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Understanding Geometric Algebra for Electromagnetic Theory

ISBN: 978-0-470-94163-8
320 pages
September 2011, Wiley-IEEE Press
Understanding Geometric Algebra for Electromagnetic Theory (0470941634) cover image
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

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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 Transformation of Basis for Any Given Vector 162

9.9 Dilations 165

9.10 Exercises 166

10. Further Spacetime Concepts 169

10.1 Review of Frames and Time Vectors 169

10.2 Frames in General 171

10.3 Maps and Grids 173

10.4 Proper Time 175

10.5 Proper Velocity 176

10.6 Relative Vectors and Paravectors 178

10.6.1 Geometric Interpretation of the Spacetime Split 179

10.6.2 Relative Basis Vectors 183

10.6.3 Evaluating Relative Vectors 185

10.6.4 Relative Vectors Involving Parameters 188

10.6.5 Transforming Relative Vectors and Paravectors to a Different Frame 190

10.7 Frame-Dependent versus Frame-Independent Scalars 192

10.8 Change of Basis for Any Object in Component Form 194

10.9 Velocity as Seen in Different Frames 196

10.10 Frame-Free Form of the Lorentz Transformation 200

10.11 Exercises 202

11. Application of the Spacetime Geometric Algebra to Basic Electromagnetics 203

11.1 The Vector Potential and Some Spacetime Splits 204

11.2 Maxwell’s Equations in Spacetime Form 208

11.2.1 Maxwell’s Free Space or Microscopic Equation 208

11.2.2 Maxwell’s Equations in Polarizable Media 210

11.3 Charge Conservation and the Wave Equation 212

11.4 Plane Electromagnetic Waves 213

11.5 Transformation of the Electromagnetic Field 217

11.5.1 A General Spacetime Split for F 217

11.5.2 Maxwell’s Equation in a Different Frame 219

11.5.3 Transformation of F by Replacement of Basis Elements 221

11.5.4 The Electromagnetic Field of a Plane Wave Under a Change of Frame 223

11.6 Lorentz Force 224

11.7 The Spacetime Approach to Electrodynamics 227

11.8 The Electromagnetic Field of a Moving Point Charge 232

11.8.1 General Spacetime Form of a Charge’s Electromagnetic Potential 232

11.8.2 Electromagnetic Potential of a Point Charge in Uniform Motion 234

11.8.3 Electromagnetic Field of a Point Charge in Uniform Motion 237

11.9 Exercises 240

12. The Electromagnetic Field of a Point Charge Undergoing Acceleration 243

12.1 Working with Null Vectors 243

12.2 Finding F for a Moving Point Charge 248

12.3 Frad in the Charge’s Rest Frame 252

12.4 Frad in the Observer’s Rest Frame 254

12.5 Exercises 258

13. Conclusion 259

14. Appendices 265

14.1 Glossary 265

14.2 Axial versus True Vectors 273

14.3 Complex Numbers and the 2D Geometric Algebra 274

14.4 The Structure of Vector Spaces and Geometric Algebras 275

14.4.1 A Vector Space 275

14.4.2 A Geometric Algebra 275

14.5 Quaternions Compared 281

14.6 Evaluation of an Integral in Equation (5.14) 283

14.7 Formal Derivation of the Spacetime Vector Derivative 284

References 287

Further Reading 291

Index 293

The IEEE Press Series on Electromagnetic Wave Theory

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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.
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"This book will benefit scientists and engineers who use electromagnetic theory in the course of their work.”  (Zentralblatt MATH, 1 May 2013)

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