Understanding Geometric Algebra for Electromagnetic Theory (IEEE Press Series on Electromagnetic Wave Theory)

319 Pages·2011·2.1 MB·English

by  John W. Arthur

Understanding
Geometric Algebra for
Electromagnetic Theory
IEEE Press
445 Hoes Lane
Piscataway, NJ 08854
IEEE Press Editorial Board
Lajos Hanzo, Editor in Chief
R. Abhari M. El-Hawary O. P. Malik
J. Anderson B.-M. Haemmerli S. Nahavandi
G. W. Arnold M. Lanzerotti T. Samad
F. Canavero D. Jacobson G. Zobrist
Kenneth Moore, Director of IEEE Book and Information Services (BIS)
IEEE Antenna Propagation Society, Sponsor
APS Liaison to IEEE Press, Robert Mailloux
Understanding
Geometric Algebra for
Electromagnetic Theory
John W. Arthur
Honorary Fellow, School of Engineering, University of Edinburgh
IEEE Antennas and Propagation Society, Sponsor
The IEEE Press Series on Electromagnetic Wave Theory
Andreas C. Cangellaris, Series Editor
IEEE PRESS
A John Wiley & Sons, Inc., Publication
Copyright © 2011 by the Institute of Electrical and Electronics Engineers, Inc.
Published by John Wiley & Sons, Inc., Hoboken, New Jersey. All rights reserved
Published simultaneously in Canada
No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form
or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as
permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior
written permission of the Publisher, or authorization through payment of the appropriate per-copy fee
to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, [phone],
fax [phone], or on the web at www.copyright.com. Requests to the Publisher for permission
should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street,
Hoboken, NJ 07030, [phone], fax [phone], or online at http://www.wiley.com/go/
permissions.
Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts
in preparing this book, they make no representations or warranties with respect to the accuracy or
completeness of the contents of this book and specifically disclaim any implied warranties of
merchantability or fitness for a particular purpose. No warranty may be created or extended by sales
representatives or written sales materials. The advice and strategies contained herein may not be
suitable for your situation. You should consult with a professional where appropriate. Neither the
publisher nor author shall be liable for any loss of profit or any other commercial damages, including
but not limited to special, incidental, consequential, or other damages.
For general information on our other products and services or for technical support, please contact our
Customer Care Department within the United States at [phone], outside the United States at
[phone] or fax [phone].
Wiley also publishes its books in a variety of electronic formats. Some content that appears in print
may not be available in electronic formats. For more information about Wiley products, visit our web
site at www.wiley.com.
Library of Congress Cataloging-in-Publication Data:
Arthur, John W., 1949–
Understanding geometric algebra for electromagnetic theory / John W. Arthur.
p. cm.—(IEEE Press series on electromagnetic wave theory ; 38)
Includes bibliographical references and index.
ISBN 978-0-470-94163-8
1. Electromagnetic theory—Mathematics. 2. Geometry, Algebraic. I. Title.
QC670.A76 2011
530.14′10151635—dc22
2011005744
Printed in the United States of America
oBook ISBN: 978-1-118-07854-3
ePDF ISBN: 978-1-118-07852-5
ePub ISBN: 978-1-118-07853-2
10 9 8 7 6 5 4 3 2 1
. . . it is a good thing to have two ways of looking at a subject, 
and to admit that there are two ways of looking at it.
James Clerk Maxwell, on addressing the question of two versions
of electromagnetic theory, one due to Michael Faraday and the
other to Wilhelm Weber, in a paper on Faraday’s lines of force
presented at Cambridge University, February 1856 [1, p. 67].

Contents
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
vii
viii    Contents
5. (3+1)D Electromagnetics  55
5.1  The Lorentz Force    55
5.2  Maxwell’s Equations in Free Space    56
5.3  Simplified 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  Classification 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