Differentiable Manifolds pp 34-78 | Cite as

# Currents

Chapter

## Abstract

In an n-dimensional manifold
for all forms

*V*, a*current*^{1}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$$T[{k_1}{\phi _1} + {k_2}{\phi _2}] = {k_1}T[{\phi _1}] + {k_2}T[{\phi _2}]$$

*ø*and*ø*of this space and for all constants*k*_{1}and*k*_{2}, 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 *x*^{1},…, *x*^{ n } such that each derivative of each coefficient of the form *ø*_{ h } (represented using *x*^{1},…, *x*^{ n }*)* tends uniformly to zero as *h*→∞, then *T* [*ø*_{ h }]→0^{2}.

## Keywords

Vector Space Compact Support Topological Vector Space Bounded Subset Chain Element
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer-Verlag Berlin Heidelberg 1984