TENSOR ANALYSISwith applications to

Geometry and Continuum Mechanics

C J Camilleri

Physical Components of Tensor Derivatives

˜∇mbik =1

hM

∂bik∂um

+bik

hIhKhM

∂hIhK

∂um−

∑ hL

hIhM

{

lim

}

blk

˜δbik . . . (u, t)

δt=

∂bik . . .

∂t+ vm ˜∇mbik . . . ,m

˜Dbik . . . (u, t)

Dt=

∂bik . . .

∂t+ vm ˜∇mbik . . . +

∑

ωimbmk . . .

˜Vb. . k . .

. i . . . (u, t)

Vt=

˜Db . ik . .

Dt+

∑

emib .mk . . −∑′

ekmb . im . .

˜£v b. k

i = vm ˜∇mbik + ωimbmk + ωkmbim + eimbmk − ekmbim Curvilinear Coordinates

Summation is understood over repeated lower-case indices. No components or summation are implied

by upper-case suffixes of the scale factors – they are just labels. At each stage an upper-case suffix

‘follows’ the value of the corresponding lower-case index. The∑

denotes the sum of all such terms, one

for each similar index of the tensor.

Electronic

Portable Document Format (PDF)

copy to

M J Roberts

of

Portsmouth, England

With best wishes

TENSOR ANALYSIS

with applications to

Geometry and Continuum Mechanics

C J Camilleri

Department of Mathematics

University of Malta

Second Edition

July 16, 2004

First Edition published in 1999 byMalta University Press, Tal-Qroqq, Msida, Malta

Second Edition in electronic-pdf format published in 2004 byC J Camilleri, Department of MathematicsUniversity of Malta, Tal-Qroqq, Msida, Malta

Typesetting and Page Layout Produced C J Camilleri.

All rights reserved. No part of this publication may be reproduced,stored in a retrieval system or transmitted in any form or by anymeans, electronic, mechanical, photocopying, recording or otherwisewithout the prior permission of the author.

Copyright c© C J Camilleri 1999 and 2004.

The text was prepared by the author using LaTEX2.09 under Windows XP.The diagrams were drawn by the author using AutoCad.

Windows XP is a trademark of Microsoft Corp.AutoCad is a trademark of Autodesk, Inc.

To my beloved wife Kate, who makes life a joy,

and my children

Anthony, Susan, David, Christopher

Contents

Preface vii

1 Notation and Definitions 11.1 Index Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Summation Convention . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Symmetry and Skew-symmetry . . . . . . . . . . . . . . . . . . 4

1.4 Elementary Algebraic Operations . . . . . . . . . . . . . . . . . 5

1.4.1 Equality of Holors – Free Index Balance . . . . . . . . . 5

1.4.2 Null Holor . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4.3 Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4.4 Product . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4.5 Contraction . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4.6 Contracted or Inner Product . . . . . . . . . . . . . . . 8

1.5 The Alternators and the Kronecker Deltas . . . . . . . . . . . . 8

Exercises 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.6 Determinant of a (3× 3) Matrix . . . . . . . . . . . . . . . . . 11

1.6.1 Useful Properties . . . . . . . . . . . . . . . . . . . . . . 11

1.6.2 Cofactors . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Exercises 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.7 Matrices and Vectors . . . . . . . . . . . . . . . . . . . . . . . . 15

1.7.1 Multiplication of Matrices . . . . . . . . . . . . . . . . . 16

1.7.2 Square Matrices . . . . . . . . . . . . . . . . . . . . . . 18

Exercises 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2 Rectilinear Coordinate Systems 252.1 Rectangular Systems and Base Vectors . . . . . . . . . . . . . . 25

