Tensor Analysis and Nonlinear Tensor Functions

Tensor Analysis and Nonlinear Tensor

Functions

by

Yu. I. Dimitrienko Bauman Moscow State Technical University,

Moscow. Russia

SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN 978-90-481-6169-0 ISBN 978-94-017-3221-5 (eBook) DOI 10.1007/978-94-017-3221-5

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Originally published by Kluwer Academic Publishers in 2002

Softcover reprint of the hardcover 1st edition 2002

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TABLE OF CONTENTS

Preface vii

Sources of Tensor Calculus ix

Introduction xv

Chapter 1. TENSOR ALGEBRA 1 1.1. Local Basis Vectors. Jacobian and Metric Matrices 1 1.2. Vector Product 12 1.3. Geometric Definition of a Tensor and Algebraic Opera-

tions on Tensors 17 1.4. Algebra of Tensor Fields 34 1.5. Eigenvalues of a Tensor 40 1.6. Symmetric, Skew-Symmetric and Orthogonal Tensors 43 1. 7. Physical Components of Tensors 50 1.8. Tensors of Higher Orders 53 1.9. Pseudotensors 59

Chapter 2. TENSORS IN LINEAR SPACES 65 2.1. Linear n-Dimensional Space 65 2.2. Matrices of the nth Order 74 2.3. Linear Transformations of n-Dimensional Spaces 85 2.4. Dual Space 93 2.5. Algebra of Tensors inn-Dimensional Linear Spaces 98 2.6. Outer Forms · 116

Chapter 3. GROUPS OF TRANSFORMATIONS 129 3.1. Linear Transformations of Coordinates 129 3.2. Transformation Groups in Three-Dimensional Euclidean

Space 142 3.3. Symmetry of Finite Bodies 146 3.4. Matrix Representation of Transformation Groups 164

Chapter 4. INDIFFERENT TENSORS AND INVARI-ANTS 169

4.1. Indifferent Tensors 169 4.2. A Number of Independent Components for Indifferent

Tensors 180 4.3. Symmetric Indifferent Tensors 192 4.4. Scalar Invariants 202 4.5. Invariants of Symmetric Second-Order Tensors 212

Chapter 5. TENSOR FUNCTIONS 227 5.1. Linear Tensor Functions 227 5.2. Scalar Functions of a Tensor Argument 253 5.3. Potential Tensor Functions 265 5.4. Quasilinear Tensor Functions 274 5.5. Spectral Resolutions of Second-Order Tensors 283 5.6. Spectral Resolutions of Quasilinear Tensor Functions 298 5.7. Nonpotential Tensor Functions 310 5.8. Differentiation of a Tensor Function with respect to a

Tensor Argument 325

v

vi TABLE OF CONTENTS

5.9. Scalar Functions of Several Tensor Arguments 328 5.10. Tensor Functions of Several Tensor Arguments 342

Chapter 6. TENSOR ANALYSIS 347 6.1. Covariant Differentiation 34 7 6.2. Differentiation of Second-Order Tensors 357 6.3. Properties of Covariant Derivatives 361 6.4. Covariant Derivatives of the Second Order 367 6.5. Differentiation in Orthogonal Curvilinear Coordinates 372

Chapter 7. GEOMETRY OF CURVES AND SUR-FACES 385

7.1. Curves in Three-Dimensional Euclidean Space 385 7.2. Surfaces in Three-Dimensional Euclidean Space 394 7.3. Curves on a Surface 413 7.4. Geometry in a Vicinity of a Surface 426 7.5 . Planar Surfaces in IR3 432

Chapter 8. TENSORS IN RIEMANNIAN SPACES AND AFFINELY CONNECTED SPACES 437

8.1. Riemannian Spaces 437 8.2. Affinely Connected Spaces 448 8.3. Riemannian Affinely Connected Spaces 456 8.4. The Riemann-Christoffel Tensor 462

Chapter 9. INTEGRATION OF TENSORS 475 9.1. Curvilinear Integrals of Tensors 475 9.2. Surface Integrals of Tensors 483 9.3. Volume Integrals of Tensors 488

