Vector Analysis and Cartesian Tensors

315 Pages·2018·18.38 MB·English

by  Donald Edward

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
CRCCRC Press
Taylor & Francis Group
Boca Raton London New York
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First published 1967, 1977, 1992 by CRC Press
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Reissued 2018 by CRC Press
© 1967, 1977, 1992 by D.E. Bourne and P.C. Kendall
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Contents
(cid:9)
Preface ix
(cid:9)
Preface to second edition xi
1 Rectangular cartesian coordinates and rotation of axes
1.1(cid:9) Rectangular cartesian coordinates(cid:9) 1
1.2 Direction cosines and direction ratios(cid:9) 5
1.3 Angles between lines through the origin(cid:9) 6
1.4(cid:9) The orthogonal projection of one line on another (cid:9) 8
1.5(cid:9) Rotation of axes(cid:9) 9
1.6 The summation convention and its use(cid:9) 14
1.7(cid:9) Invariance with respect to a rotation of the axes(cid:9) 17
1.8 Matrix notation(cid:9) 19
2 Scalar and vector algebra
2.1 Scalars(cid:9) 21
2.2 Vectors: basic notions(cid:9) 22
2.3(cid:9) Multiplication of a vector by a scalar (cid:9) 28
2.4(cid:9) Addition and subtraction of vectors(cid:9) 30
2.5(cid:9) The unit vectors i, j, k(cid:9) 34
2.6 Scalar products(cid:9) 35
2.7 Vector products(cid:9) 40
2.8 The triple scalar product(cid:9) 48
2.9 The triple vector product(cid:9) 51
2.10 Products of four vectors(cid:9) 52
2.11 Bound vectors(cid:9) 53
3 Vector functions of a real variable. Differential geometry
of curves
3.1(cid:9) Vector functions and their geometrical representation(cid:9) 55
3.2(cid:9) Differentiation of vectors(cid:9) 60
3.3 Differentiation rules(cid:9) 62
3.4 The tangent to a curve. Smooth, piecewise smooth and
simple curves(cid:9) 63
(cid:9)
vi CONTENTS
(cid:9)
3.5 Arc length 69
(cid:9)
3.6 Curvature and torsion 70
(cid:9)
3.7 Applications in kinematics 75
4 Scalar and vector fields
4.1 Regions(cid:9) 89
4.2 Functions of several variables(cid:9) 90
4.3(cid:9) Definitions of scalar and vector fields(cid:9) 96
4.4(cid:9) Gradient of a scalar field(cid:9) 97
4.5(cid:9) Properties of gradient (cid:9) 99
4.6(cid:9) The divergence and curl of a vector field(cid:9) 104
4.7 The del-operator (cid:9) 106
4.8(cid:9) Scalar invariant operators(cid:9) 110
4.9 Useful identities(cid:9) 114
4.10 Cylindrical and spherical polar coordinates(cid:9) 118
4.11 General orthogonal curvilinear coordinates(cid:9) 122
4.12 Vector components in orthogonal curvilinear coordinates(cid:9) 128
4.13 Expressions for grad 0, div F, curl F, and V2 in
orthogonal curvilinear coordinates(cid:9) 130
4.14 Vector analysis in n-dimensional space(cid:9) 136
4.15 Method of Steepest Descent (cid:9) 139
5 Line, surface and volume integrals
5.1(cid:9) Line integral of a scalar field(cid:9) 147
5.2(cid:9) Line integrals of a vector field(cid:9) 153
5.3 Repeated integrals(cid:9) 156
5.4 Double and triple integrals(cid:9) 158
5.5 Surfaces(cid:9) 172
5.6 Surface integrals(cid:9) 181
5.7 Volume integrals(cid:9) 189
6 Integral theorems
6.1 Introduction(cid:9) 195
6.2 The divergence theorem (Gauss's theorem)(cid:9) 195
6.3 Green's theorems(cid:9) 204
6.4 Stokes's theorem(cid:9) 209
6.5(cid:9) Limit definitions of div F and curl F(cid:9) 220
6.6(cid:9) Geometrical and physical significance of divergence and curl 222
7 Applications in potential theory
7.1 Connectivity(cid:9) 225
7.2 The scalar potential(cid:9) 226
7.3(cid:9) The vector potential(cid:9) 230
7.4 Poisson's equation(cid:9) 232
CONTENTS vii
7.5(cid:9) Poisson's equation in vector form(cid:9) 237
7.6 Helmholtz's theorem(cid:9) 238
7.7 Solid angles(cid:9) 239
8 Cartesian tensors
8.1 Introduction(cid:9) 244
8.2 Cartesian tensors: basic algebra(cid:9) 245
8.3 Isotropic tensors(cid:9) 250
8.4 Tensor fields(cid:9) 259
8.5 The divergence theorem in tensor field theory(cid:9) 263
9 Representation theorems for isotropic tensor functions
9.1 Introduction(cid:9) 265
9.2 Diagonalization of second order symmetrical tensors(cid:9) 266
9.3(cid:9) Invariants of second order symmetrical tensors(cid:9) 272
9.4(cid:9) Representation of isotropic vector functions(cid:9) 273
9.5(cid:9) Isotropic scalar functions of symmetrical second order
tensors(cid:9) 275
9.6 Representation of an isotropic tensor function(cid:9) 277
Appendix A Determinants(cid:9) 282
Appendix B Expressions for grad, div, curl, and V2 in cylindrical
and spherical polar coordinates(cid:9) 284
(cid:9)
Appendix C The chain rule for Jacobians 286
(cid:9)
Answers to exercises 287
(cid:9)
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.C. 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