2.2 Scalar and Vector Products . . . . . . . . . . . . . . . . . . . . 27

Exercises 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.3 Oblique Systems and Base Vectors . . . . . . . . . . . . . . . . 28

i

ii CONTENTS Tensors_cjc

2.4 Covariant and Contravariant Components . . . . . . . . . . . . 322.4.1 Relation Between Covariant and Contravariant Compo-

nents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.5 Physical Components . . . . . . . . . . . . . . . . . . . . . . . . 332.6 Scalar and Vector Products . . . . . . . . . . . . . . . . . . . . 342.7 Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.8 Relation Between Skew and Rectangular Systems . . . . . . . . 37

Exercises 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.9 The Metric Form . . . . . . . . . . . . . . . . . . . . . . . . . . 432.10 Magnitude of a Vector . . . . . . . . . . . . . . . . . . . . . . . 432.11 Angle Between Two Directions . . . . . . . . . . . . . . . . . . 44

Exercises 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3 Curvilinear Coordinate Systems 493.1 Cylindrical Polar Coordinates . . . . . . . . . . . . . . . . . . . 493.2 Spherical Polar Coordinates . . . . . . . . . . . . . . . . . . . . 503.3 General Curvilinear Coordinates . . . . . . . . . . . . . . . . . 513.4 Natural Base Vectors . . . . . . . . . . . . . . . . . . . . . . . . 523.5 Reciprocal Base Vectors . . . . . . . . . . . . . . . . . . . . . . 523.6 Components of Vectors on Local Basis . . . . . . . . . . . . . . 543.7 The Metric Tensor . . . . . . . . . . . . . . . . . . . . . . . . . 543.8 Element of Volume . . . . . . . . . . . . . . . . . . . . . . . . . 553.9 Magnitude of a Vector . . . . . . . . . . . . . . . . . . . . . . . 563.10 Angle Between Two Directions . . . . . . . . . . . . . . . . . . 56

Exercises 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.11 Background Rectangular Cartesian System . . . . . . . . . . . 593.12 Orthogonal Curvilinear Coordinates . . . . . . . . . . . . . . . 60

3.12.1 Scale Factors . . . . . . . . . . . . . . . . . . . . . . . . 61Exercises 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4 Cartesian Tensors 694.1 Transformation of Rectangular Cartesian Coordinates . . . . . 694.2 Transformation of Orthonormal Basis . . . . . . . . . . . . . . 704.3 Transformation of Vector Components . . . . . . . . . . . . . . 714.4 Cartesian Tensors of Order One . . . . . . . . . . . . . . . . . . 714.5 Scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724.6 Cartesian Tensors of Order Two . . . . . . . . . . . . . . . . . 724.7 Cartesian Tensors of order n . . . . . . . . . . . . . . . . . . . . 734.8 Elementary Algebraic Operations . . . . . . . . . . . . . . . . . 74

4.8.1 Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

Tensors_cjc CONTENTS iii

4.8.2 Product . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.8.3 Contraction . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.9 The Quotient Law . . . . . . . . . . . . . . . . . . . . . . . . . 754.10 Invariance of Tensor Equations . . . . . . . . . . . . . . . . . . 764.11 Isotropic Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . 77

Exercises 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5 General Tensors 835.1 Transformation of Curvilinear Coordinates . . . . . . . . . . . . 835.2 Transformation of Local Bases . . . . . . . . . . . . . . . . . . 855.3 Transformation of Vectors . . . . . . . . . . . . . . . . . . . . . 865.4 Tensors of Order One . . . . . . . . . . . . . . . . . . . . . . . 865.5 Associated Vectors . . . . . . . . . . . . . . . . . . . . . . . . . 885.6 Tensors of Order Two . . . . . . . . . . . . . . . . . . . . . . . 88

Exercises 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.7 Tensors of Order n . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.7.1 Lateral Position of Indices – Symmetry . . . . . . . . . 915.8 Associated Tensors . . . . . . . . . . . . . . . . . . . . . . . . . 915.9 Relative or Weighted Tensors . . . . . . . . . . . . . . . . . . . 93

