Eminently readable, completely elementary treatment begins with linear spaces and ends with analytic geometry, covering multilinear forms, tensors, linear transformation, and more. 250 problems, most with hints and answers. 1972 edition.

Author: M. A. Akivis

Publisher: Courier Corporation

ISBN: 9780486148786

Category: Mathematics

Page: 192

View: 519

Eminently readable, completely elementary treatment begins with linear spaces and ends with analytic geometry, covering multilinear forms, tensors, linear transformation, and more. 250 problems, most with hints and answers. 1972 edition.

Covers a notably broad range of topics, including some topics not generally found in linear algebra books Contains a discussion of the basics of linear algebra This is a graduate textbook covering an especially broad range of topics.

Author: Steven Roman

Publisher: Springer Science & Business Media

ISBN: 038727474X

Category: Mathematics

Page: 486

View: 127

Covers a notably broad range of topics, including some topics not generally found in linear algebra books Contains a discussion of the basics of linear algebra

While not designed as an introductory text, the book's well-chosen topics, brevity of presentation, and the author's reputation will recommend it to all students, teachers, and mathematicians working in this sector.

Author: I. M. Gelfand

Publisher: Courier Corporation

ISBN: 0486660826

Category: Mathematics

Page: 185

View: 223

Prominent Russian mathematician's concise, well-written exposition considers n-dimensional spaces, linear and bilinear forms, linear transformations, canonical form of an arbitrary linear transformation, and an introduction to tensors. While not designed as an introductory text, the book's well-chosen topics, brevity of presentation, and the author's reputation will recommend it to all students, teachers, and mathematicians working in this sector.

This convenient single-volume compilation of two texts offers both an introduction and an in-depth survey.

Author: Ray M. Bowen

Publisher: Courier Corporation

ISBN: 9780486469140

Category: Mathematics

Page: 520

View: 624

This convenient single-volume compilation of two texts offers both an introduction and an in-depth survey. Geared toward engineering and science students rather than mathematicians, its less rigorous treatment focuses on physics and engineering applications. A practical reference for professionals, it is suitable for advanced undergraduate and graduate students. 1976 edition.

The volume is well suited for pure and applied mathematicians as well as theoretical physicists and engineers.New topics added to the third edition are: mutually unbiased bases, Cayley transform, spectral theorem, nonnormal matrices, ...

Author: Hardy Yorick

Publisher: World Scientific

ISBN: 9789811202537

Category: Mathematics

Page: 388

View: 673

Our self-contained volume provides an accessible introduction to linear and multilinear algebra as well as tensor calculus. Besides the standard techniques for linear algebra, multilinear algebra and tensor calculus, many advanced topics are included where emphasis is placed on the Kronecker product and tensor product. The Kronecker product has widespread applications in signal processing, discrete wavelets, statistical physics, Hopf algebra, Yang-Baxter relations, computer graphics, fractals, quantum mechanics, quantum computing, entanglement, teleportation and partial trace. All these fields are covered comprehensively.The volume contains many detailed worked-out examples. Each chapter includes useful exercises and supplementary problems. In the last chapter, software implementations are provided for different concepts. The volume is well suited for pure and applied mathematicians as well as theoretical physicists and engineers.New topics added to the third edition are: mutually unbiased bases, Cayley transform, spectral theorem, nonnormal matrices, Gâteaux derivatives and matrices, trace and partial trace, spin coherent states, Clebsch-Gordan series, entanglement, hyperdeterminant, tensor eigenvalue problem, Carleman matrix and Bell matrix, tensor fields and Ricci tensors, and software implementations.

The First Part Of This Book Begins With An Introduction To Matrices Through Linear Transformations On Vector Spaces, Followed By A Discussion On The Algebra Of Matrices, Special Matrices, Linear Equations, The Eigenvalue Problem, Bilinear ...

