Understanding Geometric Algebr

1. Auflage 2011
ISBN: 978-0-470-94163-8
Verlag: Wiley

Buch, Englisch, 320 Seiten, Format (B × H): 167 mm x 249 mm, Gewicht: 578 g

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.

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Autoren/Hrsg.

Arthur, John W

Fachgebiete

• Naturwissenschaften Physik Elektromagnetismus Elektrizität, Elektrodynamik
• Mathematik | Informatik Mathematik Geometrie Algebraische Geometrie
• Mathematik | Informatik Mathematik Mathematische Analysis Vektoranalysis, Physikalische Felder
• Technische Wissenschaften Energietechnik | Elektrotechnik Elektrotechnik

Inhaltsverzeichnis

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

Über Autor(innen)

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.