5.9.1 Conservation of Indices in Transformation Laws . . . . . 945.10 Reduction of Relative Tensors to Tensors . . . . . . . . . . . . 955.11 Algebraic Operations . . . . . . . . . . . . . . . . . . . . . . . . 955.12 The Quotient Law . . . . . . . . . . . . . . . . . . . . . . . . . 965.13 Invariance of General Tensor Equations . . . . . . . . . . . . . 965.14 Isotropic General Tensors . . . . . . . . . . . . . . . . . . . . . 975.15 Principal Directions . . . . . . . . . . . . . . . . . . . . . . . . 98

Exercises 5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6 Differential Tensor Calculus 1036.1 General Tensor Fields . . . . . . . . . . . . . . . . . . . . . . . 1036.2 Differentiation of General Tensor Fields . . . . . . . . . . . . . 1046.3 Base Vectors and the Christoffel Symbols . . . . . . . . . . . . 105

Exercises 6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1066.4 Intrinsic Derivative of a Vector . . . . . . . . . . . . . . . . . . 1096.5 Covariant Derivative of a Vector . . . . . . . . . . . . . . . . . 110

6.5.1 The Operator ∇j . . . . . . . . . . . . . . . . . . . . . . 112Exercises 6.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.6 Differentiation of Sum and Product . . . . . . . . . . . . . . . . 1146.7 Intrinsic and Covariant Derivative of Tensors . . . . . . . . . . 1166.8 Generalized Vector Functions . . . . . . . . . . . . . . . . . . . 117

iv CONTENTS Tensors_cjc

6.8.1 Generalized Vector Identities . . . . . . . . . . . . . . . 1196.9 Commutative Properties . . . . . . . . . . . . . . . . . . . . . . 119

Exercises 6.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1236.10 Second Derivative of a Tensor . . . . . . . . . . . . . . . . . . . 126

Exercises 6.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

7 Orthogonal Curvilinear Coordinates 1337.1 Christoffel Symbols . . . . . . . . . . . . . . . . . . . . . . . . . 1337.2 Physical Components of Tensors . . . . . . . . . . . . . . . . . 1357.3 Intrinsic Derivative . . . . . . . . . . . . . . . . . . . . . . . . . 1387.4 Covariant Derivative . . . . . . . . . . . . . . . . . . . . . . . . 1387.5 The ε-Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 1397.6 Physical Components of Vector Functions . . . . . . . . . . . . 139

Exercises 7.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

8 Differential Geometry 1458.1 Space Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1458.2 Unit Tangent to Curve . . . . . . . . . . . . . . . . . . . . . . . 1468.3 Unit Normal to Curve . . . . . . . . . . . . . . . . . . . . . . . 146

Exercises 8.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1488.4 Frenet Formulæ . . . . . . . . . . . . . . . . . . . . . . . . . . . 1498.5 Parallel Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

Exercises 8.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1518.6 Surfaces Embedded in R3 . . . . . . . . . . . . . . . . . . . . . 152

Exercises 8.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

9 Continuum Mechanics 1619.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1619.2 Convected Coordinates . . . . . . . . . . . . . . . . . . . . . . . 1629.3 Time-Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . 1649.4 Transformation to Fixed-Components . . . . . . . . . . . . . . 165

Exercises 9.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1679.5 ‘Flowing’ Coordinates – Total Derivative . . . . . . . . . . . . . 168

9.5.1 Tensor Properties of Total Derivative . . . . . . . . . . . 170Exercises 9.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

9.6 Spatial Motion Following the Particle . . . . . . . . . . . . . . 1749.6.1 Equation of Continuity . . . . . . . . . . . . . . . . . . 1749.6.2 Equations of Motion . . . . . . . . . . . . . . . . . . . . 175

9.7 Instantaneous Motion – Deformation . . . . . . . . . . . . . . . 1769.8 Convected Motion - Deformation . . . . . . . . . . . . . . . . . 178

Tensors_cjc CONTENTS v

9.9 Translated Fixed-Components . . . . . . . . . . . . . . . . . . . 1809.9.1 Convected Time Integrals . . . . . . . . . . . . . . . . . 182Exercises 9.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