Chapter 10. TENSORS IN CONTINUUM MECHAN-ICS 493

10.1. Deformation Theory 493 10.2. Velocity Characteristics of Continuum Motion 508 10.3. Co-rotational Derivatives 516 10.4. Mass, Momentum and Angular Momentum Balance

Laws 524 10.5. Thermodynamic Laws 538 10.6. The Deformation Compatibility Equation 546 10.7. The Complete System of Continuum Mechanics Laws 553

Chapter 11. TENSOR FUNCTIONS IN CONTINUUM MECHANICS 555

11.1. Energetic and Quasienergetic Couples of Tensors 555 11.2. General Principles for Tensor Functions in Continuum

Mechanics 572 11.3. The Material Indifference Principle 582 11.4. The Material Symmetry Principle 600 11 .5. Tensor Functions for Nonlinear Elastic Continua 618 11.6. Tensor Functions for Nonlinear Hypoelastic Continua 646

References 653

Subject Index 655

PREFACE

Tensor calculus appeared in its present-day form thanks to Ricci , who, first of all , suggested mathematical methods for operations on systems with indices at the close of the XIX century. Although these systems had been detected before, namely in investigations of non-Euclidean geometry by Gauss , Riemann, Christoffel and of elastic bodies by Cauchy, Euler , Lagrange, Poisson (see paragraph 'Sources of Tensor Calculus') , it was Ricci who developed the convenient compact system of symbols and concepts, which is widely used nowadays in different fields of mechan­ics, physics, chemistry, crystallophysics and other sciences.

At present tensor calculus goes on developing: advance directions appear and some concepts, introduced before, are re-interpreted. That is why, in spite of existing works on tensors (see References), there is an actual need of expounding these questions. To illustrate the above, we give one example. The following questions: 'May a second-order tensor be represented visually or graphically as well as a vector in three-dimensional space?' and ' What is a dyad? ' - can cause difficulties even for readers experienced in studying of tensors.

The present book is intended for a reader beginning to study methods of tensor calculus. That is why the introduction of the book gives the well-known concept of a vector as a geometric object in three-dimensional space. On the basis of the concept, the author suggests a geometric definition of a tensor . This definition allows us to see a tensor and main operations on tensors. And only after this acquaintance with tensors, there is a formal generalized definition of a tensor in an arbitrary linear n-dimensional space. According to the definition , a tensor is introduced as an element of a factor-space relative to the special equivalence. The book presents this approach in a mathematically rigorous form (the preceding works did not take into account the role of zero vectors in the equivalence relation). It should be noted that this approach introduces the notion of a tensor as an individual object, while other existing definitions introduce not a tensor itself but only concepts related to a tensor: tensor components, or linear transformations (for a second-order tensor), or bilinear functionals etc. The principal idea, that a tensor is an individual object, is the basis of the present book.

I hope that the book is of interest also for investigators in continuum mechanics, solid physics, erystallophysics, quantum chemistry, because, besides chapters for beginners , the book expounds many problems of the tensor theory which were not resolved before. This concerns tensors specifying physical properties (they are called indifferent tensors in the book), tensor invariants relative to crystallographic groups, a theory of tensor functions and integration of tensors.

The book pays great attention to the problems of construction of nonlinear tensor bases, besides the book is the first to present the construction methods for tensor bases in a systematized form . Then with their help, tensor anisotropic nonlinear functions for all crystallographic groups are constructed as well. The classification of tensor functions is given, and the representations are shown for

Vll

viii PREFACE

most important classes of these functions . Theorems about a number of indepen­dent components of tensors for all crystallographic groups and theorems about a number of functionally independent invariants of second-order tensors (including joint invariants) appear to be correctly formulated and proved in the present book for the first time.

Several chapters are devoted to tensor analysis . Besides the traditional informa­tion on covariant differentiation, there are results concerning nonlinear differential operators applied to nonlinear solid mechanics. New results, which are of interest for geometry and general relativity, are given in Chapter 8 devoted to tensors in Riemannian and affinely connected spaces.

The last two chapters are devoted to application of tensors and tensor functions to continuum mechanics. The book is the first to give a systematized theory of co-rotational derivatives of tensors specified in moving continua and to present a systematized description of energetic and quasienergetic couples of stress and deformation tensors. These quasienergetic couples have been found by the author. With the help of these couples, four main types of continuum models are intro­duced, which cover all known models of nonlinear elastic continua and contain new models of solids including hypoelastic continua and anisotropic continua with finite deformations.