Author: A. W. Joshi

Publisher: New Age International

ISBN: 8122405630

Category: Calculus of tensors

Page: 342

View: 710

The First Part Of This Book Begins With An Introduction To Matrices Through Linear Transformations On Vector Spaces, Followed By A Discussion On The Algebra Of Matrices, Special Matrices, Linear Equations, The Eigenvalue Problem, Bilinear And Quadratic Forms, Kronecker Sum And Product Of Matrices. Other Matrices Which Occur In Physics, Such As The Rotation Matrix, Pauli Spin Matrices And Dirac Matrices, Are Then Presented. A Brief Account Of Infinite Matrices From The Point Of View Of Matrix Formulation Of Quantum Mechanics Is Also Included. The Emphasis In This Part Is On Linear Dependence And Independence Of Vectors And Matrices, Linear Combinations, Independent Parameters Of Various Special Matrices And Such Other Concepts As Help The Student In Obtaining A Clear Understanding Of The Subject. A Simplified Proof Of The Theorem That A Common Set Of Eigenvectors Can Be Found For Two Commuting Matrices Is Given. The Second Part Deals With Cartesian And General Tensors. Many Physical Situations Are Discussed Which Require The Use Of Second And Higher Rank Tensors, Such As Effective Mass Tensor, Moment Of Inertia Tensor, Stress, Strain And Elastic Constants, Piezoelectric Strain Coefficient Tensor, Etc. Einsteins Summation Convention Is Explained In Detail And Common Errors Arising In Its Use Are Pointed Out. Rules For Checking The Correctness Of Tensor Equations Are Given. This Is Followed By Four-Vectors In Special Relativity And Covarient Formulation Of Electrodynamics. This Part Comes To An End With The Concept Of Parallel Displacement Of Vectors In Riemannian Space And Covariant Derivative Of Tensors, Leading To The Curvature Tensors And Its Properties.Appendix I Has Expanded And Two New Appendices Have Been Added In This Edition.

1–3) is an introduction to tensors and their physical applications, and Part II (
Chaps. ... 1, I introduce this additional mathematics, which is just an extension of
the linear algebra you probably saw in your lower division coursework. This
material ...

Author: Nadir Jeevanjee

Publisher: Birkhäuser

ISBN: 9783319147949

Category: Science

Page: 305

View: 921

The second edition of this highly praised textbook provides an introduction to tensors, group theory, and their applications in classical and quantum physics. Both intuitive and rigorous, it aims to demystify tensors by giving the slightly more abstract but conceptually much clearer definition found in the math literature, and then connects this formulation to the component formalism of physics calculations. New pedagogical features, such as new illustrations, tables, and boxed sections, as well as additional “invitation” sections that provide accessible introductions to new material, offer increased visual engagement, clarity, and motivation for students. Part I begins with linear algebraic foundations, follows with the modern component-free definition of tensors, and concludes with applications to physics through the use of tensor products. Part II introduces group theory, including abstract groups and Lie groups and their associated Lie algebras, then intertwines this material with that of Part I by introducing representation theory. Examples and exercises are provided in each chapter for good practice in applying the presented material and techniques. Prerequisites for this text include the standard lower-division mathematics and physics courses, though extensive references are provided for the motivated student who has not yet had these. Advanced undergraduate and beginning graduate students in physics and applied mathematics will find this textbook to be a clear, concise, and engaging introduction to tensors and groups. Reviews of the First Edition “[P]hysicist Nadir Jeevanjee has produced a masterly book that will help other physicists understand those subjects [tensors and groups] as mathematicians understand them... From the first pages, Jeevanjee shows amazing skill in finding fresh, compelling words to bring forward the insight that animates the modern mathematical view...[W]ith compelling force and clarity, he provides many carefully worked-out examples and well-chosen specific problems... Jeevanjee’s clear and forceful writing presents familiar cases with a freshness that will draw in and reassure even a fearful student. [This] is a masterpiece of exposition and explanation that would win credit for even a seasoned author.” —Physics Today "Jeevanjee’s [text] is a valuable piece of work on several counts, including its express pedagogical service rendered to fledgling physicists and the fact that it does indeed give pure mathematicians a way to come to terms with what physicists are saying with the same words we use, but with an ostensibly different meaning. The book is very easy to read, very user-friendly, full of examples...and exercises, and will do the job the author wants it to do with style.” —MAA Reviews

The definition of a tensor comes only in Chapter 6 – when the reader is ready for it. While this text maintains a consistent level of rigor, it takes great care to avoid formalizing the subject.