9.10 Commutative Properties of Derivatives . . . . . . . . . . . . . . 1859.11 Physical Components of Derivatives . . . . . . . . . . . . . . . 1859.12 Physical Translated Fixed-Components . . . . . . . . . . . . . . 189

Exercises 9.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1919.13 Constitutive Equations of State . . . . . . . . . . . . . . . . . . 1939.14 Generalized Equations of State . . . . . . . . . . . . . . . . . . 196

Exercises 9.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

A Rheology – An Overview 201A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201A.2 Rheological Study Cycle . . . . . . . . . . . . . . . . . . . . . . 201A.3 Idealized Materials with Simple Properties . . . . . . . . . . . . 202

A.3.1 Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . 202A.3.2 Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

A.4 Materials with Complicated Properties . . . . . . . . . . . . . . 203A.4.1 Elastico-Viscous Liquids . . . . . . . . . . . . . . . . . . 204A.4.2 Visco-Elastic Solids . . . . . . . . . . . . . . . . . . . . 207

References 209

Answers and Notes 213

Index 225

vi CONTENTS Tensors_cjc

Preface

This book is based on a course of lectures given by the author at the Universityof Malta, and its main objective is to present an introductory course in tensormethods adapted to the needs of students in the physical and engineering sci-ences or in applied mathematics. The classical, component, local approach1 isemployed and the organization of the book is intended to firstly familiarize thestudent with index operations, through the use of the summation conventionand the permutation symbols, and then to use these methods in the algebraand calculus of tensors which are defined in terms of appropriate transforma-tion rules for their components denoted by indexed systems. Application togeometry and mechanics is made by including related exercises throughoutthe text, particularly in the last two chapters where tensors are applied todifferential geometry and mechanics of continuous material.

Prerequisites include an elementary knowledge of the calculus of ordinaryfunctions of several variables including the chain rule for partial differentiation,and a basic knowledge of vector algebra such as that found in the author’s bookon vector analysis [1] to which the present work is a follow up.

The emphasis of this book is on analytical techniques as revealed by thelarge number of exercises it contains. These exercises are to be taken as anintegral part of the text as suggested by their positioning within the chap-ters. Those of the problem-type are supplied with answers and should providepractice in manipulation, the rest are meant to encourage the student to de-rive some of the important standard results for himself, as extensions of thepreceding sections, which are often required and quoted later. At times theseare accompanied by instructive remarks.

The outstanding difficulty in introducing tensor notation is the amount ofindices involved and Chapter1 may appear to be unnecessarily detailed andpossibly overwhelming for first-time readers. However, it is my experience

1It is felt that students will have deeper understanding of the modern, noncomponent,global treatment of the subject after mastering the component approach; after all in solvingreal physical or engineering problems calculations are carried out on components.

vii

viii Tensors_cjc

that the student who takes the challenge of working through the exercises forhimself right at the beginning of the course, will soon realize how much easierit is to write tensor notation than to read it and will appreciate the advantagesit offers in writing complex expressions compactly. Moreover, it is hoped thatthe student also appreciates the simplification offered by the index notation ifit supplants the traditional notation of vector analysis and matrix theory.

The treatment of vectors in terms of basis along rectilinear coordinates inChapter2 provides a gentle introduction of the Riemannian metric 3-space vialocal basis along curvilinear coordinate directions considered in Chapter3.

In Chapter4 tensors are introduced in the Cartesian framework to be de-veloped more generally in the next two chapters. This may appear to involveduplication, but in my opinion this is offset by a definite pedagogical value.Physics and engineering students will welcome this lucid introduction of ten-sors to what may normally be considered a complex aspect of mathematics.

The two middle chapters make up the heart of the book. Chapter5 dealswith the algebra of absolute and relative (weighted) tensors; considers howthese are related and how associated tensors are related, addressing the im-portance of the lateral position of the indices unless the tensor is symmetric.Chapter6 deals with the calculus of tensors in Riemannian 3-space where themetric is a function of position but not of time. It considers curvature and thecondition for Euclidean 3-space.

