Generalized Vector and Dyadic Analysis: Applied Mathematics in Field Theory, 2nd Ed. (IEEE Press Series on Electromagnetic Wave Theory)

209 Pages·1997·11.63 MB·English

by  Chen-To Tai

Generalized Vector
and Dyadic Analysis
IEEE/OUP Series on Electromagnetic Wave Theory
The IEEElOUP Series on Electromagnetic Wave Theory consists of new titles as well as
reprintings and revisions of recognized classics that maintain long-term archival significance
in electromagnetic waves and applications.
Series Editor Associate Editors
Donald G. Dudley Electromagnetic Theory, Scattering,
University of Arizona and Diffraction
Ehud Heyman
Tel-Aviv University
Advisory Board
Robert E. Collin
Differential Equation Methods
Case Western Reserve University
Andreas C. Cangellaris
University of Arizona
Akira Ishimaru
University of Washington
Integral Equation Methods
Donald R. Wilton
D. S. Jones
University of Houston
University of Dundee
Antennas, Propagation, and Microwaves
David R. Jackson
University of Houston
Books in the Series
Chew, W. C., Waves and Fields in Inhomogeneous Media
Christopolous, C., The Transmission-line Modeling Method: TLM
Clemmow, P. C., The Plane Wave Spectrum Representation ofElectromagnetic Fields
Collin, R. E., Field Theory ofGuided Waves, Second Edition
Dudley, D. G., Mathematical Foundations for Electromagnetic Theory
Elliott, R. 5., Electromagnetics: History, Theory, and Applications
Felsen, L. B., and Marcuvitz, N., Radiation and Scattering of Waves
Harrington, R. F., Field Computation by Moment Methods
Ishimaru, A., Wave Propagation and Scattering in Random Media
Jones, D. 5., Methods in Electromagnetic Wave Propagation, Second Edition
Lindell, I. V., Methods for Electromagnetic Field Analysis
Tai, C.-T., Dyadic Green Functions in Electromagnetic Theory, Second Edition
Van Bladel, J., Singular Electromagnetic Fields and Sources
Wait, J., Electromagnetic Waves in Stratified Media
Generalized Vector
and Dyadic Analysis
Applied Mathematics in Field Theory
Second Edition
-----..,--
Chen-To Tai
Professor Emeritus
Radiation Laboratory
Department ofElectrical Engineering
and Computer Science
University ofMichigan
IEEE Antennas & Propagation Society, Sponsor
~ IEEE
~ PRESS
WILEY~
•
The Institute of Electrical INTERSCIENCE
and Electronics Engineers, Inc.
New York
IEEE Press
445 Hoes Lane, RO. Box 1331
Piscataway, NJ 08855-1331
IEEE Press Editorial Board
Roger F. Hoyt, Editor in Chief
J. B. Anderson S. Furni P. Laplante
P. M. Anderson A. H. Haddad R. S. Muller
M. Eden R. Herrick W. D. Reeve
M. E. El-Hawary G. F. Hoffnagle D. J. Wells
S. Kartalopoulos
Ken Moore, Director, IEEE Press
John Griffin, Senior Acquisitions Editor
Lisa Dayne, Assistant Editor
Linda Matarazzo, Assistant Editor
Denise Phillip, Associate Production Editor
IEEE Antennas & Propagation Society. Sponsor
AP-S Liaison to IEEE Press, Robert Mailloux
Oxford University Press,
Walton Street, Oxford OX2 6DP
Oxford New York
Athens Auckland Bangkok Bombay
Calcutta Cape Town Dar es Salaam Delhi
Florence Hong Kong Istanbul Karachi
Kuala Lumpur Madras Madrid Melbourne
Mexico City Nairobi Paris Singapore
Taipei Tokyo Toronto
and associated companies in
Berlin Ibadan
Oxford is a trade mark of Oxford University Press
In Memory
of
Professor Dr. Yeh Chi-Sun
(1898-1977)
This book and other books may be purchased at a discount from the
publisher when ordered in bulk quantities. Contact:
IEEE Press Marketing
Attn: Special Sales
445 Hoes Lane, ~O. Box 1331
Piscataway, NJ 08855-1331
Fax: [phone]
For more information about IEEE PRESS products,
visit the IEEE Home Page: http/lwww.ieee.orgl
© 1997 by the Institute of Electrical and Electronics Engineers, Inc.
