Vector Analysis and Cartesian Tensors

313 Pages·1992·8.88 MB·English

by  D. E. Bourne

Vector Analysis
and
Cartesian Tensors
Vector Analysis
and
Cartesian Tensors
Third edition
D.E. Bourne
Department of Applied and Computational Mathematics
University of Sheffield, UK
and
P.C. Kendall
Department of Electronic and Electrical Engineering
University of Sheffield, UK
SPRINGER-SCIENCE+BUSINESS MEDIA, B.Y.
First edition 1967
Reprinted 1982, 1983, 1984, 1985, 1986, 1988, 1990, 1991
Second edition 1977/81
© 1967, 1977, 1992 D.E. Bourne and P.C. Kendall
Originally published by Chapman & Hall in 1992
ISBN 978-0-412-42750-3 ISBN 978-1-4899-4427-6 (eBook)
DOI 10.1007/978-1-4899-4427-6
Typeset in 10/12 pt Times by Pure Tech Corporation, India
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The publisher makes no representation, express or implied, with regard to the
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A catalogue record for this book is available from the British Library
Library of Congress Cataloging-in-Publication data
Bourne, Donald Edward.
Vector analysis and cartesian tensors / D.E. Bourne and P.C. Kendall.—
3rd ed.
p. cm.
Includes bibliographical references and index.
I. Vector analysis. 2. Calculus of tensors. I. Kendall, P.C. (Peter Calvin)
II. Title.
QA433.B63 1992 91-28636
515'.63—dc20 CIP
Contents
Preface ix
Preface to second edition Xl
1 Rectangular cartesian coordinates and rotation of axes
1.1 Rectangular cartesian coordinates 1
1.2 Direction cosines and direction ratios 5
1.3 Angles between lines through the origin 6
1.4 The orthogonal projection of one line on another 8
1.5 Rotation of axes 9
1.6 The summation convention and its use 14
1.7 Invariance with respect to a rotation of the axes 17
1.8 Matrix notation 19
2 Scalar and vector algebra
2.1 Scalars 21
2.2 Vectors: basic notions 22
2.3 Multiplication of a vector by a scalar 28
2.4 Addition and subtraction of vectors 30
2.5 The unit vectors i, j, k 34
2.6 Scalar products 35
2.7 Vector products 40
2.8 The triple scalar product 48
2.9 The triple vector product 51
2.10 Products of four vectors 52
2.11 Bound vectors 53
3 Vector functions of a real variable. Differential geometry
of curves
3.1 Vector functions and their geometrical representation 55
3.2 Differentiation of vectors 60
3.3 Differentiation rules 62
3.4 The tangent to a curve. Smooth, piecewise smooth and
simple curves 63
I ______________________
I~ C__O _N_ T_E__NT_ S_ _____________________~
VI
3.5 Arc length 69
3.6 Curvature and torsion 70
3.7 Applications in kinematics 75
4 Scalar and vector fields
4.1 Regions 89
4.2 Functions of several variables 90
4.3 Definitions of scalar and vector fields 96
4.4 Gradient of a scalar field 97
4.5 Properties of gradient 99
4.6 The divergence and curl of a v~ctor field 104
4.7 The del-operator 106
4.8 Scalar invariant operators 110
4.9 Useful identities 114
4.10 Cylindrical and spherical polar coordinates 118
4.11 General orthogonal curvilinear coordinates 122
4.12 Vector components in orthogonal curvilinear coordinates 128
4.13 Expressions for grad fl, div F, curl F, and V2 in
orthogonal curvilinear coordinates 130
4.14 Vector analysis in n-dimensional space 136
4.15 Method of Steepest Descent 139
5 Line, surface and volume integrals
5.1 Line integral of a scalar field 147
5.2 Line integrals of a vector field 153
5.3 Repeated integrals 156
5.4 Double and triple integrals 158
5.5 Surfaces 172
5.6 Surface integrals 181
5.7 Volume integrals 189
6 Integral theorems
6.1 Introduction 195
6.2 The divergence theorem (Gauss's theorem) 195
6.3 Green's theorems 204
6.4 Stokes's theorem 209
6.5 Limit definitions of div F and curl F 220
6.6 Geometrical and physical significance of divergence and curl 222
7 Applications in potential theory
7.1 Connectivity 225
7.2 The scalar potential 226
7.3 The vector potential 230
7.4 Poisson's equation 232
II
CONTENTS Vll
7.5 Poisson's equation in vector form 237
7.6 Helmholtz's theorem 238
7.7 Solid angles 239
8 Cartesian tensors
8.1 Introduction 244
8.2 Cartesian tensors: basic algebra 245
8.3 Isotropic tensors 250
8.4 Tensor fields 259
8.5 The divergence theorem in tensor field theory 263
9 Representation theorems for isotropic tensor functions
9.1 Introduction 265
9.2 Diagonalization of second order symmetrical tensors 266
9.3 Invariants of second order symmetrical tensors 272
9.4 Representation of isotropic vector functions 273
9.5 Isotropic scalar functions of symmetrical second order
tensors 275
9.6 Representation of an isotropic tensor function 277
Appendix A Determinants 282
Appendix B Expressions for grad, diy, curl, and V2 in cylindrical
and spherical polar coordinates 284
Appendix C The chain rule for Jacobians 286
Answers to exercises 287
Index 299
Preface
We are very grateful to Chapman & Hall for their offer to reset this book
completely. This has given us the opportunity to include small but important
teaching points which have accumulated over a long period, to improve the
notation and the diagrams, and to introduce some new material. The kinema
tics section has been extended to deal with the existence and nature of angular
velocity, including rotating frames of reference and the concept of the Coriolis