Physical components of vectors and tensors referred to local curvilinearcoordinate directions are shown to represent quantities having the physicaldimensions of the field and therefore capable of immediate physical interpre-tation. In the case of orthogonal curvilinear coordinates, general formulaeare derived in Chapter7 for the physical components of tensors including theintrinsic and covariant derivatives.

Chapter8 provides a brief introduction to local differential geometry ofcurves and surfaces in Riemannian 3-space. Again the student is encouraged toobtain most of the standard results for himself by applying the basic methodsand concepts of tensors. This chapter may be read immediately after Chapter6or it may be omitted completely.

An introduction to the theory of deformation and flow of homogeneouscontinuous material – a subject otherwise known as ‘rheology’ – is given inChapter9 as a natural application of tensor methods. The method of keep-ing the material element ‘fixed’ by introducing a convected coordinate system(drawn in the material and deforming continuously with it) is discussed, lead-ing to Oldroyd’s material and convected time derivatives and the Lie deriva-tive.

Tensors_cjc ix

General formulae2 for the physical components of these derived tensors referredto local orthogonal curvilinear coordinate directions are established which for-mulae should prove to be invaluable for research scientists. This chapter issupplemented by an overview of rheology in Appendix A.

I have tried to provide as complete an index as possible and in some casesthe number of pages indicated is rather large. An underlined page numberindicates the most important reference to whatever is being indexed, its defi-nition or its first appearance in the text shown in bold type.

I am thankful to my colleagues for their support and encouragement, par-ticularly Dr D. Buhagiar who read the first draft and Dr J. Muscat whoscrutinized the first and final drafts, made valid criticisms and suggestions,and many encouraging remarks.

I would also like to express my gratitude to the students who sufferedthrough the many versions of the classroom editions of the text and for theirhelp in checking the exercises.

Last but certainly not least, I wish to thank my wife for her unlimitedpatience, tolerance and continuous encouragement.

The book’s faults, errors, and other imperfections are entirely my ownparticularly since I have typed the text and typeset it myself using LaTEX.

C J Camilleri

Department of MathematicsUniversity of MaltaJuly, 2004.

2The novel notation used enhances compactness and eliminates the necessity to introduceleft and right physical components in the case of non-orthogonal coordinates as suggestedby Truesdell [10].

Published by C J Camilleri

This book provides a concise butcomplete treatment of tensor analysis.It should meet the needs of mostundergraduate students in the physicaland engineering sciences or in appliedmathematics.The classical, component, local approachis employed and the organization of thebook is intended to firstly familiarizethe student with index operations,through the use of the summationconvention and the permutationsymbols, and then to use these methodsin the algebra and calculus of tensors.The emphasis of this book is onanalytical techniques as revealed by thelarge number of exercises it contains.An introduction to the theory ofdeformation and flow of homogeneouscontinuous material is given as a naturalapplication of tensor methods. Themethod of keeping the material element‘fixed’ by introducing a convectedcoordinate system (drawn in thematerial and deforming continuouslywith it) is discussed, leading toOldroyd’s material and convected timederivatives and the Lie derivative.General formulæ for the physicalcomponents of these derived tensorsreferred to local orthogonal curvilinearcoordinate directions are establishedwhich formulæ should prove to beinvaluable for research scientists. Thisis supplemented by an overview ofrheology.

CONTENTS

Preface

1. Notation and Definitions

2. Rectilinear Coordinates Systems

3. Curvilinear Coordinate Systems

4. Cartesian Tensors

5. General Tensors

6. Differential Tensor Calculus

7. Orthogonal Curvilinear Coordinates

8. Differential Geometry

9. Continuum Mechanics

APPENDICES

A. Rheology – An Overview

B. References

Answers and Notes

Index

THE AUTHOR

C J CAMILLERI is professor emeritusin the Department of Mathematicsat the University of Malta.