The book is constructed by the mathematical principle: there are definitions, theorems , proofs and exercises at the end of each paragraph . The beginning and the end of each proof are denoted by symbols Y and & , respectively.

The indexless form of tensors is preferable in the book, that allows us to formu­late different relationships in mechanics and physics compactly without overload­ing a physical essence of phenomena. At the same time, there are corresponding component and matrix representations of tensor relationships , when they are ap­propriate.

I would like to thank Professor B.E.Pobedrya (Moscow Lomonosov State Uni­versity), Professor A.G.Gorshkov and Professor D.V.Tarlakovskii (Moscow Avia­tion Institute), Professor V.S.Zarubin (Moscow Bauman State Technical Univer­sity) for fruitful discussions and valuable advice on different problems in the book.

I am very grateful to Dr.lrina D .Dimitrienko (Department of Mechanics and Mathematics at Moscow Lomonosov State University) , who translated the book into English and prepared the camera-ready typescript .

I hope that the book proves to be useful for graduates and post-graduates of mathematical and natural-scientific departments of universities and for investigat­ors and academic scientists working in mathematics and also in solid mechanics, physics , general relativity, crystallophysics and quantum chemistry of solids.

Yuriy Dimitrienko

SOURCES OF TENSOR CALCULUS •

The predecessors of tensors were vectors, matrices and systems without indices. Archimedes (287-212 B.C.) added forces acting on a body by the parallelogram rule, i.e. he introduced intuitively special objects which were characterized not only by a value but also by a direction. This basic principle for the development of vector calculus remained the only one for a long time. The Holland mathematician and engineer S.Stevin (1548-1620) , who is considered to be a creator of the concept of a vector value, actually re-discovered once again the law of addition of forces by the parallelogram rule . This law was also formulated by !.Newton (1642-1727) in 'Principia mathematica' side by side with the laws of a motion of bodies.

The next important step in the development of vector calculus was made only in the XIX century by the Irish mathematician W.Hamilton (1805-1865), who extended a theory of quaternion-hypercomplex numbers, introduced in 1845 the term 'vector' (from Latin 'vector', i.e. carrying) and also the terms: 'scalar', 'scalar product' , 'vector product', and gave a definition of these operations.

At the same time, G.Grassmann (1809-1877) created a theory of outer products (this concept was introduced in 1844) , which is known nowadays as Grassmann's algebra. The English scientist W.Clifford (1845-1879) merged Hamilton's and Grassmann's approaches, but a final connection of quaternions, Grassmann's al­gebra and vector algebra was established only at the close of the XIX century by J.W.Gibbs (1839-1903).

The geometric image of a vector as a straight-line segment with arrow appeared to be used for the first time thanks to Hamilton, and in 1853 the French mathe­matician O.Cauchy (1789-1857) introduced the concept of a radius-vector and the corresponding notation i .

In the XIX century, mathematicians actively began to use one more object, namely a matrix being the predecessor of a tensor. The first appearance of ma­trices is connected with Old Chinese mathematicians, who in the II century B.C . applied matrices to writing systems of linear equations. The matrix expression of algebraic equations and the up-to-date matrix calculus were developed by the Eng­lish mathematician A.Cayley (1821-1895), who introduced in 1841, in particular,

*This brief historical sketch does not pretend to embrace the whole history of a development of tensor calculus and other sciences connected with tensor calculus; the purpose of the sketch is to acquaint a beginning reader with some stages of the development and with names of the scientists whose efforts promoted the creation of the up-to-date tensor calculus .

IX

X SOURCES OF TENSOR CALCULUS

the notation for the determinant being used nowadays:

Many basic results in the theory of linear algebraic equation systems were obtained by the German mathematician L.Kronecker (1823-1894).

During the XIX century, systems with indices appeared in different fields of mathematics. For example, these were quadratic forms in algebra (this theory was developed by A.Cayley, S.Lie (1842-1899) and others) , quadratic differential forms in geometry, which are known as the first and the second quadratic forms of a surface and the square of elementary segment length nowadays.

