Electrical Circuits as a Paradigm in Homology and Cohomology

Eberhard Zeidler
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Eberhard Zeidler: Max Planck Institute for Mathematics in the Sciences

Chapter 22 in Quantum Field Theory III: Gauge Theory, 2011, pp 1009-1026 from Springer

Abstract: Abstract We want to show that: (i) Electric currents J are 1-cycles: ∂ J=0. (ii) Voltages V are 1-coboundaries: V=−dU (U is the electrostatic potential). (iii) There exists a duality relation between electric currents and voltages: 〈V|J〉=0 (orthogonality). (iv) If the electrical circuit is connected, then we get β 0=1 for the zeroth Betti number. In the general case, β 0 is equal to the number of connectivity components of the electrical circuit. (v) If the electrical circuit has s 0 nodes and s 1 connections, then the Euler characteristic is given by χ=s 0−s 1. (vi) This yields the first Betti number β 1=β 0−χ. (vii) The space of electric currents is a linear space of dimension β 1. Modern computers are based on huge electrical circuits. In this section, we would like to study the theory of electrical circuits as a paradigm for important generalizations in modern physics and mathematics.

Keywords: Electrical Circuit; Cohomology Group; Homology Group; Euler Characteristic; Betti Number (search for similar items in EconPapers)
Date: 2011
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DOI: 10.1007/978-3-642-22421-8_23

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