345 East 47th Street, New York, NY 10017-2394
All rights reserved. No part ofthis book may be reproduced in any form,
nor may it be stored in a retrieval system or transmitted in any form,
without written permission/rom the publisher.
IEEE ISBN 0-7803-3413-2
IEEE Order Number: PC5685
OUP ISBN 0 19 856546 1
Library of Congress Cataloging-in-Publication Data
Tai, Chen-To (date)
Generalized vector and dyadic analysis: applied mathematics in
field theory / Chen-To Tai -2nd ed.
p. ern.
Includes bibliographical references and index.
ISBN 0-7803-3413-2 (cloth)
1. Vector analysis. I. Title.
QA433.T3 1997
515'.63-dc21 96-29863
CIP
Contents
Preface to the Second Edition xi
Preface to the First Edition xiii
Acknowledgments for the First Edition xv
1 Vector and Dyadic Algebra 1
1-1 Representations ofVector Functions
1-2 Products and Identities 4
1-3 Orthogonal Transformation of Vector
Functions 8
1-4 Transform of Vector Products 14
1-5 Definition of Dyadics and Tensors 16
1-6 Classification of Dyadics 17
1-7 Products Between Vectors and Dyadics 19
2 Coordinate Systems 23
2-1 General Curvilinear System (GCS) 23
2-2 Orthogonal Curvilinear System (OCS) 28
vii
viii Contents
2-3 Derivatives of Unit Vectors in OCS 33
2-4 Dupin Coordinate System 35
2-5 Radii of Curvature 37
3 Line Integrals, Surface Integrals, and Volume
Integrals 43
3-1 Differential Length, Area, and Volume 43
3-2 Classification of Line Integrals 44
3-3 Classification of Surface Integrals 48
3-4 Classification of Volume Integrals 56
4 Vector Analysis in Space 58
4-1 Symbolic Vector And Symbolic Vector
Expressions 58
4-2 Differential Formulas ofthe Symbolic Expression
in the Orthogonal Curvilinear Coordinate System
for Gradient, Divergence, and Curl 61
4-3 Invariance of the Differential Operators 65
4-4 Differential Formulas of the Symbolic
Expression in the General Curvilinear
System 69
4-5 Alternative Definitions of Gradient
and Curl 75
4-6 The Method of Gradient 78
4-7 Symbolic Expressions with Two Functions
and the Partial Symbolic Vectors 81
4-8 Symbolic Expressions with Double Symbolic
Vectors 86
4-9 Generalized Gauss Theorem in Space 91
4-10 Scalar and Vector Green's Theorems 93
4-11 Solenoidal Vector, Irrotational Vector,
and Potential Functions 95
5 Vector Analysis on Surface 99
5-1 Surface SymbolicVector and Symbolic Expression
for a Surface 99
5-2 Surface Gradient, Surface Divergence, and Surface
Curl 101
5-2-1 Surface Gradient 101
Con~n~ ~
5-2-2 Surface Divergence 102
5-2-3 Surface Curl 103
5-3 Relationship Between the Volume and Surface
Symbolic Expressions 104
5-4 Relationship Between Weatherbum's Surface
Functions and the Functions Defined
in the Method of Symbolic Vector 104
5-5 Generalized Gauss Theorem for a Surface 106
5-6 Surface Symbolic Expressions with a Single
Symbolic Vector and Two Functions 111
5-7 Surface Symbolic Expressions with Two Surface
Symbolic Vectors and a Single Function 113
6 Vector Analysis of Transport Theorems 116
6-1 Helmholtz Transport Theorem 116
6-2 Maxwell Theorem and Reynolds Transport
Theorem 119
7 Dyadic Analysis 121
7-1 Divergence and Curl of Dyadic Functions
and Gradient of Vector Functions 121
7-2 Dyadic Integral Theorems 124
8 A Historical StUdy of Vector Analysis 127
8-1 Introduction 127
8-2 Notations and Operators 129
8-2-1 Past and Present Notations
in Vector Analysis 129
8-2-2 QuatemionAnalysis 131
8-2-3 Operators 132
8-3 The Pioneer Works of J. Willard Gibbs
(1839-1903) 135
8-3-1 Two Pamphlets Printed in 1881
and 1884 135
8-3-2 Divergence and Curl Operators
and Their New Notations 138
8-4 Book by Edwin Bidwell Wilson Founded
Upon the Lectures of J. Willard Gibbs 141