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Electromagnetic theory and computation: a topological approach. (English) Zbl 1096.78001
Mathematical Sciences Research Institute Publications 48. Cambridge: Cambridge University Press (ISBN 0-521-80160-5/hbk; 978-0-511-75633-7/ebook). ix, 278 p. (2004).
The present book is based on the thesis of R. Kotiuga but has been re-edited and complemented with recent references. Thus, this important contribution of R. Kotiuga to Computational Electromagnetism with finite elements linking Maxwell’s equations and algebraic topology gets accessible now to a broader audience. The book is addressed not only to mathematicians but also to electrical engineers with a deep theoretical interest in Maxwell’s theory and its discretisation. The composition of the book reflects the broad readership and carefully introduces algebraic topology with homology and cohomology, chains and cochains, duality theory for manifolds, differential forms and the Hodge operator to the engineer as well as the electromagnetic theory to the mathematician. The author puts a lot of effort also in motivating the topological idea of chains and co-chains, e.g.. Illustrations and a good collection of examples throughout the chapters help to understand the mathematical apparatus. The reader would perhaps wish even more links to the recent literature but this is secondary.
All in all it is a book on a high scientific level. Its topic is of great importance and actuality in Computational Electromagnetism. The mathematician will enjoy reading the book, the engineer probably needs some more effort but both will get a very good insight in the link between Maxwell’s equations in continuous and discrete form.

MSC:
78-02 Research exposition (monographs, survey articles) pertaining to optics and electromagnetic theory
78M10 Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory
78A25 Electromagnetic theory (general)
78A30 Electro- and magnetostatics
78-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to optics and electromagnetic theory
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