Table Of ContentUniversitext
Juan Ramon Ruiz-Tolosa
Enrique Castillo
From Vectors
to Tensors
Springer
Authors
Juan Ramon Ruiz-Tolosa
Enrique Castillo
Universidad Cantabria
Depto. Matematica Aplicada
Avenida de los Castros
39005 Santander, Spain
castie@unican,es
Library of Congress Control Number: 20041114044
Mathematics Subject Classification (2000): 15A60,15A72
ISBN3-540-22887-X Springer Berlin Heidelberg New York
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To the memory of Bernhard Riemann and Albert Einstein
Preface
It is true that there exist many books dedicated to linear algebra and some
what fewer to multilinear algebra, written in several languages, and perhaps
one can think that no more books are needed. However, it is also true that in
algebra many new results are continuously appearing, different points of view
can be used to see the mathematical objects and their associated structures,
and different orientations can be selected to present the material, and all of
them deserve publication.
Under the leadership of Juan Ramon Ruiz-Tolosa, Professor of multilin
ear algebra, and the collaboration of Enrique Castillo, Professor of applied
mathematics, both teaching at an engineering school in Santander, a tensor
textbook has been born, written from a practical point of view and free from
the esoteric language typical of treatises written by algebraists, who are not
interested in descending to numerical details. The balance between follow
ing this line and keeping the rigor of classical theoretical treatises has been
maintained throughout this book.
The book assumes a certain knowledge of linear algebra, and is intended as
a textbook for graduate and postgraduate students and also as a consultation
book. It is addressed to mathematicians, physicists, engineers, and applied
scientists with a practical orientation who are looking for powerful tensor
tools to solve their problems.
The book covers an existing chasm between the classic theory of tensors
and the possibility of solving tensor problems with a computer. In fact, the
computational algebra is formulated in matrix form to facilitate its implemen
tation on computers.
The book includes 197 examples and end-of-chapter exercises, which makes
it specially suitable as a textbook for tensor courses. This material combines
classic matrix techniques together with novel methods and in many cases the
questions and problems are solved using different methods. They confirm the
applied orientation of the book.
A computer package, written in Mathematica, accompanies this book,
available on: http://personales.unican.es/castie/tensors. In it, most of the
novel methods developed in the book have been implemented. We note that
existing general computer software packages (Mathematica, Mathlab, etc.) for
tensors are very poor, up to the point that some problems cannot be dealt
VIII Preface
with using computers because of the lack of computer programs to perform
these operations.
The main contributions of the book are:
1. The book employs a new technique that permits one to extend (stretch)
the tensors, as one-column matrices, solve on these matrices the desired
problems, and recover the initial format of the tensor (condensation). This
technique, applied in all chapters, is described and used to solve matrix
equations in Chapter 1.
2. An important criterion is established in Chapter 2 for all the components
of a tensor to have a given ordering, by the definition of a unique canonical
tensor basis. This permits the mentioned technique to be applied.
3. In Chapter 3, factors are illustrated that have led to an important con
fusion in tensor books due to inadequate notation of tensors or tensor
operations.
4. In addition to dealing with the classical topics of tensor books, new tensor
concepts are introduced, such as the rotation of tensors, the transposer
tensor, the eigentensors, and the permutation tensor structure, in Chapter
5.
5. A very detailed study of generalized Kronecker deltas is presented in Chap
ter 8.
6. Chapter 10 is devoted to mixed exterior algebras, analyzing the problem
of change-of-basis and the exterior product of this kind of tensors.
7. In Chapter 11 the rules for the "Euclidean contraction" are given in detail.
This chapter ends by introducing the geometric concepts to tensors.
8. The orientation and polar tensors in Euclidean spaces are dealt with in
Chapter 12.
9. In Chapter 13 the Gram matrices G(r) are established to connect exterior
tensors.
10. Chapter 14 is devoted to Euclidean tensors in E^(R), affine geometric
tensors (homographies), and some important tensors in physics and me
chanics, such as the stress and strain tensors, the elastic tensor and the
inertial moment tensor. It is shown how tensors allow one to solve very
interesting practical problems.
In summary, the book is not a standard book on tensors because of its
orientation, the many novel contributions included in it, the careful notation
and the stretching-condensing techniques used for most of the transformations
used in the book. We hope that our readers enjoy reading this book, discover
a new world, and acquire stimulating ideas for their applications and new
contributions and research.
The authors want to thank an anonimous French referee for the careful
reading of the initial manuscript, and to Jeffrey Boys for the copyediting of
the final manuscript.