Author: Pavel Grinfeld

Publisher: Springer Science & Business Media

ISBN: 9781461478676

Category: Mathematics

Page: 302

View: 116

This textbook is distinguished from other texts on the subject by the depth of the presentation and the discussion of the calculus of moving surfaces, which is an extension of tensor calculus to deforming manifolds. Designed for advanced undergraduate and graduate students, this text invites its audience to take a fresh look at previously learned material through the prism of tensor calculus. Once the framework is mastered, the student is introduced to new material which includes differential geometry on manifolds, shape optimization, boundary perturbation and dynamic fluid film equations. The language of tensors, originally championed by Einstein, is as fundamental as the languages of calculus and linear algebra and is one that every technical scientist ought to speak. The tensor technique, invented at the turn of the 20th century, is now considered classical. Yet, as the author shows, it remains remarkably vital and relevant. The author’s skilled lecturing capabilities are evident by the inclusion of insightful examples and a plethora of exercises. A great deal of material is devoted to the geometric fundamentals, the mechanics of change of variables, the proper use of the tensor notation and the discussion of the interplay between algebra and geometry. The early chapters have many words and few equations. The definition of a tensor comes only in Chapter 6 – when the reader is ready for it. While this text maintains a consistent level of rigor, it takes great care to avoid formalizing the subject. The last part of the textbook is devoted to the Calculus of Moving Surfaces. It is the first textbook exposition of this important technique and is one of the gems of this text. A number of exciting applications of the calculus are presented including shape optimization, boundary perturbation of boundary value problems and dynamic fluid film equations developed by the author in recent years. Furthermore, the moving surfaces framework is used to offer new derivations of classical results such as the geodesic equation and the celebrated Gauss-Bonnet theorem.

Still, with no surprise, non-linear algebra is naturally and efficiently embedded
into poly-linear one. • Tensors are associated with functions on the dual spaces
in an obvious way. Generic tensor is associated T(v1,..., vp ; u∗1 ,..., u∗q) = ∑
with ...

Author: Morozov Alexei

Publisher: World Scientific Publishing Company

ISBN: 9789813107007

Category: Mathematics

Page: 288

View: 111

This unique text presents the new domain of consistent non-linear counterparts for all basic objects and tools of linear algebra, and develops an adequate calculus for solving non-linear algebraic and differential equations. It reveals the non-linear algebraic activity as an essentially wider and diverse field with its own original methods, of which the linear one is a special restricted case.This volume contains a detailed and comprehensive description of basic objects and fundamental techniques arising from the theory of non-linear equations, which constitute the scope of what should be called non-linear algebra. The objects of non-linear algebra are presented in parallel with the corresponding linear ones, followed by an exposition of specific non-linear properties treated with the use of classical (such as the Koszul complex) and original new tools. This volume extensively uses a new diagram technique and is enriched with a variety of illustrations throughout the text. Thus, most of the material is new and is clearly exposed, starting from the elementary level. With the scope of its perspective applications spreading from general algebra to mathematical physics, it will interest a broad audience of physicists; mathematicians, as well as advanced undergraduate and graduate students.

In every orthogonal cartesian system in Ez a three dimensional cartesian tensor T
= T ( z ) of the second order is represented by a 3 - by - 3 matrix A = A ( z ) , where
z is a point in Ez . A constant tensor is a tensor independent of position , i.e. ...

AN INTRODUCTION TO LINEAR ALGEBRA AND TENSORS, M. A. Akivis and
V. V. Goldberg. (63545-7) VECTORS, TENSORS AND THE BASIC EQUATIONS
OF FLUID MECHANICS, Rutherford Aris. (661 10-5) ASYMPTOTIC
EXPANSIONS ...

Author: Martin Davis

Publisher: Courier Corporation

ISBN: 9780486151069

Category: Mathematics

Page: 288

View: 544

Classic graduate-level introduction to theory of computability. Discusses general theory of computability, computable functions, operations on computable functions, Turing machines self-applied, unsolvable decision problems, applications of general theory, mathematical logic, Kleene hierarchy, more.

Author: Heinrich W. GuggenheimerPublish On: 2012-04-27

AN INTRODUCTION TO LINEAR ALGEBRA AND TENSORS, M. A. Akivis and
V. V. Goldberg. (63545-7) VECTORS, TENSORS AND THE BASIC EQUATIONS
OF FLUID MECHANICS, Rutherford Aris. (66110-5) ASYMPTOTIC EXPANSIONS
...

Author: Heinrich W. Guggenheimer

Publisher: Courier Corporation

ISBN: 9780486157207

Category: Mathematics

Page: 400

View: 915

This text contains an elementary introduction to continuous groups and differential invariants; an extensive treatment of groups of motions in euclidean, affine, and riemannian geometry; more. Includes exercises and 62 figures.

3.2 General Linear Algebra This section considers the structure matrix of general linear algebra or its subalgebra. General linear algebra is an important Lie algebra. We first introduce Lie algebra. Definition 3.2. (1) Let V be a vector space
over ...

