An Introduction to Vector Analysis MBTHUBN'S MONOGRAPHS ON PHYSICAL SUBIBCTS General Editors: B. 1. WORSNOP, B.Sc., Ph.D. G. K. T. CONN, M.A., Ph.D.
An Introduction to Vector Analysis For Physicists and Engineers B. HAGUE D.SC PH.D., F.C.G.!. Prolessor 01 Eleetrical Engineering at the UnJuersity 01 Glasgow 1946-1960 RBVISED BY D. MARTIN M.A. B.SC., PH.D. Senior Leeturer in Mathematics at the University 01 Gwgow METHUEN & CO. LTD. and SCIENCE PAPERBACKS
First published 1939 6th edition published 1970 by Methuen & Co Ltd 11 New Fetter Lane, London E.C.4 Hardback SBN 416 15700 9 First published as a Science Paperback 1970 Reprinted 1973 Paperback SBN 412207303 Copyright both editions 1970 Mrs S.T. Mackay Haverhill, Suffolk This book is available both as a hardbound and as a paperback edition. The paperback edition is sold subject to the condition that it shall not, by way of trade or otherwise, be lent, resold, hired-out, or otherwise circulated without the publisher's prior consent in any form ofbinding or cover other than that in which it is published and without a similar condition being imposed on the subsequent purchaser. ISBN-13: 978-0-412-20730-3 DOI: 10.1 007/978-94-009-5841-8 e-isbn-13: 978-94-009-5841-8
Contents PREFACE TO THE REVISED EDITION PREFACE page vü ix DEFINITIONS. ADDITION OF VECTORS 1. ScaIar and Veetor Quantities. 2. Graphical Representation of Veetors. 3. Addition and Subtraction of Veetors. 4. Components of a Veetor. 5. Geometrical Applications. 6. Scalar and Veetor Fields. Miscellaneous Exercises I. 2 PRODUCTS OF VECTORS 1. General. 2. The Scalar Product. 3. The Veetor Product. 4. Veetor Area. 5. Application to Veetor Products. 6. Products of Three Veetors. 7. Line and Surface Integra}s as Scalar Products. Miscellaneous Exercises II. 3 THE DIFFERENTIA TION OF VECTORS 1. Scalar Differentiation. 2. Differentiation of Sums and Products. 3. Partial Differentiation. MisceUaneous Exercises III. 4 THE OPERA TOR V AND ITS USES 1. The Operator V. 2. The Gradient of a Scalar Field. 3. The Divergence of a Veetor Field. 4. The Operator div grad. 5. The Operator Va with Veetor Operand. 6. The Curl of a Veetor Field. 7. Simple Examples of the Curl of a Veetor Field. 8. Divergence of a Veetor Product. 9. Divergence and Curl of SA. 10. The IS 36 41
vi CONTENTS Operator eurl grad. 11. The Operator grad div. 12. The Operator div eurl. 13. The Operator eurl eurl. 14. The Vector Field grad (klr). 15. Vector Operators in Terms of Polar Co-ordinates. Miscellaneous Exercises IV. S INTEGRAL THEOREMS 1. The Divergence Theorem of Gauss. 2. Gauss's Theorem and the Inverse Square Law. 3. Green's Theorem. 4. Stokes's Theorem. 5. Alternative Definitions of Divergence and Curl. 6. Oassification of Vector Fields. Miscellaneous Exercises V. 6 THE SCALAR POTENTIAL FIELD 1. General Properties. 2. The Inverse Square Law. Point Sources. 3. Volume Distributions. 4. Multi-valued Potentials. 7 THE VECTOR POTENTIAL FIELD 1. The Magnetic Field of a Steady Current. 2. The Vector PotentiaI. 3. Linear Currents. 4. Simple Examples of Vector Potential. 8 THE ELECTROMAGNETIC FIELD EQUATIONS OF MAXWELL 1. General. 2. Maxwell's Equations. 3. Energy Considerations. Miscellaneous Exercises vm. ANSWERS TO EXERCISES BIBLIOGRAPHY INDEX 70 91 101 110 116 117 119
Preface to the Revised Edition The principal changes that I have made in preparing this revised edition of the book are the following. (i) Carefuily selected worked and unworked examples have been added to six of the chapters. These examples have been taken from class and degree examination papers set in this University and I am grateful to the University Court for permission to use them. (ii) Some additional matter on the geometrieai application of veetors has been incorporated in Chapter 1. (iii) Chapters 4 and 5 have been combined into one chapter, some material has been rearranged and some further material added. (iv) The chapter on int~gral theorems, now Chapter 5, has been expanded to include an altemative proof of Gauss's theorem, a treatmeot of Green's theorem and a more extended discussioo of the classification of vector fields. (v) The only major change made in what are now Chapters 6 and 7 is the deletioo of the discussion of the DOW obsolete pot