The Very Basics of Tensors

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Author: Nils K. Oeijord

Publisher: iUniverse

ISBN: 059535694X

Category: Mathematics

Page: 137

View: 7094

Tensor calculus is a generalization of vector calculus, and comes near of being a universal language in physics. Physical laws must be independent of any particular coordinate system used in describing them. This requirement leads to tensor calculus. The only prerequisites for reading this book are a familiarity with calculus (including vector calculus) and linear algebra, and some knowledge of differential equations.

Vectors, Tensors and the Basic Equations of Fluid Mechanics

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Author: Rutherford Aris

Publisher: Courier Corporation

ISBN: 048613489X

Category: Mathematics

Page: 320

View: 2092

Introductory text, geared toward advanced undergraduate and graduate students, applies mathematics of Cartesian and general tensors to physical field theories and demonstrates them in terms of the theory of fluid mechanics. 1962 edition.

Vector and Tensor Analysis with Applications

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Author: A. I. Borisenko,I. E. Tarapov

Publisher: Courier Corporation

ISBN: 0486131904

Category: Mathematics

Page: 288

View: 8040

Concise, readable text ranges from definition of vectors and discussion of algebraic operations on vectors to the concept of tensor and algebraic operations on tensors. Worked-out problems and solutions. 1968 edition.

A Brief on Tensor Analysis

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Author: James G. Simmonds

Publisher: Springer Science & Business Media

ISBN: 1441985220

Category: Mathematics

Page: 114

View: 1952

In this text which gradually develops the tools for formulating and manipulating the field equations of Continuum Mechanics, the mathematics of tensor analysis is introduced in four, well-separated stages, and the physical interpretation and application of vectors and tensors are stressed throughout. This new edition contains more exercises. In addition, the author has appended a section on Differential Geometry.

Tensors, Differential Forms, and Variational Principles

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Author: David Lovelock,Hanno Rund

Publisher: Courier Corporation

ISBN: 048613198X

Category: Mathematics

Page: 400

View: 7549

Incisive, self-contained account of tensor analysis and the calculus of exterior differential forms, interaction between the concept of invariance and the calculus of variations. Emphasis is on analytical techniques. Includes problems.

Tensor Calculus

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Author: J. L. Synge,A. Schild

Publisher: Courier Corporation

ISBN: 048614139X

Category: Mathematics

Page: 336

View: 7050

Fundamental introduction of absolute differential calculus and for those interested in applications of tensor calculus to mathematical physics and engineering. Topics include spaces and tensors; basic operations in Riemannian space, curvature of space, more.

A Most Incomprehensible Thing

Notes Towards a Very Gentle Introduction to the Mathematics of Relativity

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Author: Peter Collier

Publisher: Incomprehensible Books

ISBN: 0957389469

Category: Science

Page: 274

View: 3316

A clear and enjoyable guide to the mathematics of relativity To really understand relativity – one of the cornerstones of modern physics – you have to get to grips with the mathematics. This user-friendly self-study guide is aimed at the general reader who is motivated to tackle that not insignificant challenge. The book is written using straightforward and accessible language, with clear derivations and explanations as well as numerous fully solved problems. For those with minimal mathematical background, the first chapter provides a crash course in foundation mathematics. The reader is then taken gently by the hand and guided through a wide range of fundamental topics, including Newtonian mechanics; the Lorentz transformations; tensor calculus; the Einstein field equations; the Schwarzschild solution (which gives a good approximation of the spacetime of our Solar System); simple black holes and relativistic cosmology. Following the historic 2015 LIGO (Laser Interferometer Gravitational-Wave Observatory) detection, there is now an additional chapter on gravitational waves, ripples in the fabric of spacetime that potentially provide a revolutionary new way to study the universe. Special relativity helps explain a huge range of non-gravitational physical phenomena and has some strangely counter-intuitive consequences. These include time dilation, length contraction, the relativity of simultaneity, mass-energy equivalence and an absolute speed limit. General relativity, the leading theory of gravity, is at the heart of our understanding of cosmology and black holes. Understand even the basics of Einstein's amazing theory and the world will never seem the same again. March 2017. This third edition has been revised to make the material even more accessible to the enthusiastic general reader who seeks to understand the mathematics of relativity.

Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers

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Author: Hung Nguyen-Schäfer,Jan-Philip Schmidt

Publisher: Springer

ISBN: 3662484978

Category: Technology & Engineering

Page: 376

View: 8503

This book comprehensively presents topics, such as Dirac notation, tensor analysis, elementary differential geometry of moving surfaces, and k-differential forms. Additionally, two new chapters of Cartan differential forms and Dirac and tensor notations in quantum mechanics are added to this second edition. The reader is provided with hands-on calculations and worked-out examples at which he will learn how to handle the bra-ket notation, tensors, differential geometry, and differential forms; and to apply them to the physical and engineering world. Many methods and applications are given in CFD, continuum mechanics, electrodynamics in special relativity, cosmology in the Minkowski four-dimensional spacetime, and relativistic and non-relativistic quantum mechanics. Tensors, differential geometry, differential forms, and Dirac notation are very useful advanced mathematical tools in many fields of modern physics and computational engineering. They are involved in special and general relativity physics, quantum mechanics, cosmology, electrodynamics, computational fluid dynamics (CFD), and continuum mechanics. The target audience of this all-in-one book primarily comprises graduate students in mathematics, physics, engineering, research scientists, and engineers.

Introduction to Diffusion Tensor Imaging

And Higher Order Models

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Author: Susumu Mori,J-Donald Tournier

Publisher: Academic Press

ISBN: 0123984076

Category: Medical

Page: 140

View: 7242

The concepts behind diffusion tensor imaging (DTI) are commonly difficult to grasp, even for magnetic resonance physicists. To make matters worse, a many more complex higher-order methods have been proposed over the last few years to overcome the now well-known deficiencies of DTI. In Introduction to Diffusion Tensor Imaging: And Higher Order Models, these concepts are explained through extensive use of illustrations rather than equations to help readers gain a more intuitive understanding of the inner workings of these techniques. Emphasis is placed on the interpretation of DTI images and tractography results, the design of experiments, and the types of application studies that can be undertaken. Diffusion MRI is a very active field of research, and theories and techniques are constantly evolving. To make sense of this constantly shifting landscape, there is a need for a textbook that explains the concepts behind how these techniques work in a way that is easy and intuitive to understand—Introduction to Diffusion Tensor Imaging fills this gap. Extensive use of illustrations to explain the concepts of diffusion tensor imaging and related methods Easy to understand, even without a background in physics Includes sections on image interpretation, experimental design, and applications Up-to-date information on more recent higher-order models, which are increasingly being used for clinical applications

Tensor Calculus for Engineers and Physicists

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Author: Emil de Souza Sánchez Filho

Publisher: Springer

ISBN: 331931520X

Category: Technology & Engineering

Page: 345

View: 3904

This textbook provides a rigorous approach to tensor manifolds in several aspects relevant for Engineers and Physicists working in industry or academia. With a thorough, comprehensive, and unified presentation, this book offers insights into several topics of tensor analysis, which covers all aspects of n-dimensional spaces. The main purpose of this book is to give a self-contained yet simple, correct and comprehensive mathematical explanation of tensor calculus for undergraduate and graduate students and for professionals. In addition to many worked problems, this book features a selection of examples, solved step by step. Although no emphasis is placed on special and particular problems of Engineering or Physics, the text covers the fundamentals of these fields of science. The book makes a brief introduction into the basic concept of the tensorial formalism so as to allow the reader to make a quick and easy review of the essential topics that enable having the grounds for the subsequent themes, without needing to resort to other bibliographical sources on tensors. Chapter 1 deals with Fundamental Concepts about tensors and chapter 2 is devoted to the study of covariant, absolute and contravariant derivatives. The chapters 3 and 4 are dedicated to the Integral Theorems and Differential Operators, respectively. Chapter 5 deals with Riemann Spaces, and finally the chapter 6 presents a concise study of the Parallelism of Vectors. It also shows how to solve various problems of several particular manifolds.

Tensors

The Mathematics of Relativity Theory and Continuum Mechanics

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Author: Anadi Jiban Das

Publisher: Springer Science & Business Media

ISBN: 0387694692

Category: Science

Page: 290

View: 7184

Here is a modern introduction to the theory of tensor algebra and tensor analysis. It discusses tensor algebra and introduces differential manifold. Coverage also details tensor analysis, differential forms, connection forms, and curvature tensor. In addition, the book investigates Riemannian and pseudo-Riemannian manifolds in great detail. Throughout, examples and problems are furnished from the theory of relativity and continuum mechanics.

Tensor Categories

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Author: Pavel Etingof,Shlomo Gelaki,Dmitri Nikshych,Victor Ostrik

Publisher: American Mathematical Soc.

ISBN: 1470434415

Category:

Page: 344

View: 2721

Is there a vector space whose dimension is the golden ratio? Of course not—the golden ratio is not an integer! But this can happen for generalizations of vector spaces—objects of a tensor category. The theory of tensor categories is a relatively new field of mathematics that generalizes the theory of group representations. It has deep connections with many other fields, including representation theory, Hopf algebras, operator algebras, low-dimensional topology (in particular, knot theory), homotopy theory, quantum mechanics and field theory, quantum computation, theory of motives, etc. This book gives a systematic introduction to this theory and a review of its applications. While giving a detailed overview of general tensor categories, it focuses especially on the theory of finite tensor categories and fusion categories (in particular, braided and modular ones), and discusses the main results about them with proofs. In particular, it shows how the main properties of finite-dimensional Hopf algebras may be derived from the theory of tensor categories. Many important results are presented as a sequence of exercises, which makes the book valuable for students and suitable for graduate courses. Many applications, connections to other areas, additional results, and references are discussed at the end of each chapter.

