Understanding Geometric Algebra for Electromagnetic Theory

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

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