Author: Cheng Daizhan

Publisher: World Scientific

ISBN: 9789814458016

Category: Mathematics

Page: 612

View: 426

A generalization of Conventional Matrix Product (CMP), called the Semi-Tensor Product (STP), is proposed. It extends the CMP to two arbitrary matrices and maintains all fundamental properties of CMP. In addition, it has a pseudo-commutative property, which makes it more superior to CMP. The STP was proposed by the authors to deal with higher-dimensional data as well as multilinear mappings. After over a decade of development, STP has been proven to be a powerful tool in dealing with nonlinear and logical calculations.This book is a comprehensive introduction to the theory of STP and its various applications, including logical function, fuzzy control, Boolean networks, analysis and control of nonlinear systems, amongst others.

Building on a first course in linear algebra, this book offers readers a deeper understanding of abstract structures, matrix decompositions, multilinearity, and tensors.

Author: Nathaniel Johnston

Publisher: Springer

ISBN: 3030528146

Category: Mathematics

Page: 496

View: 927

This textbook emphasizes the interplay between algebra and geometry to motivate the study of advanced linear algebra techniques. Matrices and linear transformations are presented as two sides of the same coin, with their connection motivating inquiry throughout the book. Building on a first course in linear algebra, this book offers readers a deeper understanding of abstract structures, matrix decompositions, multilinearity, and tensors. Concepts draw on concrete examples throughout, offering accessible pathways to advanced techniques. Beginning with a study of vector spaces that includes coordinates, isomorphisms, orthogonality, and projections, the book goes on to focus on matrix decompositions. Numerous decompositions are explored, including the Shur, spectral, singular value, and Jordan decompositions. In each case, the author ties the new technique back to familiar ones, to create a coherent set of tools. Tensors and multilinearity complete the book, with a study of the Kronecker product, multilinear transformations, and tensor products. Throughout, “Extra Topic” sections augment the core content with a wide range of ideas and applications, from the QR and Cholesky decompositions, to matrix-valued linear maps and semidefinite programming. Exercises of all levels accompany each section. Advanced Linear and Matrix Algebra offers students of mathematics, data analysis, and beyond the essential tools and concepts needed for further study. The engaging color presentation and frequent marginal notes showcase the author’s visual approach. A first course in proof-based linear algebra is assumed. An ideal preparation can be found in the author’s companion volume, Introduction to Linear and Matrix Algebra.

CONTENTS 9.1 Introduction to Tensor Products . ... 300 9.3 The Tensor Algebra .
... Finally, we show how a linear transformation from a vector space V to a vector
space W induces a linear transformation on the exterior algebra and its ...

Author: Bruce Cooperstein

Publisher: CRC Press

ISBN: 9781439829691

Category: Mathematics

Page: 364

View: 873

Advanced Linear Algebra focuses on vector spaces and the maps between them that preserve their structure (linear transformations). It starts with familiar concepts and then slowly builds to deeper results. Along with including many exercises and examples, each section reviews what students need to know before studying the material. The book first introduces vector spaces over fields as well as the fundamental concepts of linear combinations, span of vectors, linear independence, basis, and dimension. After covering linear transformations, it discusses the algebra of polynomials with coefficients in a field, concentrating on results that are consequences of the division algorithm. The author then develops the whole structure theory of a linear operator on a finite dimensional vector space from a collection of some simple results. He also explores the entire range of topics associated with inner product spaces, from the Gram–Schmidt process to the spectral theorems for normal and self-adjoint operators on an inner product space. The text goes on to rigorously describe the trace and determinant of linear operators and square matrices. The final two chapters focus on bilinear forms and tensor products and related material. Designed for advanced undergraduate and beginning graduate students, this textbook shows students the beauty of linear algebra. It also prepares them for further study in mathematics.

... differential and integral calculus, differential equations, functions of several
variables, partial derivatives, and an introduction to linear algebra. Matrices are
reviewed briefly, and determinants, vectors, and tensors of order two are
described.

