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Introduction to
Tensor Calculus
and
Continuum Mechanics
by J.H. Heinbockel
Department of Mathematics and Statistics
Old Dominion University
79654899.001.png
PREFACE
This is an introductory text which presents fundamental concepts from the subject
areas of tensor calculus, differential geometry and continuum mechanics. The material
presented is suitable for a two semester course in applied mathematics and is flexible
enoughtobepresented toeitherupperlevelundergraduate orbeginninggraduatestudents
majoring in applied mathematics, engineering or physics. The presentation assumes the
studentshavesomeknowledgefromtheareasofmatrixtheory,linearalgebraandadvanced
calculus. Each section includes many illustrative worked examples. At the end of each
section there is a large collection of exercises which range in di culty. Many new ideas
are presented in the exercises and so the students should be encouraged to read all the
exercises.
Thepurposeofpreparingthesenotesistocondenseintoanintroductorytextthebasic
definitions and techniques arising in tensor calculus, differential geometry and continuum
mechanics. In particular, the material is presented to (i) develop a physical understanding
of the mathematical concepts associated with tensor calculus and (ii) develop the basic
equations of tensor calculus, differential geometry and continuum mechanics which arise
in engineering applications. From these basic equations one can go on to develop more
sophisticated models of applied mathematics. The material is presented in an informal
manner and uses mathematics which minimizes excessive formalism.
The material has been divided into two parts. The first part deals with an introduc-
tion to tensor calculus and differential geometry which covers such things as the indicial
notation, tensor algebra, covariant differentiation, dual tensors, bilinear and multilinear
forms, special tensors, the Riemann Christoffel tensor, space curves, surface curves, cur-
vature and fundamental quadratic forms. The second part emphasizes the application of
tensoralgebraandcalculustoawidevarietyofappliedareasfromengineering andphysics.
The selected applications are from the areas of dynamics, elasticity, fluids and electromag-
netic theory. The continuum mechanics portion focuses on an introduction of the basic
concepts fromlinearelasticityand fluids. TheAppendix Acontainsunitsofmeasurements
from the Systeme International d’Unites along with some selected physical constants. The
Appendix B contains a listing of Christoffel symbols of the second kind associated with
various coordinate systems. The Appendix C is a summary of useful vector identities.
J.H. Heinbockel, 1996
Copyright c
1996 by J.H. Heinbockel. All rights reserved.
Reproduction and distribution of these notes is allowable provided it is for non-profit
purposes only.
INTRODUCTION TO
TENSOR CALCULUS
AND
CONTINUUM MECHANICS
PART 1: INTRODUCTION TO TENSOR CALCULUS
§ 1.1 INDEX NOTATION .................. 1
Exercise 1.1 .......................... 28
§ 1.2 TENSOR CONCEPTS AND TRANSFORMATIONS .... 35
Exercise 1.2 ........................... 54
§
1.3 SPECIAL TENSORS .................. 65
Exercise 1.3 ........................... 101
§
1.4 DERIVATIVE OF A TENSOR .............. 108
Exercise 1.4 ........................... 123
§
1.5 DIFFERENTIAL GEOMETRY AND RELATIVITY .... 129
Exercise 1.5 ........................... 162
PART 2: INTRODUCTION TO CONTINUUM MECHANICS
§ 2.1 TENSOR NOTATION FOR VECTOR QUANTITIES .... 171
Exercise 2.1 ........................... 182
§
2.2 DYNAMICS ...................... 187
Exercise 2.2 ........................... 206
§
2.3 BASIC EQUATIONS OF CONTINUUM MECHANICS ... 211
Exercise 2.3 ........................... 238
§
2.4 CONTINUUM MECHANICS (SOLIDS) ......... 243
Exercise 2.4 ........................... 272
§
2.5 CONTINUUM MECHANICS (FLUIDS) ......... 282
Exercise 2.5 ........................... 317
2.6 ELECTRIC AND MAGNETIC FIELDS .......... 325
Exercise 2.6 ........................... 347
BIBLIOGRAPHY ..................... 352
APPENDIX A UNITS OF MEASUREMENT ....... 353
APPENDIX B CHRISTOFFEL SYMBOLS OF SECOND KIND 355
APPENDIX C VECTOR IDENTITIES .......... 362
INDEX .......................... 363
§
1
PART 1: INTRODUCTION TO TENSOR CALCULUS
A scalar field describes a one-to-onecorrespondencebetween a single scalarnumber and a point. An n-
dimensional vector field is described by a one-to-one correspondence between n-numbers and a point. Let us
generalize these concepts by assigning n -squared numbers to a single point or n -cubed numbers to a single
point. When these numbers obey certain transformation laws they become examples of tensor fields. In
general, scalar fields are referred to as tensor fields of rank or order zero whereas vector fields are called
tensor fields of rank or order one.
Closely associated with tensor calculus is the indicial or index notation. In section 1 the indicial
notation is defined and illustrated. We also define and investigate scalar, vector and tensor fields when they
are subjected to various coordinate transformations. It turns out that tensors have certain properties which
are independent of the coordinate system used to describe the tensor. Because of these useful properties,
we can use tensors to represent various fundamental laws occurring in physics, engineering, science and
mathematics. These representations are extremely useful as they are independent of the coordinate systems
considered.
1.1 INDEX NOTATION
Two vectors A and B can be expressed in the component form
A = A 1 e 1 + A 2 e 2 + A 3 e 3 and
B = B 1 e 1 + B 2 e 2 + B 3 e 3 ,
where e 1 , e 2 and e 3 are orthogonal unit basis vectors. Often when no confusion arises, the vectors A and
B are expressed for brevity sake as number triples. For example, we can write
A =( A 1 ,A 2 ,A 3 ) ad B =( B 1 ,B 2 ,B 3 )
where it is understood that only the components of the vectors A and B are given. The unit vectors would
be represented
e 1 =(1 , 0 , 0) ,
e 2 =(0 , 1 , 0) ,
e 3 =(0 , 0 , 1) .
Astillshorternotation,depictingthevectors A and B istheindexorindicialnotation. Intheindexnotation,
the quantities
A i , i =1 , 2 , 3d B p , p =1 , 2 , 3
representthe components ofthevectors A and B. This notationfocuses attentiononly onthe components of
the vectors and employs a dummy subscript whose rangeover the integers is specified. The symbol A i refers
to all of the components of the vector A simultaneously. The dummy subscript i can have any of the integer
values 1 , 2or3 . For i = 1 we focus attention on the A 1 component of the vector A. Setting i =2focuses
attention on the second component A 2 of the vector A and similarly when i = 3 we can focus attention on
the third component of A. The subscript i is a dummy subscript and may be replaced by another letter, say
p , so long as one specifies the integer values that this dummy subscript can have.
§

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