funetioo. (vi) A small part of Chapter 8 on Maxwell's equations has been rewritten to give a fuller account of the use of scalar and veetor potentials in eleetromagnetic theory, and the units employed have been changed to the m.k.s. system. (vü) The final chapter of the originai book, which dealt very brietly with tensors, has been omitted. A more detailed aecount would now be required and in any case excellent introductory treatments can be found in the books by Temple and by Lichnerowicz published in this series. In revising a book written by someone else, it is difficult to know to what extent to make changes. In this case, there are some matters which I should have treated in a different way had I been writing a hook on vectors myself, but, since I felt it right to try to preserve the distinctive character of the originai editioo, I refrained from
vill VEeToR ANALYSIS making the relevant alterations. Professor Hague adopted an essentially physical standpoint throughout and this was one of the virtues of the book. Finally, I should like to say that it is a pleasure to have had the opportunity of revising a hook by one whom I knew so weil and for whom I always had such high regard; Bernard Hague was greatly admired in this University not only as an excellent scientist but also as a perfect gentieman in the correct sense of that much abused word. University of Glasgow, Glasgow, W.2. D. MARTIN
Preface Veetor Analysis is the natural means of expression for the threedimensional problems of physics and engineering, because its coneiseness and freedom from mathematical detail enable the relationships between the various physical quantities to be kept clearly in view. Sinee the pioneer work of Gibbs and of Heaviside an increasing number of text-books and scientific papers on physical and teehnical subjeets have made use of veetor methods, until it has now become almost essential for any advanced worker in these scienees to have some knowledge of veetor analysis. Much good work can be done with the aid of a very few elementary principles. It is the objeet of this monograph to give an introduction to these principles and to explain them from a physical standpoint, so that they may be easily available to the busy physicist or engineer approaching the subjeet for the first time. Such workers are usually so much occupied by their major task as to lack the time necessary to enable them to seek out such principles as they need to use from the more comprehensive treatises which aim at mathematical completeness. The monograph is not, therefore, intended for the reader with purely mathematical interests, whose more rigorous and systematic requirements are fully satisfied elsewhere. For these reasons the outlook adopted is almost entirely physical; geometrical matters and questions of an exclusively mathematical interest are limited to essentials. Formal proofs of invarianee and conditions of continuity in veetor processes are replaced by an appea) to physical intuition. Purely analytical topics of an advanced kind, such as Green's theorem, are justifiably omitted from such an introductory treatment as this monograph aims to present. The reader who later wishes to amend his knowledge in these and other respeets will find ample material in the standard books listed in the Bibliography.
x VECTOR ANALYSIS The reader's attention is drawn in particular to two features. First, the use of trimetric projection by Gough's method (Engineering, Vol. 143, p. 458, 1937) for eertain of the three-dimensional diagrams, a purpose for which this method is admirably suited. Second, the inclusion of a chapter giving a brief sketch of the elementary properties of tensors and dyadics in their relation to veetors. This is a subject which usually puzzles and often repels physics and engineering students because of the abstract mathematical way in which it is generally brought to their notiee. The monograph is based on a course of leetures given a few years ago to post-graduate electrica1 engineering students in the Polytechnic Institute of Brooklyn, New York. I am particularly grateful to my coiieague Dr A. J. Small for his valued assistance in reading the manuscript and proofs, and for numerous suggestions. GLASGOW June 1938 D.HAGUE