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This book aims to disseminate geometric algebra as a straightforward mathematical tool set for working with and understanding classical electromagnetic theory. It's target readership is anyone who has some knowledge of electromagnetic theory, predominantly ordinary scientists and engineers who use it in the course of their work, or postgraduate students and senior undergraduates who are seeking to broaden their knowledge and increase their understanding of the subject. It is assumed that the reader is not a mathematical specialist and is neither familiar with geometric algebra or its application to electromagnetic theory. The modern approach, geometric algebra, is the mathematical tool set we should all have started out with and once the reader has a grasp of the subject, he or she cannot fail to realize that traditional vector analysis is really awkward and even misleading by comparison. Professors can request a solutions manual by email: pressbooks@ieee.org
Auteur
JOHN W. ARTHUR earned his PhD from Edinburgh University in 1974 for research into light scattering in crystals. He has been involved in academic research, the microelectronics industry, and corporate R&D. Dr. Arthur has published various research papers in acclaimed journals, including IEEE Antennas and Propagation Magazine. His 2008 paper entitled "The Fundamentals of Electromagnetic Theory Revisited" received the 2010 IEEE Donald G. Fink Prize for Best Tutorial Paper. A senior member of the IEEE, Dr. Arthur was elected a fellow of the Royal Society of Edinburgh and of the United Kingdom's Royal Academy of Engineering in 2002. He is currently an honorary fellow in the School of Engineering at the University of Edinburgh.
Texte du rabat
Provides insight into classical electromagnetic theory through geometric algebra
This practical book promotes the understanding of geometric algebra as a straightforward mathematical toolset for both working with and appreciating the fundamentals of electromagnetic theory. Taking a non-axiomatic, less formal tutorial approach, the text introduces new ideas gradually and goes into considerable detail in giving explanations and working out equations.
After an introduction to geometric algebra, the book shows how to apply it to some basic concepts. Then, the essential toolset is developed, allowing for the application of geometric algebra more generallyfor example, in any dimension of space. After applying the toolset to fundamental electromagnetics in the usual (3+1)D situation where space and time are separate entities, the book prepares the ground for a full 4D treatment in which they are treated equally as spacetime vectors. Through geometric algebra, the reader will discover how to tackle the electromagnetic theory of moving charges in a systematic yet uncomplicated way.
In later chapters, the book provides a self-contained primer on the spacetime approach that seeks to avoid the usual conceptual difficulties of special relativity. While this involves some intriguing subtleties, its application is straightforward and readers will see how the toolset unifies previously separate ideas under a single theme: Coulomb's Law + Spacetime = Classical Electromagnetic Theory. The electromagnetic field of an accelerating charge is worked through in detail to show how the toolset is applied.
Most chapters include exercises. There are figures and tables with detailed captions as well as various appendices that offer explanatory information and background material. In particular, a glossary provides an at-a-glance explanation of key terms and symbols.
This book will benefit scientists and engineers who use electromagnetic theory in the course of their work, including those who teach the subject; graduate students and senior undergraduates studying electromagnetics; and electromagnetic theorists.
Problems and solutions materials are available by sending an email to pressbooks@ieee.org
Contenu
Preface xi
Reading Guide xv
1. Introduction 1
2. A Quick Tour of Geometric Algebra 7
2.1 The Basic Rules of a Geometric Algebra 16
2.2 3D Geometric Algebra 17
2.3 Developing the Rules 19
2.3.1 General Rules 20
2.3.2 3D 21
2.3.3 The Geometric Interpretation of Inner and Outer Products 22
2.4 Comparison with Traditional 3D Tools 24
2.5 New Possibilities 24
2.6 Exercises 26
3. Applying the Abstraction 27
3.1 Space and Time 27
3.2 Electromagnetics 28
3.2.1 The Electromagnetic Field 28
3.2.2 Electric and Magnetic Dipoles 30
3.3 The Vector Derivative 32
3.4 The Integral Equations 34
3.5 The Role of the Dual 36
3.6 Exercises 37
4. Generalization 39
4.1 Homogeneous and Inhomogeneous Multivectors 40
4.2 Blades 40
4.3 Reversal 42
4.4 Maximum Grade 43
4.5 Inner and Outer Products Involving a Multivector 44
4.6 Inner and Outer Products between Higher Grades 48
4.7 Summary So Far 50
4.8 Exercises 51
5. (3+1)D Electromagnetics 55
5.1 The Lorentz Force 55
5.2 Maxwell's Equations in Free Space 56
5.3 Simplifi ed Equations 59
5.4 The Connection between the Electric and Magnetic Fields 60
5.5 Plane Electromagnetic Waves 64
5.6 Charge Conservation 68
5.7 Multivector Potential 69
5.7.1 The Potential of a Moving Charge 70
5.8 Energy and Momentum 76
5.9 Maxwell's Equations in Polarizable Media 78
5.9.1 Boundary Conditions at an Interface 84
5.10 Exercises 88
6. Review of (3+1)D 91
7. Introducing Spacetime 97
7.1 Background and Key Concepts 98
7.2 Time as a Vector 102
7.3 The Spacetime Basis Elements 104
7.3.1 Spatial and Temporal Vectors 106
7.4 Basic Operations 109
7.5 Velocity 111
7.6 Different Basis Vectors and Frames 112
7.7 Events and Histories 115
7.7.1 Events 115
7.7.2 Histories 115
7.7.3 Straight-Line Histories and Their Time Vectors 116
7.7.4 Arbitrary Histories 119
7.8 The Spacetime Form of 121
7.9 Working with Vector Differentiation 123
7.10 Working without Basis Vectors 124
7.11 Classifi cation of Spacetime Vectors and Bivectors 126
7.12 Exercises 127
8. Relating Spacetime to (3+1)D 129
8.1 The Correspondence between the Elements 129
8.1.1 The Even Elements of Spacetime 130
8.1.2 The Odd Elements of Spacetime 131
8.1.3 From (3+1)D to Spacetime 132
8.2 Translations in General 133
8.2.1 Vectors 133
8.2.2 Bivectors 135
8.2.3 Trivectors 136
8.3 Introduction to Spacetime Splits 137
8.4 Some Important Spacetime Splits 140
8.4.1 Time 140
8.4.2 Velocity 141
8.4.3 Vector Derivatives 142
8.4.4 Vector Derivatives of General Multivectors 144
8.5 What Next? 144
8.6 Exercises 145
9. Change of Basis Vectors 147
9.1 Linear Transformations 147
9.2 Relationship to Geometric Algebras 149
9.3 Implementing Spatial Rotations and the Lorentz Transformation 150
9.4 Lorentz Transformation of the Basis Vectors 153
9.5 Lorentz Transformation of the Basis Bivectors 155
9.6 Transformation of the Unit Scalar and Pseudoscalar 156
9.7 Reverse Lorentz Transformation 156
9.8 The Lorentz Transformation with Vectors in Component Form 158
9.8.1 Transformation of a Vector versus a Transformation of Basis 158 9.8.2 Transf...