Tensor Analysis and Continuum Mechanics

Author: Y.R. Talpaert

Publisher: Springer Science & Business Media

ISBN: 9401599882

Category: Mathematics

Page: 591

View: 576

This book is designed for students in engineering, physics and mathematics. The material can be taught from the beginning of the third academic year. It could also be used for self study, given its pedagogical structure and the numerous solved problems which prepare for modem physics and technology. One of the original aspects of this work is the development together of the basic theory of tensors and the foundations of continuum mechanics. Why two books in one? Firstly, Tensor Analysis provides a thorough introduction of intrinsic mathematical entities, called tensors, which is essential for continuum mechanics. This way of proceeding greatly unifies the various subjects. Only some basic knowledge of linear algebra is necessary to start out on the topic of tensors. The essence of the mathematical foundations is introduced in a practical way. Tensor developments are often too abstract, since they are either aimed at algebraists only, or too quickly applied to physicists and engineers. Here a good balance has been found which allows these extremes to be brought closer together. Though the exposition of tensor theory forms a subject in itself, it is viewed not only as an autonomous mathematical discipline, but as a preparation for theories of physics and engineering. More specifically, because this part of the work deals with tensors in general coordinates and not solely in Cartesian coordinates, it will greatly help with many different disciplines such as differential geometry, analytical mechanics, continuum mechanics, special relativity, general relativity, cosmology, electromagnetism, quantum mechanics, etc ..
Release

Tensor Analysis and Continuum Mechanics

Author: Wilhelm Flügge

Publisher: Springer Science & Business Media

ISBN: 3642883826

Category: Technology & Engineering

Page: 207

View: 8385

Through several centuries there has been a lively interaction between mathematics and mechanics. On the one side, mechanics has used mathemat ics to formulate the basic laws and to apply them to a host of problems that call for the quantitative prediction of the consequences of some action. On the other side, the needs of mechanics have stimulated the development of mathematical concepts. Differential calculus grew out of the needs of Newtonian dynamics; vector algebra was developed as a means . to describe force systems; vector analysis, to study velocity fields and force fields; and the calcul~s of variations has evolved from the energy principles of mechan ics. In recent times the theory of tensors has attracted the attention of the mechanics people. Its very name indicates its origin in the theory of elasticity. For a long time little use has been made of it in this area, but in the last decade its usefulness in the mechanics of continuous media has been widely recognized. While the undergraduate textbook literature in this country was becoming "vectorized" (lagging almost half a century behind the development in Europe), books dealing with various aspects of continuum mechanics took to tensors like fish to water. Since many authors were not sure whether their readers were sufficiently familiar with tensors~ they either added' a chapter on tensors or wrote a separate book on the subject.
Release

Tensor Algebra and Tensor Analysis for Engineers

With Applications to Continuum Mechanics

Author: Mikhail Itskov

Publisher: Springer Science & Business Media

ISBN: 3540939075

Category: Technology & Engineering

Page: 247

View: 8674

There is a large gap between engineering courses in tensor algebra on one hand, and the treatment of linear transformations within classical linear algebra on the other. This book addresses primarily engineering students with some initial knowledge of matrix algebra. Thereby, mathematical formalism is applied as far as it is absolutely necessary. Numerous exercises provided in the book are accompanied by solutions enabling autonomous study. The last chapters deal with modern developments in the theory of isotropic and anisotropic tensor functions and their applications to continuum mechanics and might therefore be of high interest for PhD-students and scientists working in this area.
Release