force. A new section on the application of vector analysis in optimization
theory has been added, giving a simple approach to the method of steepest
descent, which students have found stimulating. New examples and exercises
have been added, and some deleted.
D.E. BOURNE
P.e.
KENDALL
Preface to second edition
The most popular textbook approach to vector analysis begins with the defini
tion of a vector as an equivalence class of directed line segments - or, more
loosely, as an entity having both magnitude and direction. This approach is no
doubt appealing because of its apparent conceptual simplicity, but it is fraught
with logical difficulties which need careful handling if they are to be properly
resolved. Consequently, students often have difficulty in understanding fully
the early parts of vector algebra and many rapidly lose confidence. Another
disadvantage is that subsequent developments usually make frequent appeal
to geometrical intuition and much care is needed if analytical requirements
are not to be obscured or overlooked. For example, it is seldom made clear
that the definitions of the gradient of a scalar field and the divergence and curl
of a vector field imply that these fields are continuously differentiable, and
hence that the mere existence of the appropriate first order partial derivatives
is insufficient.
The account of vector analysis presented in this volume is based upon the
definition of a vector in terms of rectangular cartesian components which
satisfy appropriate rules of transformation under changes of axes. This ap
proach has now been used successfully for ten years in courses given from the
first year onwards to undergraduate mathematicians and scientists, and offers
several advantages. The rules for addition and subtraction of vectors, for
finding scalar and vector products and differentiation are readily grasped, and
the ability to handle vectors so easily gives the student immediate confidence.
The later entry into vector field theory takes place naturally with gradient,
divergence and curl being defined in their cartesian forms. This avoids the
alternative, more sophisticated, definitions involving limits of integrals. An
other advantage of the direct treatment of vectors by components is that
introducing the student at a later stage to tensor analysis is easier. At that stage
tensors are seen as a widening of the vector concept and no mental readjust
ment is necessary.
The approach to vectors through rectangular cartesian components does not
obscure the intuitive idea of a vector as an entity with magnitude and direction.
The notion emerges as an almost immediate consequence of the definition and
xu I L-I _______P _R_E_F_A_C_E_T_O_S_E_C_O__ND _E_ D_IT_I_O_N_ ______- --'
is more soundly based, inasmuch as both the magnitude and direction then
have precise analytical interpretations. The familiar parallelogram law of ad
dition also follows easily.
The essential background ideas associated with rotations of rectangular
cartesian coordinate axes are introduced in Chapter I at a level suitable for
undergraduates beginning their first year. The second and third chapters deal
with the basic concepts of vector algebra and differentiation of vectors, res
pectively; applications to the differential geometry of curves are also given in
preparation for later work.
Vector field theory begins in Chapter 4 with the definitions of gradient,
divergence and curl. We also show in this chapter how orthogonal curvilinear
coordinate systems can be handled within the framework of rectangular car
tesian theory.
An account of line, surface and volume integrals is given in the fifth chapter
in preparation for the integral theorems of Gauss, Stokes and Green which are
discussed in Chapter 6. The basic approach to vectors that we have adopted
enables rigorous proofs to be given which are nevertheless within the grasp
of the average student. Chapter 7 deals with some applications of vector
analysis in potential theory and presents proofs of the principal theorems.
Chapters g and 9, on cartesian tensors, have been added to this second
edition in response to the suggestion that it would be useful to have between
two covers most of the vector and tensor analysis that undergraduates require.
The case for adding this material is strengthened by the fact that the approach
to vectors in the early chapters makes the transition to tensors quite straight
forward. Chapter 8 deals with the basic algebra and calculus of cartesian
tensors, including an account of isotropic tensors of second, third and fourth
order. Chapter 9 briefly discusses those properties of second order tensors
which have risen to importance in continuum mechanics over the last twenty
years. Some theorems on invariants and the representation of isotropic tensor
functions are proved.
We warmly acknowledge the many useful comments from students and
colleagues who have worked with the first edition. They have enabled us to
make improvements to the original text. We particularly thank the following:
Dr G.T. Kneebone and Professor L. Mirsky for their early interest in the first
edition; Professor A. Jeffrey and Thomas Nelson and Sons Ltd without whom
this new edition would not have appeared.
D.E. BOURNE
P.e.
KENDALL