Santander, Juan Ramon Rmz-Tolosa
September 30, 2004 Enrique Castillo
Contents
Part I Basic Tensor Algebra
Tensor Spaces 3
1.1 Introduction 3
1.2 Dual or reciprocal coordinate frames in affine Euclidean spaces 3
1.3 Different types of matrix products 8
1.3.1 Definitions 8
1.3.2 Properties concerning general matrices 10
1.3.3 Properties concerning square matrices 11
1.3.4 Properties concerning eigenvalues and eigenvectors ... 12
1.3.5 Properties concerning the Schur product 13
1.3.6 Extension and condensation of matrices 13
1.3.7 Some important matrix equations 17
1.4 Special tensors 26
1.5 Exercises . 30
Introduction to Tensors 33
2.1 Introduction 33
2.2 The triple tensor product linear space 33
2.3 Einstein's summation convention 36
2.4 Tensor analytical representation 37
2.5 Tensor product axiomatic properties 38
2.6 Generalization 40
2.7 Illustrative examples , 41
2.8 Exercises 46
Homogeneous Tensors 47
3.1 Introduction 47
3.2 The concept of homogeneous tensors 47
3.3 General rules about tensor notation 48
3.4 The tensor product of tensors 50
X Contents
3.5 Einstein's contraction of the tensor product 54
3.6 Matrix representation of tensors 56
3.6.1 First-order tensors 56
3.6.2 Second-order tensors 57
3.7 Exercises 61
4 Change-of-basis in Tensor Spaces 65
4.1 Introduction 65
4.2 Change of basis in a third-order tensor product space ....... 65
4.3 Matrix representation of a change-of-basis in tensor spaces ... 67
4.4 General criteria for tensor character 69
4.5 Extension to homogeneous tensors 72
4.6 Matrix operation rules for tensor expressions 74
4.6.1 Second-order tensors (matrices) 74
4.6.2 Third-order tensors 77
4.6.3 Fourth-order tensors 78
4.7 Change-of-basis invariant tensors: Isotropic tensors 80
4.8 Main isotropic tensors 80
4.8.1 The null tensor 80
4.8.2 Zero-order tensor (scalar invariant) 80
4.8.3 Kronecker's delta 80
4.9 Exercises 106
5 Homogeneous Tensor Algebra: Tensor Homomorphisms Ill
5.1 Introduction Ill
5.2 Main theorem on tensor contraction Ill
5.3 The contracted tensor product and tensor homomorphisms... 113
5.4 Tensor product applications 119
5.4.1 Common simply contracted tensor products 119
5.4.2 Multiply contracted tensor products 120
5.4.3 Scalar and inner tensor products 120
5.5 Criteria for tensor character based on contraction 122
5.6 The contracted tensor product in the reverse sense: The
quotient law 124
5.7 Matrix representation of permutation homomorphisms 127
5.7.1 Permutation matrix tensor product types in K"^ . . . .. 127
5.7.2 Linear span of precedent types 129
5.7.3 The isomers of a tensor 137
5.8 Matrices associated with simply contraction homomorphisms . 141
5.8.1 Mixed tensors of second order (r = 2): Matrices 141
5.8.2 Mixed tensors of third order (r = 3) 141
5.8.3 Mixed tensors of fourth order (r = 4) 142
5.8.4 Mixed tensors of fifth order (r = 5) 143
5.9 Matrices associated with doubly contracted homomorphisms . 144
5.9.1 Mixed tensors of fourth order (r = 4) 144
Contents XI
5.9.2 Mixed tensors of fifth order (r = 5) 145
5.10 Eigentensors 159
5.11 Generahzed multihnear mappings 165
5.11.1 Theorems of simihtude with tensor mappings 167
5.11.2 Tensor mapping types 168
5.11.3 Direct n-dimensional tensor endomorphisms 169
5.12 Exercises 183
Part II Special Tensors
6 Symmetric Homogeneous Tensors: Tensor Algebras 189
6.1 Introduction 189
6.2 Symmetric systems of scalar components 189
6.2.1 Symmetric systems with respect to an index subset .. 190
6.2.2 Symmetric systems. Total symmetry 190
6.3 Strict components of a symmetric system 191
6.3.1 Number of strict components of a symmetric system
with respect to an index subset 191
6.3.2 Number of strict components of a symmetric system . 192
6.4 Tensors with symmetries: Tensors with branched symmetry,
symmetric tensors 193
6.4.1 Generation of symmetric tensors 194
6.4.2 Intrinsic character of tensor symmetry: Fundamental
theorem of tensors with symmetry 197
6.4.3 Symmetric tensor spaces and subspaces. Strict
components associated with subspaces 204
6.5 Symmetric tensors under the tensor algebra perspective 206
6.5.1 Symmetrized tensor associated with an arbitrary
pure tensor 210
6.5.2 Extension of the symmetrized tensor associated with
a mixed tensor 210
6.6 Symmetric tensor algebras: The (8)5 product 212
6.7 Illustrative examples 214
6.8 Exercises 220
7 Anti-symmetric Homogeneous Tensors, Tensor and Inner
Product Algebras 225
7.1 Introduction 225
7.2 Anti-symmetric systems of scalar components 225
7.2.1 Anti-symmetric systems with respect to an index
subset 226
7.2.2 Anti-symmetric systems. Total anti-symmetry 228
7.3 Strict components of an anti-symmetric system and with
respect to an index subset 228
