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.

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.

Preface.

Reading Guide.

1. Introduction.

2. A Quick Tour of Geometric Algebra.

2.1 The Basic Rules Geometric Algebra.

2.2 3D Geometric Algebra.

2.3 Developing the Rules.

2.4 Comparison with Traditional 3D Tools.

2.5 New Possibilities.

2.6 Exercises.

3. Applying the Abstraction.

3.1 Space and Time.

3.2 Electromagnetics.

3.3 The Vector Derivative.

3.4 The Integral Equations.

3.5 The Role of the Dual.

3.6 Exercises.

4. Generalisation.

4.1 Homogeneous and Inhomogeneous Multivectors.

4.2 Blades.

4.3 Reversal.

Understanding Geometric Algebra for Electromagnetic Theory.

4.4 Maximum Grade.

4.5 Inner and Outer Products Involving a Multivector.

4.6 Inner and Outer Products between Higher Grades.

4.7 Summary so Far.

4.8 Exercises.

5. (3+1)D Electromagnetics.

5.1 The Lorentz Force.

5.2 Maxwell’s Equations in Free Space.

5.3 Simplified Equations.

5.4 The Connexion between the Electric and Magnetic Fields.

5.5 Plane Electromagnetic Waves.

5.6 Charge Conservation.

5.7 Multivector Potential.

5.8 Energy and Momentum.

5.9 Maxwell’s Equations on Polarisable Media.

5.10 Exercises.

6. Review of (3+1)D.

7. Introducing Spacetime.

7.1 Background and Key Concepts.

7.2 Time as a Vector.

7.3 The Spacetime Basis Elements.

7.4 Basic Operations.

7.5 Velocity.

7.6 Different Basis Vectors and Frames.

7.7 Events and Hstories.

Understanding Geometric Algebra for Electromagnetic Theory.

7.8 The Spacetime Form of.

7.9 Working with Vector Differentiation.

7.10 Working without Basis Vectors.

7.11 Classification of Spacetime Vectors and Bivectors.

7.12 Exercises.

8. Relating Spacetime to (3+1)D.

8.1 The Correspondence between the Elements.

8.2 Translations in General.

8.3 Introduction to Spacetime Splits.

8.4 Some Important Spacetime Splits.

8.5 What Next?

8.6 Exercises.

9. Change of Basis Vectors.

9.1 Linear transformations.

Understanding Geometric Algebra for Electromagnetic Theory.

9.2 Relationship to Geometric Algebras.

9.3 Implementing Spatial Rotations and the Lorentz Transformation.

9.4 Lorentz Transformation of the Basis Vectors.

9.5 Lorentz Transformation of the Basis Bivectors.

9.6 Transformation of the Unit Scalar and Pseudoscalar.

9.7 Reverse Lorentz Transformation.

9.8 The Lorentz Transformation with Vectors in Component Form.

9.9 Dilations.

9.10 Exercises.

10. Further Spacetime Concepts.

10.1 Review of Frames and Time Vectors.

10.2 Frames in General.

10.3 Maps and Grids.

10.4 Proper Time.

10.5 Proper Velocity.

10.6 Relative Vectors and Paravectors.

10.7 Frame Dependent v. Frame Independent Scalars.

10.8 Change of Basis for any Object in Component Form.

10.9 Velocity as Seen in Different Frames.

10.10 Frame Free Form of the Lorentz Transformation.

10.11 Exercises.

Understanding Geometric Algebra for Electromagnetic Theory.

11. Application of Spacetime Geometric Algebra to Basic Electromagnetics.

11.1 The Spacetime Approach to Electrodynamics.

11.2 The Vector Potential and some Spacetime Splits.

11.3 Maxwell’s Equations in Spacetime Form.

11.4 Charge Conservation and the Wave Equation.

11.5 Plane Electromagnetic Waves.

11.6 Transformation of the Electromagnetic Field.

11.7 Lorentz Force.

11.8 The Electromagnetic Field of a Moving Point Charge.

11.9 Exercises.

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

12.1 Working with Null Vectors.

12.2 Finding F for a Moving Point Charge.

12.3 Frad in the Charge’s Rest Frame.

12.4 Frad in the Observer’s Rest Frame.

12. 5 Exercises.

13. Conclusion.

14. Appendices.

14.1 Glossary.

14.2 Axial v True Vectors.

Understanding Geometric Algebra for Electromagnetic Theory.

14.3 Complex Numbers and the 2D Geometric Algebra.

14.4 The Structure of Vector Spaces and Geometric Algebras.

14.5 Quaternions Compared.

14.6 Evaluation of an Integral in Equation (5.14).

14.7 Formal Derivation of the Spacetime Vector Derivative.

15. Table and Figure Captions.

16. Further Reading on Geometric Algebra.

17. References.

18. Tables and Figures.