Foreword
Every science elaborates tools for the description of its objects of study.
In classical mechanics we make extensive use of vectorial quantities: forces,
moments, positions, velocities, momenta. Confining ourselves to a single
coordinate frame, we can regard a vector as a fixed column matrix. The definitive
trait of a vector quantity, however, is its objectivity; a vector does not
depend on our choice of coordinate frame. This means that as soon as the
components of a force are specified in one frame, the components of that
force relative to any other frame can be found through the use of appropriate
transformation rules.
But vector quantities alone do not suffice for the description of continuum
media. The stress and strain at a point inside a body are also objective
quantities; however, the specification of each of these relative to a given
frame requires a square matrix of elements. Under changes of frame these
elements transform according to rules different from those mentioned above.
Stress and strain are examples of tensors of the second rank. We could go
on to cite other objective quantities that occur in the mechanics of continua.
The set of elastic moduli associated with Hooke's law comprise a tensor of
the fourth rank; as such, these moduli obey yet another set of transformation
rules. Despite the differences that exist between the transformation laws
for the various types of objective quantities, they all fit into a unified
scheme: the theory of tensors.
Tensor theory not only relieves our memory from a huge burden, but enables
us to carry out differential operations with ease. This is the case even
in curvilinear coordinate systems. Through the unmatched simplicity and brevity
it affords, tensor analysis has attained the status of a general language
that can be spoken across the various areas of continuum physics. A full
comprehension of this language has become necessary for those working in
electromagnetism, the theory of relativity, or virtually any other field-theoretic
discipline. More modem books on physical subjects invariably contain appendices
in which various vector and tensor identities are listed. These may suffice
when one wishes to verify the steps in a development, but can leave one in
doubt as to how the facts were established or, a fortiori, how they could
be adapted to other circumstances. On the other hand, a comprehensive treatment
of tensors (e.g., involving a full excursion into multilinear algebra) is
necessarily so large as to be flatly inappropriate for the student or practicing
engineer.
Hence the need for a treatment of tensor theory that does justice to the
subject and is friendly to the practitioner. The authors of the present book
have met these objectives with a presentation that is simple, clear, and
sufficiently detailed. The concise text explains practically all those formulas
needed to describe objects in three-dimensional space. Occurrences in physics
are mentioned when helpful, but the discussion is kept largely independent
of application area in order to appeal to the widest possible audience. A
final chapter on the properties of curves and surfaces has been included;
a brief introduction to the study of these properties can be considered as
an informative and natural extension of tensor theory.
I.I. Vorovich
Professor of Mechanics and Mathematics
Rostov State University, Russia
Fellow of Russian Academy of Sciences
Preface
Originally a vector was regarded as an arrow of a certain length that could
represent a force acting on a material point. Over a period of many years,
this naive viewpoint evolved into the modern interpretation of the notion
of vector and its extension to tensors. It was found that the use of vectors
and tensors led to a proper description of certain properties and behaviors
of real natural objects: those aspects that do not depend on the coordinate
systems we introduce in space. This independence means that if we define
such properties using one coordinate system, then in another system we can
recalculate these characteristics using valid transformation rules. The ease
with which a given problem can be solved often depends on the coordinate
system employed. So in applications we must apply various coordinate systems,
derive corresponding equations, and understand how to recalculate results
in other systems. This book provides the tools necessary for such calculation.
Many physical laws are cumbersome when written in coordinate form but become
compact and attractive looking when written in tensorial form. Such compact
forms are easy to remember, and can be used to state complex physical boundary
value problems. It is conceivable that soon an ability to merely formulate
statements of boundary value problems will be regarded as a fundamental skill
for the practitioner. Indeed, computer software is slowly advancing toward
the point where the only necessary input data will be a coordinate-free statement
of a boundary value problem; presumably the user will be able to initiate
a solution process in a certain frame and by a certain method (analytical,
numerical, or mixed), or simply ask the computer algorithm to choose the
best frame and method. In this way vectors and tensors will become important
elements of the macro-language for the next generation of software in engineering
and applied mathematics.
We would like to thank the editorial staff at World Scientific — especially
Mr. Kwang-Wei Tjan and Ms. Sook-Cheng Lim — for their assistance in the production
of this book. Professor Byron C. Drachman of Michigan State University provided
useful commentary on the manuscript in its initial stages. Lastly, Natasha
Lebedeva and Beth Lannon-Cloud deserve thanks for their patience and support.
Department of Mechanics and Mathematics
L.P. Lebedev
Rostov State University, Russia
&
Department of Mathematics
National University of Colombia, Colombia
Department of Electrical and Computer Engineering
M.J. Cloud
Lawrence Technological University, USA