the union of the surfaces they represents. Multiplication by a scalar represents a change in the multiplicity of the surface. In particular multiplication by represents the change of orientation of the surface.
We denote with the vector subspace of of currents with compact support.
A stronger norm on the space of currents is the mass norm. First of all we define the mass norm of a -form as
So if is a simple -form, then its mass norm is the usual norm of its coefficient. We hence define the mass of a current as
The mass of a currents represents the area of the generalized surface.
An intermediate norm, is the flat norm defined by
Notice that two currents are close in the mass norm if they coincide apart from a small part. On the other hand the are close in the flat norm if they coincide up to a small deformation.
Recall that so that the following defines a -current:
In particuar every signed measure with finite mass is a -current:
Let be the coordinates in . Then the following defines a -current:
|Date of creation||2013-03-22 14:27:39|
|Last modified on||2013-03-22 14:27:39|
|Last modified by||paolini (1187)|