Synopses & Reviews
This book presents a fresh, original exposition of the foundations of classical electrodynamics in the tradition of the so-called metric-free approach. The fundamental structure of classical electrodynamics is described in the form of six axioms: (1) electric charge conservation, (2) existence of the Lorentz force, (3) magnetic flux conservation, (4) localization of electromagnetic energy-momentum, (5) existence of an electromagnetic spacetime relation, and (6) splitting of the electric current into material and external pieces. The first four axioms require an arbitrary 4-dimensional differentiable manifold. The fifth axiom characterizes spacetime as the environment in which the electromagnetic field propagates -- a research topic of considerable interest -- and in which the metric tensor of spacetime makes its appearance, thus coupling electromagnetism and gravitation. Repeated emphasis is placed on interweaving the mathematical definitions of physical notions and the actual physical measurement procedures. The tool for formulating the theory is the calculus of exterior differential forms, which is explained in sufficient detail, along with the corresponding computer algebra programs. Prerequisites for the reader include a knowledge of elementary electrodynamics (with Maxwell's equations), linear algebra and elementary vector analysis; some knowledge of differential geometry would help. Foundations of Classical Electrodynamics unfolds systematically at a level suitable for graduate students and researchers in mathematics, physics, and electrical engineering.
Review
"[The authors] ...have stressed the phenomena underlying the axioms chosen and the operational interpretation of the quantities introduced. In this, they have clearly succeeded." --Mathematical Reviews "Throughout this book, the rationalized MKS system of units is used, making analysis more intelligible, and there are many diagrams which are of great help in understanding the text. Each part of the book is followed by a copious list of references.... Also, in appropriate places there are indications how computer algebra (REDUCE/EXCALC) can be used.... The printing and appearance of the book are excellent.... It can be warmly recommended." --Zentralblatt Math
Synopsis
In this book we display the fundamental structure underlying classical electro- dynamics, i. e., the phenomenological theory of electric and magnetic effects. The book can be used as a textbook for an advanced course in theoretical electrodynamics for physics and mathematics students and, perhaps, for some highly motivated electrical engineering students. We expect from our readers that they know elementary electrodynamics in the conventional (1 + 3)-dimensional form including Maxwell's equations. More- over, they should be familiar with linear algebra and elementary analysis, in- cluding vector analysis. Some knowledge of differential geometry would help. Our approach rests on the metric-free integral formulation of the conservation laws of electrodynamics in the tradition of F. Kottler (1922), E. Cartan (1923), and D. van Dantzig (1934), and we stress, in particular, the axiomatic point of view. In this manner we are led to an understanding of why the Maxwell equa- tions have their specific form. We hope that our book can be seen in the classical tradition of the book by E. J. Post (1962) on the Formal Structure of Electro- magnetics and of the chapter "Charge and Magnetic Flux" of the encyclopedia article on classical field theories by C. Truesdell and R. A. Toupin (1960), in- cluding R. A. Toupin's Bressanone lectures (1965); for the exact references see the end of the introduction on page 11. .
Table of Contents
Preface * Introduction * Part A--Mathematics: Some Exterior Calculus * Why Exterior Differential Forms? * A.1. Algebra * A.2. Exterior Calculus * A.3. Integration on a Manifold * Part B--Axioms of Classical Electrodynamics * B.1. Electric Charge Conservation * B.2. Lorentz Force Density * B.3. Magnetic Flux Conservation * B.4. Basic Classical Electrodynamics Summarized, Example * B.5. Electromagnetic Energy-Momentum Current and Action* Part C--More Mathematics * C.1. Linear Connection * C.2. Metric * Part D--The Maxwell--Lorentz Spacetime Relation * D.1. A Linear Relation Between H and F * D.2. Propagation of Electromagnetic Waves: Quartic Wave Surface * D.3. First Constraint: Electric/Magnetic Reciprocity * D.4. Second Constraint: Vanishing Skewon Field and Light Cone * D.5. Extracting the Metric by an Alternative Method * D.6. Fifth Axiom: Maxwell--Lorentz Spacetime Relation * Part E--Electrodynamic in Vacuum and in Matter * E.1. Standard Maxwell--Lorentz Theory in a Vacuum * E.2. Electrodynamic Spacetime Relations Beyond Locality and Linearity * E.3. Electrodynamics in Matter, Constitutive Law * Electrodynamics of Moving Continua * Outlook * References * Author Index * Subject Index