Understanding Geometric Algebra for Electromagnetic Theory

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Edition: 1st
Format: Hardcover
Pub. Date: 2011-09-13
Publisher(s): Wiley-IEEE Press
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Summary

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 either 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.

Author Biography

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 RD. 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.

Table of Contents

Prefacep. xi
Reading Guidep. xv
Introductionp. 1
A Quick Tour of Geometric Algebrap. 7
The Basic Rules of a Geometric Algebrap. 16
3D Geometric Algebrap. 17
Developing the Rulesp. 19
General Rulesp. 20
3Dp. 21
The Geometric Interpretation of Inner and Outer Productsp. 22
Comparison with Traditional 3D Toolsp. 24
New Possibilitiesp. 24
Exercisesp. 26
Applying the Abstractionp. 27
Space and Timep. 27
Electromagneticsp. 28
The Electromagnetic Fieldp. 28
Electric and Magnetic Dipolesp. 30
The Vector Derivativep. 32
The Integral Equationsp. 34
The Role of the Dualp. 36
Exercisesp. 37
Generalizationp. 39
Homogeneous and Inhomogeneous Multivectorsp. 40
Bladesp. 40
Reversalp. 42
Maximum Gradep. 43
Inner and Outer Products Involving a Multivectorp. 44
Inner and Outer Products between Higher Gradesp. 48
Summary So Farp. 50
Exercisesp. 51
(3+l)D Electromagneticsp. 55
The Lorentz Forcep. 55
Maxwell's Equations in Free Spacep. 56
Simplified Equationsp. 59
The Connection between the Electric and Magnetic Fieldsp. 60
Plane Electromagnetic Wavesp. 64
Charge Conservationp. 68
Multivector Potentialp. 69
The Potential of a Moving Chargep. 70
Energy and Momentump. 76
Maxwell's Equations in Polarizable Mediap. 78
Boundary Conditions at an Interfacep. 84
Exercisesp. 88
Review of (3+l)Dp. 91
Introducing Spacetimep. 97
Background and Key Conceptsp. 98
Time as a Vectorp. 102
The Spacetime Basis Elementsp. 104
Spatial and Temporal Vectorsp. 106
Basic Operationsp. 109
Velocityp. 111
Different Basis Vectors and Framesp. 112
Events and Historiesp. 115
Eventsp. 115
Historiesp. 115
Straight-Line Histories and Their Time Vectorsp. 116
Arbitrary Historiesp. 119
The Spacetime Form of Vp. 121
Working with Vector Differentiationp. 123
Working without Basis Vectorsp. 124
Classification of Spacetime Vectors and Bivectorsp. 126
Exercisesp. 127
Relating Spacetime to (3+l)Dp. 129
The Correspondence between the Elementsp. 129
The Even Elements of Spacetimep. 130
The Odd Elements of Spacetimep. 131
From (3+l)D to Spacetimep. 132
Translations in Generalp. 133
Vectorsp. 133
Bivectorsp. 135
Trivectorsp. 136
Introduction to Spacetime Splitsp. 137
Some Important Spacetime Splitsp. 140
Timep. 140
Velocityp. 141
Vector Derivativesp. 142
Vector Derivatives of General Multivectorsp. 144
What Next?p. 144
Exercisesp. 145
Change of Basis Vectorsp. 147
Linear Transformationsp. 147
Relationship to Geometric Algebrasp. 149
Implementing Spatial Rotations and the Lorentz Transformationp. 150
Lorentz Transformation of the Basis Vectorsp. 153
Lorentz Transformation of the Basis Bivectorsp. 155
Transformation of the Unit Scalar and Pseudoscalarp. 156
Reverse Lorentz Transformationp. 156
The Lorentz Transformation with Vectors in Component Formp. 158
Transformation of a Vector versus a Transformation of Basisp. 158
Transformation of Basis for Any Given Vectorp. 162
Dilationsp. 165
Exercisesp. 166
Further Spacetime Conceptsp. 169
Review of Frames and Time Vectorsp. 169
Frames in Generalp. 171
Maps and Gridsp. 173
Proper Timep. 175
Proper Velocityp. 176
Relative Vectors and Paravectorsp. 178
Geometric Interpretation of the Spacetime Splitp. 179
Relative Basis Vectorsp. 183
Evaluating Relative Vectorsp. 185
Relative Vectors Involving Parametersp. 188
Transforming Relative Vectors and Paravectors to a Different Framep. 190
Frame-Dependent versus Frame-Independent Scalarsp. 192
Change of Basis for Any Object in Component Formp. 194
Velocity as Seen in Different Framesp. 196
Frame-Free Form of the Lorentz Transformationp. 200
Exercisesp. 202
Application of the Spacetime Geometric Algebra to Basic Electromagneticsp. 203
The Vector Potential and Some Spacetime Splitsp. 204
Maxwell's Equations in Spacetime Formp. 208
Maxwell's Free Space or Microscopic Equationp. 208
Maxwell's Equations in Polarizable Mediap. 210
Charge Conservation and the Wave Equationp. 212
Plane Electromagnetic Wavesp. 213
Transformation of the Electromagnetic Fieldp. 217
A General Spacetime Split for Fp. 217
Maxwell's Equation in a Different Framep. 219
Transformation of F by Replacement of Basis Elementsp. 221
The Electromagnetic Field of a Plane Wave Under a Change of Framep. 223
Lorentz Forcep. 224
The Spacetime Approach to Electrodynamicsp. 227
The Electromagnetic Field of a Moving Point Chargep. 232
General Spacetime Form of a Charge's Electromagnetic Potentialp. 232
Electromagnetic Potential of a Point Charge in Uniform Motionp. 234
Electromagnetic Field of a Point Charge in Uniform Motionp. 237
Exercisesp. 240
The Electromagnetic Field of a Point Charge Undergoing Accelerationp. 243
Working with Null Vectorsp. 243
Finding F for a Moving Point Chargep. 248
Frad in the Charge's Rest Framep. 252
Frad in the Observer's Rest Framep. 254
Exercisesp. 258
Conclusionp. 259
Appendicesp. 265
Glossaryp. 265
Axial versus True Vectorsp. 273
Complex Numbers and the 2D Geometric Algebrap. 274
The Structure of Vector Spaces and Geometric Algebrasp. 275
A Vector Spacep. 275
A Geometric Algebrap. 275
Quaternions Comparedp. 281
Evaluation of an Integral in Equation (5.14)p. 283
Formal Derivation of the Spacetime Vector Derivativep. 284
Referencesp. 287
Further Readingp. 291
Indexp. 293
Table of Contents provided by Ingram. All Rights Reserved.

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