Abstract
Currents are defined abstractly as dual spaces to differential forms and differential forms are defined abstractly as dual spaces to linear spaces of k-vectors. This double duality makes it sometimes difficult to recognize a specific example as a current or to give a current geometric meaning. Operators on currents are usually defined by dualizing analytic operators on differential forms. We give direct geometric definitions for a large class of currents and their operators. Chainlets are limits of polyhedral chains taken with respect to a norm. Integrals of differential forms over polyhedra have well defined limits to chainlets and a full exterior calculus has been established for chain-let domains. ([H2], [M], [114]) Every chainlet is thus a current. In this paper we specify which currents T correspond uniquely to chainlets, giving such T direct, geometric representation.
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Harrison, J. (2004). Geometric Representations of Currents and Distributions. In: Bandt, C., Mosco, U., Zähle, M. (eds) Fractal Geometry and Stochastics III. Progress in Probability, vol 57. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7891-3_12
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DOI: https://doi.org/10.1007/978-3-0348-7891-3_12
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9612-2
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