[D1] - Gauss’s law and residue calculus in the framework of de Rham cohomology

P. Mertikopoulos. Major thesis, University of Athens, 2003.

Abstract

In this dissertation,our aim is to incorporate Gauss’s famous law in electrodynamics and some of Cauchy’s results in residue calculus within the general setting of de Rham cohomology, which will be shown to provide a most natural environment for their development. Namely, it will be demonstrated that forms defined and closed on punctured spaces can be assigned a residue similar to the one defined in complex analysis. This can be achieved in any number of dimensions and is, in essence, an extension of Gauss’s three-dimensional law: monopole fields generate the de Rham cohomology groups of those spaces and with their help we will outline a method for evaluating certain kinds of multiple improper integrals.

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