The outstanding German scientist K.F.Gauss (1777-1855) is rightfully consid­ered to be a founder of the surface theory. Many important results in this field were obtained by B.Riemann (1826-1866), who extended the surface theory for then-dimensional case, and also by E.Beltrami (1835-1900) , F.Klein (1849-1925), G.Lame (1795-1870) . In 1869 E.B.Christoffel (1829-1900) considered transforma­tions of the quadratic forms ds 2 = I:JL,v gJLvdx!Ldxv and established a tensor law of their transformation for the first time, and then introduced the concept of deriva­tives of vector values , which were transformed by the tensor law (they are called covariant derivatives nowadays).

In the XVIII century the efforts of outstanding mathematicians and mechani­cians: L.Euler (1707-1783) , J .Lagrange (1736-1813), P.Laplace (1749-1827) , S.Poisson (1781-1840) , O.Cauchy (1789-1857) , M.V.Ostrogradskii (1801-1861) re­sulted in the creation of a theory of a motion and equilibrium of elastic bodies (elasticity theory) , which became one more source for the appearance of systems with indices (components of stresses and strains). Stress components were denoted by Xx, Xy, Xz, Yx, Yy , Yz , Zx, Zy, Zz, and they were considered as projections of forces , acting on the sides of an elementary cube, onto coordinate axes . The operations on such systems with indices were rather awkward, contained many repetitions up to the cyclic change of notation. However, only at the close of the XIX century scientists succeeded in understanding the internal unity of formu­lae containing systems with indices and in finding a new mathematical apparatus which would made the operations on the systems compact and suitable.

For the first time, for vector values this problem was solved by the American physicist and mathematician J.W.Gibbs, who created the vector algebra with the operations of addition, scalar and vector multiplication and showed its connec­tion with the theory of quaternions and Grassmann 's algebra. Moreover, Gibbs developed the up-to-date vector analysis (the theory of differential calculus of vector fields) and the language of vector calculus, where there were both com­ponent and indexless forms of relationships. In particular, he gave appropriate representations for the operations of divergence and curl on vector fields. These distinguished results obtained by Gibbs can be compared with the introduction of algebraic symbolics by F.Vieta (1540-1603), which has been used during last

SOURCES OF TENSOR CALCULUS XI

400 years. The vector algebra and analysis created by Gibbs are also widely used in contemporary physics and mechanics. His work 'Elements of Vector Analysis' published in 1881-1884 was the first text-book on vector calculus, and up-to-date corresponding courses actually follow the text-book.

Gibbs was a great enthusiast in disseminating the vector calculus to different fields of natural sciences. In particular, it was Gibbs who created the contemporary vector represe11tation of electromagnetism equations obtained by J .Maxwell (1831-1879), and Maxwell himself used the method of quaternions. Although there was a criticism of the method, the vector calculus by Gibbs was widely employed by physicists , and from the beginning of the XX century Maxwell's theory has been practically always used in the Gibbs form .

However, in those fields of science, where there were systems with a larger number of indices than for vectors (> 1): in geometry, in elasticity theory, in crystallophysics, - the vector calculus by Gibbs proved to be powerless, and, for example, to write equations of the elasticity theory in 1889 he used the notation Xx, Xy, Xz, Yx etc. himself.

The Italian mathematician Ricci (1853-1925) succeeded in solving the problem of generalizing the vector calculus for systems with an arbitrary number of in­dices. In his works, in 1886-1901 Ricci created a new apparatus called the absolute differential calculus for algebraic and differential operations on covariant and con­travariant systems of order>. (so Ricci named tensor components by Xr 1 r 2 . . . r ... and xr1 r 2 . . . r ... ) . With the help of this apparatus, Ricci established basic results in differ­ential geometry of n-dimensional spaces. The calculus originated by Ricci affected geometry and physics so considerably, that for some time the theory was called Ricci's calculus. This calculus with some modifications is widely used nowadays as well.

The application of the theory of absolute differentiation for Riemannian spaces was realized by the outstanding Italian mathematician T.Levi-Civita (1873-1942) who was a colleague and co-author of Ricci in several basic works. In particular, Levi-Civita established the rule of index convolution and introduced the symbol c;ijk, named after him, that played together with Kronecker's symbol Oij an im­portant role in tensor calculus. The concept of a parallel carry introduced by Levi-Civita for vectors and tensors in Riemannian spaces is of great importance as well.