Applications of Tensor Analysis in Continuum Mechanics

Author: Cloud Michael J,Eremeyev Victor A,Lebedev Leonid P

Publisher: World Scientific

ISBN: 9813238984

Category: Science

Page: 428

View: 1299

A tensor field is a tensor-valued function of position in space. The use of tensor fields allows us to present physical laws in a clear, compact form. A byproduct is a set of simple and clear rules for the representation of vector differential operators such as gradient, divergence, and Laplacian in curvilinear coordinate systems. The tensorial nature of a quantity permits us to formulate transformation rules for its components under a change of basis. These rules are relatively simple and easily grasped by any engineering student familiar with matrix operators in linear algebra. More complex problems arise when one considers the tensor fields that describe continuum bodies. In this case general curvilinear coordinates become necessary. The principal basis of a curvilinear system is constructed as a set of vectors tangent to the coordinate lines. Another basis, called the dual basis, is also constructed in a special manner. The existence of these two bases is responsible for the mysterious covariant and contravariant terminology encountered in tensor discussions. This book provides a clear, concise, and self-contained treatment of tensors and tensor fields. It covers the foundations of linear elasticity, shell theory, and generalized continuum media, offers hints, answers, and full solutions for many of the problems and exercises, and Includes a handbook-style summary of important tensor formulas. The book can be useful for beginners who are interested in the basics of tensor calculus. It also can be used by experienced readers who seek a comprehensive review on applications of the tensor calculus in mechanics.
Release

Tensors

The Mathematics of Relativity Theory and Continuum Mechanics

Author: Anadi Jiban Das

Publisher: Springer Science & Business Media

ISBN: 0387694692

Category: Science

Page: 290

View: 7991

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.
Release

Tensor Analysis

Author: Fridtjov Irgens

Publisher: Springer

ISBN: 3030034127

Category: Science

Page: 385

View: 8302

This book presents tensors and tensor analysis as primary mathematical tools for engineering and engineering science students and researchers. The discussion is based on the concepts of vectors and vector analysis in three-dimensional Euclidean space, and although it takes the subject matter to an advanced level, the book starts with elementary geometrical vector algebra so that it is suitable as a first introduction to tensors and tensor analysis. Each chapter includes a number of problems for readers to solve, and solutions are provided in an Appendix at the end of the text. Chapter 1 introduces the necessary mathematical foundations for the chapters that follow, while Chapter 2 presents the equations of motions for bodies of continuous material. Chapter 3 offers a general definition of tensors and tensor fields in three-dimensional Euclidean space. Chapter 4 discusses a new family of tensors related to the deformation of continuous material. Chapter 5 then addresses constitutive equations for elastic materials and viscous fluids, which are presented as tensor equations relating the tensor concept of stress to the tensors describing deformation, rate of deformation and rotation. Chapter 6 investigates general coordinate systems in three-dimensional Euclidean space and Chapter 7 shows how the tensor equations discussed in chapters 4 and 5 are presented in general coordinates. Chapter 8 describes surface geometry in three-dimensional Euclidean space, Chapter 9 includes the most common integral theorems in two- and three-dimensional Euclidean space applied in continuum mechanics and mathematical physics.
Release

Tensor Analysis and Nonlinear Tensor Functions

Author: Yuriy I. Dimitrienko

Publisher: Springer Science & Business Media

ISBN: 9401732213

Category: Mathematics

Page: 662

View: 6797

Tensor Analysis and Nonlinear Tensor Functions embraces the basic fields of tensor calculus: tensor algebra, tensor analysis, tensor description of curves and surfaces, tensor integral calculus, the basis of tensor calculus in Riemannian spaces and affinely connected spaces, - which are used in mechanics and electrodynamics of continua, crystallophysics, quantum chemistry etc. The book suggests a new approach to definition of a tensor in space R3, which allows us to show a geometric representation of a tensor and operations on tensors. Based on this approach, the author gives a mathematically rigorous definition of a tensor as an individual object in arbitrary linear, Riemannian and other spaces for the first time. It is the first book to present a systematized theory of tensor invariants, a theory of nonlinear anisotropic tensor functions and a theory of indifferent tensors describing the physical properties of continua. The book will be useful for students and postgraduates of mathematical, mechanical engineering and physical departments of universities and also for investigators and academic scientists working in continuum mechanics, solid physics, general relativity, crystallophysics, quantum chemistry of solids and material science.
Release

Ohne Panik Strömungsmechanik!

