Understanding Geometric Algebra for Electromagnetic Theory

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.

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

• Preface xi Reading Guide xv1. Introduction 12. A Quick Tour of Geometric Algebra 72.1 The Basic Rules of a Geometric Algebra 162.2 3D Geometric Algebra 172.3 Developing the Rules 192.3.1 General Rules 202.3.2 3D 212.3.3 The Geometric Interpretation of Inner and Outer Products 222.4 Comparison with Traditional 3D Tools 242.5 New Possibilities 242.6 Exercises 263. Applying the Abstraction 273.1 Space and Time 273.2 Electromagnetics 283.2.1 The Electromagnetic Field 283.2.2 Electric and Magnetic Dipoles 303.3 The Vector Derivative 323.4 The Integral Equations 343.5 The Role of the Dual 363.6 Exercises 374. Generalization 394.1 Homogeneous and Inhomogeneous Multivectors 404.2 Blades 404.3 Reversal 424.4 Maximum Grade 434.5 Inner and Outer Products Involving a Multivector 444.6 Inner and Outer Products between Higher Grades 484.7 Summary So Far 504.8 Exercises 515. (3+1)D Electromagnetics 555.1 The Lorentz Force 555.2 Maxwell’s Equations in Free Space 565.3 Simplifi ed Equations 595.4 The Connection between the Electric and Magnetic Fields 605.5 Plane Electromagnetic Waves 645.6 Charge Conservation 685.7 Multivector Potential 695.7.1 The Potential of a Moving Charge 705.8 Energy and Momentum 765.9 Maxwell’s Equations in Polarizable Media 785.9.1 Boundary Conditions at an Interface 845.10 Exercises 886. Review of (3+1)D 917. Introducing Spacetime 977.1 Background and Key Concepts 987.2 Time as a Vector 1027.3 The Spacetime Basis Elements 1047.3.1 Spatial and Temporal Vectors 1067.4 Basic Operations 1097.5 Velocity 1117.6 Different Basis Vectors and Frames 1127.7 Events and Histories 1157.7.1 Events 1157.7.2 Histories 1157.7.3 Straight-Line Histories and Their Time Vectors 1167.7.4 Arbitrary Histories 1197.8 The Spacetime Form of ∇ 1217.9 Working with Vector Differentiation 1237.10 Working without Basis Vectors 1247.11 Classifi cation of Spacetime Vectors and Bivectors 1267.12 Exercises 1278. Relating Spacetime to (3+1)D 1298.1 The Correspondence between the Elements 1298.1.1 The Even Elements of Spacetime 1308.1.2 The Odd Elements of Spacetime 1318.1.3 From (3+1)D to Spacetime 1328.2 Translations in General 1338.2.1 Vectors 1338.2.2 Bivectors 1358.2.3 Trivectors 1368.3 Introduction to Spacetime Splits 1378.4 Some Important Spacetime Splits 1408.4.1 Time 1408.4.2 Velocity 1418.4.3 Vector Derivatives 1428.4.4 Vector Derivatives of General Multivectors 1448.5 What Next? 1448.6 Exercises 1459. Change of Basis Vectors 1479.1 Linear Transformations 1479.2 Relationship to Geometric Algebras 1499.3 Implementing Spatial Rotations and the Lorentz Transformation 1509.4 Lorentz Transformation of the Basis Vectors 1539.5 Lorentz Transformation of the Basis Bivectors 1559.6 Transformation of the Unit Scalar and Pseudoscalar 1569.7 Reverse Lorentz Transformation 1569.8 The Lorentz Transformation with Vectors in Component Form 1589.8.1 Transformation of a Vector versus a Transformation of Basis 1589.8.2 Transformation of Basis for Any Given Vector 1629.9 Dilations 1659.10 Exercises 16610. Further Spacetime Concepts 16910.1 Review of Frames and Time Vectors 16910.2 Frames in General 17110.3 Maps and Grids 17310.4 Proper Time 17510.5 Proper Velocity 17610.6 Relative Vectors and Paravectors 17810.6.1 Geometric Interpretation of the Spacetime Split 17910.6.2 Relative Basis Vectors 18310.6.3 Evaluating Relative Vectors 18510.6.4 Relative Vectors Involving Parameters 18810.6.5 Transforming Relative Vectors and Paravectors to a Different Frame 19010.7 Frame-Dependent versus Frame-Independent Scalars 19210.8 Change of Basis for Any Object in Component Form 19410.9 Velocity as Seen in Different Frames 19610.10 Frame-Free Form of the Lorentz Transformation 20010.11 Exercises 20211. Application of the Spacetime Geometric Algebra to Basic Electromagnetics 20311.1 The Vector Potential and Some Spacetime Splits 20411.2 Maxwell’s Equations in Spacetime Form 20811.2.1 Maxwell’s Free Space or Microscopic Equation 20811.2.2 Maxwell’s Equations in Polarizable Media 21011.3 Charge Conservation and the Wave Equation 21211.4 Plane Electromagnetic Waves 21311.5 Transformation of the Electromagnetic Field 21711.5.1 A General Spacetime Split for F 21711.5.2 Maxwell’s Equation in a Different Frame 21911.5.3 Transformation of F by Replacement of Basis Elements 22111.5.4 The Electromagnetic Field of a Plane Wave Under a Change of Frame 22311.6 Lorentz Force 22411.7 The Spacetime Approach to Electrodynamics 22711.8 The Electromagnetic Field of a Moving Point Charge 23211.8.1 General Spacetime Form of a Charge’s Electromagnetic Potential 23211.8.2 Electromagnetic Potential of a Point Charge in Uniform Motion 23411.8.3 Electromagnetic Field of a Point Charge in Uniform Motion 23711.9 Exercises 24012. The Electromagnetic Field of a Point Charge Undergoing Acceleration 24312.1 Working with Null Vectors 24312.2 Finding F for a Moving Point Charge 24812.3 Frad in the Charge’s Rest Frame 25212.4 Frad in the Observer’s Rest Frame 25412.5 Exercises 25813. Conclusion 25914. Appendices 26514.1 Glossary 26514.2 Axial versus True Vectors 27314.3 Complex Numbers and the 2D Geometric Algebra 27414.4 The Structure of Vector Spaces and Geometric Algebras 27514.4.1 A Vector Space 27514.4.2 A Geometric Algebra 27514.5 Quaternions Compared 28114.6 Evaluation of an Integral in Equation (5.14) 28314.7 Formal Derivation of the Spacetime Vector Derivative 284References 287Further Reading 291Index 293The IEEE Press Series on Electromagnetic Wave Theory