For a theory of relativity originated at the border of the XIX and XX centuries, the apparatus of absolute differential calculus proved to be rather convenient and promising, and at the same time the further development of the calculus proceeded together with working out a physical basis of the theory. So in 1913 A.Einstein (1879-1955) and M.Grossmann applied the absolute differential calculus to the relativity theory and gravitation theory, and in 1916 in remarks for his paper, Einstein suggested 'for simplicity' to omit the sum symbol for the cases where summation was taken over twice repeated indices. Since then the rule has been widely used and called Einstein's agreement on summation nowadays.

xii SOURCES OF TENSOR CALCULUS

Approximately at the same time, the new calculus was started to apply to the elasticity theory and crystallophysics to describe the properties of crystals. Here, first of all, we should mention the German scientist W.Voigt, who introduced in 1898 -1903 the term 'tensor' (from Latin 'tensus', i.e. tense, stress) just for a description of mechanical stresses . Voigt was one of the first scientists who gave a matrix representation for components of second- and fourth-order tensors specifying the physical properties for different types of crystals. The term 'tensor' was apprehended not only in the elasticity theory but also in geometry and physics to denote covariant and contravariant systems. So since 1913 Einstein used this term in his works.

The further development of tensor calculus at the beginning of the XX century was realized by many scientists. Among them we should mention once again T.Levi-Civita and the Holland mathematician J .Schooten, who published in 1927 and in 1924, respectively, the first specialized text-books on tensor calculus. The book by J.Schooten was 'Der Ricci-Kalkul'. In this and next books , in particular, he put in order the index arrangement rules for tensors and also suggested some geometric images of tensors.

Nevertheless, like in the case of vector analysis, there was some criticism of the tensor calculus. The criticism was mainly reduced to the deciphering of tensor formulae, that needed additional efforts when different physical relationships were analyzed (it should be noted that this criticism appears sometimes up to now) . However, expenditures for studying the tensor methods are covered in further work with tensor calculus.

Works of the outstanding German mathematician H.Weyl (1885-1955) are of great importance as well. Weyland O.Veblen developed the approach to definition of a tensor, which was based on consideration of quadratic forms, thus the algebraic approach to definition of tensors was introduced.

Many results for the theory of tensors were obtained by E.B.Wilson (who pub­lished in 1913 the well-known text-book on vector analysis) , F.D.Murnaghan (who, in particular, introduced Kronecker's generalized symbols) , E.Cartan (who de­veloped a theory of outer differential forms), R .Weitzenbock, G.Vitali and also E.Cartan and J.Schooten (who worked out a theory of a space of absolute par­allelism) , J.L .Synge, T.Y.Thomas , P.Appell, L.P.Eisenhart, C.Weatherburn , and also I.S.Sokolnikoff and A.J .McConnell, who wrote excellent text-books on tensor calculus. L.Brillouin and A.E.H.Love achieved a great success in application of tensors to problems of the elasticity theory.

The Russian scientists P.K.Rashevskii, A.P.Shirokov, V.F.Kagan, N.E.Kochin , N.E.Efimov, I.N.Vekua, B.E.Pobedrya, V.V.Lohin and many others considerably contributed to the development of tensor calculus. For example, I.N.Vekua worked out a theory of covariant differentiation in complex curvilinear coordinates, and B.E.Pobedrya introduced a spectral resolution of tensors and on its base developed a theory of nonlinear tensor functions.

Introducing the indexless form of tensor relationships was of great importance.

SOURCES OF TENSOR CALCULUS xiii

This form appeared in the middle of the XX century in works on continuum mechanics by Rivlin, Eriksen, Noll, Adkins, Green , Smith, Truesdell, Lurier. The indexless form , introduced by Gibbs for vectors, allows us with the help of the special mathematical language to write all physical laws as simple, compact and objective (i.e. independent of the choice of a coordinate system) expressions, where indices do not overload a physical essence of the laws. The up-to-date tensor calculus uses all these three above-mentioned forms of relationships: component, indexless and matrix.

