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More About This Textbook
Overview
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.
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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