Author: Stephen C. Cowin

Publisher: Springer Science & Business Media

ISBN: 9780387499857

Category: Technology & Engineering

Page: 682

View: 902

The structures of living tissues are continually changing due to growth and response to the tissue environment, including the mechanical environment. Tissue Mechanics is an in-depth look at the mechanics of tissues. Tissue Mechanics describes the nature of the composite components of a tissue, the cellular processes that produce these constituents, the assembly of the constituents into a hierarchical structure, and the behavior of the tissue’s composite structure in the adaptation to its mechanical environment. Organized as a textbook for the student needing to acquire the core competencies, Tissue Mechanics will meet the demands of advanced undergraduate or graduate coursework in Biomedical Engineering, as well as, Chemical, Civil, and Mechanical Engineering. Key features: Detailed Illustrations Example problems, including problems at the end of sections A separate solutions manual available for course instructors A website (http://tissue-mechanics.com/) that has been established to provide supplemental material for the book, including downloadable additional chapters on specific tissues, downloadable PowerPoint presentations of all the book's chapters, and additional exercises and examples for the existing chapters. About the Authors: Stephen C. Cowin is a City University of New York Distinguished Professor, Departments of Biomedical and Mechanical Engineering, City College of the City University of New York and also an Adjunct Professor of Orthopaedics, at the Mt. Sinai School of Medicine in New York, New York. In 1985 he received the Society of Tulane Engineers and Lee H. Johnson Award for Teaching Excellence and a recipient of the European Society of Biomechanics Research Award in 1994. In 1999 he received the H. R. Lissner medal of the ASME for contributions to biomedical engineering. In 2004 he was elected to the National Academy of Engineering (NAE) and he also received the Maurice A. Biot medal of the American Society of Civil Engineers (ASCE). Stephen B. Doty is a Senior Scientist at Hospital for Special Surgery, New York, New York and Adjunct Professor, School of Dental and Oral Surgery, Columbia University, New York, NY. He has over 100 publications in the field of anatomy, developmental biology, and the physiology of skeletal and connective tissues. His honors include several commendations for participation in the Russian/NASA spaceflights, the Spacelab Life Science NASA spaceflights, and numerous Shuttle missions that studied the influence of spaceflight on skeletal physiology. He presently is on the scientific advisory board of the National Space Biomedical Research Institute, Houston, Texas.

Unusual in its extensive use of applications in physics to clarify each topic, this comprehensice volume should be of particular interest to advanced undergraduates and graduates in mathematics and physics, and to lecturers in linear and ...

Author: P. K. Suetin

Publisher: CRC Press

ISBN: 9056990497

Category: Mathematics

Page: 320

View: 962

This advanced textbook on linear algebra and geometry covers a wide range of classical and modern topics. Differing from existing textbooks in approach, the work illustrates the many-sided applications and connections of linear algebra with functional analysis, quantum mechanics and algebraic and differential geometry. The subjects covered in some detail include normed linear spaces, functions of linear operators, the basic structures of quantum mechanics and an introduction to linear programming. Also discussed are Kahler's metic, the theory of Hilbert polynomials, and projective and affine geometries. Unusual in its extensive use of applications in physics to clarify each topic, this comprehensice volume should be of particular interest to advanced undergraduates and graduates in mathematics and physics, and to lecturers in linear and multilinear algebra, linear programming and quantum mechanics.

Tensors. A.1 Introduction and Rationale The purpose of this appendix is to
present the notation and most of the ... integral calculus, differential equations,
functions of several variables, partial derivatives, and an introduction to linear algebra.

Author: Stephen C. Cowin

Publisher: Springer Science & Business Media

ISBN: 9781461450252

Category: Technology & Engineering

Page: 425

View: 146

Continuum Mechanics of Anisotropic Materials(CMAM) presents an entirely new and unique development of material anisotropy in the context of an appropriate selection and organization of continuum mechanics topics. These features will distinguish this continuum mechanics book from other books on this subject. Textbooks on continuum mechanics are widely employed in engineering education, however, none of them deal specifically with anisotropy in materials. For the audience of Biomedical, Chemical and Civil Engineering students, these materials will be dealt with more frequently and greater accuracy in their analysis will be desired. Continuum Mechanics of Anisotropic Materials' author has been a leader in the field of developing new approaches for the understanding of anisotropic materials.

Examines general Cartesian coordinates, the cross product, Einstein's special theory of relativity, bases in general coordinate systems, maxima and minima of functions of two variables, line integrals, integral theorems, and more. 1963 ...

Author: Robert C. Wrede

Publisher: Courier Corporation

ISBN: 9780486137117

Category: Mathematics

Page: 418

View: 624

Examines general Cartesian coordinates, the cross product, Einstein's special theory of relativity, bases in general coordinate systems, maxima and minima of functions of two variables, line integrals, integral theorems, and more. 1963 edition.