At present the tensor calculus is closely connected with other fields of math­ematics, in particular , with a theory of invariants, with a theory of groups and representations and with a theory of indifferent tensors. The theory of algebraic invariants, which appeared in the XIX century, is widely used in mechanics and physics nowadays. The theory of groups, originated by Galois (1811-1832) , was actively applied in the XIX century in natural sciences to describe the properties of crystals' symmetry. With its help, 32 crystallographic groups were established, and in 1848 Braver found 14 translational groups corresponding to crystal lat­tices named after him. In 1890-1894 the Russian scientist E.S .Fedorov and inde­pendently Schoenflies introduced 230 space groups of crystals' symmetry. After the creation of a theory of group representations, developed mainly by Frobe­nius (1849-1918), Schur (1885-1955) and Burnside (1852-1927), the group theory proved to be of great importance for quantum physics. At present the theory of representations is one of rapidly developing fields in mathematics. Some methods of the theory of representations, used for description of the properties of indifferent tensors, are given in this book.

A theory of indifferent tensors (in other words, of tensors with outer symmetry by A.V.Shubnikov or of material tensors specifying the physical properties: elas­ticity, thermal expansion, heat conduction, piezoelectric effect, electric conduction and many others) actively began to develop in the XX century following the basic works by Voigt .

The Russian scientists A.V.Shubnikov and his followers, Yu.I.Sirotin, N.V.Belov, I.S .Zheludev, F.I.Fedorov, P.Behterev, N.G.Chentsov, S.G.Lehnitskii and M.P.Shaskolskaya made valuable contributions to the science. Efforts of these and many other researchers largely developed a theory of linear properties of anisotropic media (crystals, monocrystals, composite materials, wood and others) . Nevertheless, many important questions remain unclear up to now.

Only in 1983-1984 the Polish scientist Ya.Rychlevski succeeded in reducing the fourth-order tensor of elastic moduli to a diagonal form and in investigating its properties.

From the middle of the XX century scientists actively began to develop a theory of nonlinear tensor functions and functionals , whose origin was the famous theorem by Hamilton-Cayley. This theory allows us to describe such nonlinear properties of continua as anisotropic plasticity, creepness , nonlinear viscosity and viscoplasticity, nonlinear diffusion, magnetization diagram, nonlinear optical properties etc. Basic

xiv SOURCES OF TENSOR CALCULUS

results in this developing domain were obtained by R.Rivlin, F .Smith, A.Spencer, A. Green, G.Adkins and others. They established representations mainly for scalar or algebraic functions of tensors in different groups of symmetry. Another and more general approach, based on construction of tensor bases, was applied by the Russian scientists: Yu.V.Sirotin, V.V.Lohin, B.E.Pobedrya, G.N.Maloletkin and V.L.Fomin. The fifth chapter of the present book is devoted to this promising direction.

It should be noted in conclusion that the tensor calculus is a necessary tool for many advanced natural-scientific directions in physics, mechanics, quantum chemistry and crystallophysics. Many up-to-date problems in a quantum theory of relativity, a theory of joined fields, a theory of nanostructures etc. can be resolved with the help of methods of tensor calculus.

INTRODUCTION

A. Geometric Definition of a Vector

Tensor calculus is the development of vector calculus, therefore, let us remind the simplest definitions of vectors and operations on them.

Using the axiomatics of elementary geometry (where the concepts of a point, straight line, segment, length, angle etc. have been introduced), a directed straight-line segment connecting two points 0 and M of space is called a vec­tor a . One of the points ( 0) is said to be the origin, and another point (M) is the vector end. Vectors may be shown by arrows (Figure 0.1). The length of a vector a is the distance between its origin and its end, it is denoted by lal. The straight line, passing through a vector a , is called the line of the vector action.

The definition given above is said to be geometric, because it introduces a vector a as a geometric object. There are other definitions of a vector, which will be considered below.

Fig. 0.1. Geometric definition of a vector Fig. 0. 2. Geometric representation of ad­dition of vectors

Using the geometric definition, we can introduce the operation of addition of two vectors a and b having a common origin 0: the sum of two such vectors is the vector c = a + b coinciding with the diagonal of the parallelogram constructing on the vectors a and b with the origin at the same point 0 (Figure 0.2) .

The zero vector 0 in addition with any vector a gives the vector a again: a+ 0 =a.