TENSORS made easy with SOLVED PROBLEMS

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Author: Giancarlo Bernacchi

Publisher: Lulu.com

ISBN: 1326292536

Category: Science

Page: 218

View: 2338

-- New January 2018 REVISED RELEASE -- A friendly and non-formal approach to a subject of abstract mathematics that has important applications in physics, especially in General Relativity, but also in other fields. The purpose of the book is mainly dida

Introduction to Tensor Products of Banach Spaces

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Author: Raymond A. Ryan

Publisher: Springer Science & Business Media

ISBN: 1447139038

Category: Mathematics

Page: 226

View: 479

This is the first ever truly introductory text to the theory of tensor products of Banach spaces. Coverage includes a full treatment of the Grothendieck theory of tensor norms, approximation property and the Radon-Nikodym Property, Bochner and Pettis integrals. Each chapter contains worked examples and a set of exercises, and two appendices offer material on summability in Banach spaces and properties of spaces of measures.

Tensor Calculus

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Author: Uday Chand De,Absos Ali Shaikh,Joydeep Sengupta

Publisher: Alpha Science Int'l Ltd.

ISBN: 9781842651902

Category: Mathematics

Page: 170

View: 3262

This work covers all the basic topics of tensor analysis in a lucid and clear language and is aimed at both the undergraduate and postgraduate in Civil, Mechanical and Aerospace Engineering and in Engineering Physics.

The Metric Theory of Tensor Products

Grothendieck's Résumé Revisited

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Author: Joseph Diestel,Alexandre Grothendieck,Jan H. Fourie,Johan Swart

Publisher: American Mathematical Soc.

ISBN: 9780821872697

Category: Mathematics

Page: 278

View: 3635

Famed mathematician Alexander Grothendieck, in his Resume, set forth his plan for the study of the finer structure of Banach spaces. He used tensor products as a foundation upon which he built the classes of operators most important to the study of Banach spaces and established the importance of the "local" theory in the study of these operators and the spaces they act upon. When Lintenstrauss and Pelczynski addressed his work at the rebirth of Banach space theory, they shed his Fundamental Inequality in the trappings of operator ideals by shedding the tensorial formulation. The authors of this book, however, feel that there is much of value in Grothendieck's original formulations in the Resume and here endeavor to "expose the Resume" by presenting most of Grothendieck's arguments using the mathematical tools that were available to him at the time.

A Student's Guide to Vectors and Tensors

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Author: Daniel A. Fleisch

Publisher: Cambridge University Press

ISBN: 1139503944

Category: Science

Page: N.A

View: 692

Vectors and tensors are among the most powerful problem-solving tools available, with applications ranging from mechanics and electromagnetics to general relativity. Understanding the nature and application of vectors and tensors is critically important to students of physics and engineering. Adopting the same approach used in his highly popular A Student's Guide to Maxwell's Equations, Fleisch explains vectors and tensors in plain language. Written for undergraduate and beginning graduate students, the book provides a thorough grounding in vectors and vector calculus before transitioning through contra and covariant components to tensors and their applications. Matrices and their algebra are reviewed on the book's supporting website, which also features interactive solutions to every problem in the text where students can work through a series of hints or choose to see the entire solution at once. Audio podcasts give students the opportunity to hear important concepts in the book explained by the author.

Introduction to Tensor Analysis and the Calculus of Moving Surfaces

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Author: Pavel Grinfeld

Publisher: Springer Science & Business Media

ISBN: 1461478677

Category: Mathematics

Page: 302

View: 3595

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.

Elements for Physics

Quantities, Qualities, and Intrinsic Theories

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Author: Albert Tarantola

Publisher: Springer Science & Business Media

ISBN: 3540311076

Category: Science

Page: 266

View: 2180

Reviews and extends the theory of Lie groups, develops differential geometry, proposing compact definitions of torsion and of curvature, and adapts the usual notion of linear tangent application to the intrinsic point of view proposed for physics. Uses a unifying illustration: two simple theories are studied with some detail, the theory of heat conduction and the theory of linear elastic media. Shows that the resulting equations derived in this manner differ quantitatively and qualitatively from those usually presented.

Vectors, Tensors and the Basic Equations of Fluid Mechanics

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Author: Rutherford Aris

Publisher: Courier Corporation

ISBN: 048613489X

Category: Mathematics

Page: 320

View: 3321

Introductory text, geared toward advanced undergraduate and graduate students, applies mathematics of Cartesian and general tensors to physical field theories and demonstrates them in terms of the theory of fluid mechanics. 1962 edition.