We can define geometrically the second operation on vectors, namely multipli­cation of a vector a by a real coefficient '¢: this product 'lj;a is a vector situated along the same line as a but having the length wlal and the direction coinciding with the direction of a if'¢> 0, and the opposite direction if'¢ < 0 (Figure 0.3).

XV

xvi INTRODUCTION

I I I

lblcos~

a Fig. 0. 3. Geometric representation of multiplication of a vector by a coefficient

Fig. 0.4. Geometric representation of scal­ar multiplication of vectors

The third main operation on vectors is scalar multiplication of two vectors a and b, which is defined as a real number equal to the product of the vector a length, the vector b length and cosine of the angle between the vectors (Figure 0.4), it is denoted by

a· b =!alibi cosrp. (0.1)

Nonzero vectors a and b are called orthogonal, if their scalar product is equal to zero:

a· b = 0. (0.2)

As follows from (0.1) and (0.2), the angle rp between two orthogonal vectors is equal to 90° (Figure 0.5).

The system of three mutually orthogonal vectors e 1 e 2e 3 with unit lengths, the action lines of which are situated along three mutually perpendicular straight lines (Figure 0.6) is of great importance. This set is called the orthonormal (Cartesian) basis.

We may always put an arbitrary vector a in correspondence with the diagonal of a rectangular parallelepiped, whose edges are situated on the action lines of vectors ei having a common origin with the vector a (Figure 0.7). Therefore, we can always resolve the vector a for the basis vectors ei as follows :

(0.3)

This equation may be rewritten in another form

(0.4)

Here there is summation over repeated indices (Einstein 's rule) . The relationships (0.3) and (0.4) are called the resolutions of the vector a for

the basis e i, and the values ai are the coordinates of the vector a in the basis e i .

INTRODUCTION xvii

b

a

Fig. 0. 5. Orthogonal vectors

e3 :::---,71

...... / I ......

1 a21 I I

~ ' I I e2 ,.,...,

a

Fig . 0. 7. Resolution of a vector for the Cartesian basis

Fig. 0. 6. The Cartesian basis

If we take another orthonormal basis e~ with the same origin 0 , then for the basis we can construct another paral­lelepiped with the vector a along its di­agonal. Then a can be resolved for the basis e~ as well:

(0 .5)

Here a'i are the components of the vec­tor a in the basis e~ , and , in general,

The important property of vectors, namely their invariance (i .e. they are in­dependent of the choice of a basis), follows from the relationships (0.4) and (0.5) , but at the same time vector components may change.

B. Representation of Physical Values by Vectors

Thus, a vector introduced above in the geometric way has three attributes: its origin, length and direction. Many parameters describing physical objects are characterized by the same attributes and may be shown by vectors. In this case, the length of a vector is equal to the magnitude of a physical parameter, which is measured in a certain scale. Such physical parameters are the following: a radius­vector describing the location of a material point with respect to a fixed geometric point; a velocity of the motion of a point; a force acting onto a point, a force moment and many others. The set of all vectors showing some physical value is said to constitute a vector space, if in the set the operations of addition and

xviii INTRODUCTION

multiplication by a coefficient are defined. If the operation of scalar multiplication is defined here as well, then this set constitutes a Euclidean space (more rigorous definitions will be given below).

C. Three Categories of Vectors

Vectors showing physical values may be divided into three categories: free , glancing and fastened.

If a vector physical value remains unchanged in going from one point o f space to another, then the value is described by a free vector. We may add and multiply free vectors even if they have no common origin. To make this, we should match their origins beforehand without changing lengths and directions of the vectors. Free vectors are, for example, the Cartesian basis vectors ei (an arbitrary point 0 may be taken as their origin) .

With the help of glancing vectors we can show vector physical values remaining without changes in going to any point of a straight line coinciding with the direction of a physical value. In the operations of addition and multiplication, glancing vectors may be carried in a parallel way only along their action lines. The example of a g lancing vector is a force vector acting onto a material point.

Fixed vectors show vector physical values determined only at a fixed point of space. These vectors are the following: a velocity vector of a moving material point, a force moment, a radius-vector x describing the location of a material point with respect to some geometric point 0 etc.

Below we will consider, as a rule, free vectors.