Ein Lernbuch zur Prüfungsvorbereitung, zum Auffrischen und Nachschlagen mit Cartoons von Oliver Romberg

Author: Jann Strybny

Publisher: Springer-Verlag

ISBN: 3834883417

Category: Technology & Engineering

Page: 256

View: 9260

Das Verständnis der Strömungsmechanik (Hydromechanik) ist in Technik und Umweltwissenschaften unverzichtbar. Das wichtigste ist in diesem Buch unkonventionell auf den Punkt gebracht, mit Karikaturen von Oliver Romberg, dem Koautor von "Keine Panik vor Mechanik!", mit "Wozuseiten", und "Totenkopfkapiteln". Für Studierende kein Übungsbuch, kein Lehrbuch, sondern ein Lernbuch, für Praktiker im Berufsleben systematisch sortiert zum Auffrischen und Nachschlagen von Überschlagsrechnungen - alles lösbar mit Bleistift und Papier. Im Mittelpunkt stehen Beispielaufgaben, nicht hunderte, sondern 55, dafür nicht nur mit Endergebnissen sondern ganz ausführlich. Die zum Verständnis eines Themas mindestens notwendige Theorie wird in einfachen Worten als roter Faden erläutert, auf niemals mehr als vier Seiten am Stück. Begonnen wird auf Schulniveau und die Themenpalette reicht bis in das Fachstudium der Universitäten - vom Druck zur Navier-Stokes-Gleichung. Dreh- und Angelpunkt des ganzen sind die nur 45 kurzen Formeln im vorderen/hinteren Umschlag. Das Buch verfügt über zwei Einteilungen, eine nach den klassischen Themen der Hydromechanik und eine zweite in 21 Lektionen für 21 Abende. Von Assistenten kann es als Übungskonzept für zwei Semester genutzt werden, denn jede Lektion ist gezielt so aufgebaut, dass sie das fertige Tafelbild für 90 Minuten bildet.
Release

Elements of Continuum Mechanics and Thermodynamics

Author: Joanne L. Wegner,James B. Haddow

Publisher: Cambridge University Press

ISBN: 0521866324

Category: Science

Page: 278

View: 9898

Provides a complete course in continuum mechanics with examples and exercises and a chapter on continuum thermodynamics.
Release

Introduction to Tensor Analysis and the Calculus of Moving Surfaces

Author: Pavel Grinfeld

Publisher: Springer Science & Business Media

ISBN: 1461478677

Category: Mathematics

Page: 302

View: 1067

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.
Release

Mechanics, Tensors & Virtual Works

Author: Yves Talpaert

Publisher: Cambridge International Science Pub

ISBN: 9781898326113

Category: Science

Page: 460

View: 6564

Most of the text comes from this level courses that the author taught at universities and engineering schools. In the particular case where such a course cannot be taught to engineers, a lot of introduced matters constitute the mathematical and mechanical bases of applied engineering mechanics. The various chapters connect the notions of mechanics of first and second year with the ones which are developed in more specialized subjects as continuum mechanics at first, and fluid-dynamics, quantum mechanics, special relativity, general relativity, electromagnetism, stellar dynamics, celestial mechanics, meteorology, applied differential geometry, and so on. This book is the ideal mathematical and mechanical preparation for the above mentioned specialized disciplines. This is a course of Analytical Mechanics which synthesizes the notions of first level mechanics and leads to the various mentioned disciplines by introducing mathematical concepts as tensor and virtual work methods. Analytical mechanics is not only viewed as a self-sufficient mathematical discipline, but as a subject of mechanics preparing for theories of physics and engineering too.
Release

Manifolds, Tensor Analysis, and Applications

Author: Ralph Abraham,J.E. Marsden,Tudor Ratiu

Publisher: Springer Science & Business Media

ISBN: 0387967907

Category: Mathematics

Page: 656

View: 9463

The purpose of this book is to provide core material in nonlinear analysis for mathematicians, physicists, engineers, and mathematical biologists. The main goal is to provide a working knowledge of manifolds, dynamical systems, tensors, and differential forms. Some applications to Hamiltonian mechanics, fluid me chanics, electromagnetism, plasma dynamics and control thcory arc given in Chapter 8, using both invariant and index notation. The current edition of the book does not deal with Riemannian geometry in much detail, and it does not treat Lie groups, principal bundles, or Morse theory. Some of this is planned for a subsequent edition. Meanwhile, the authors will make available to interested readers supplementary chapters on Lie Groups and Differential Topology and invite comments on the book's contents and development. Throughout the text supplementary topics are given, marked with the symbols ~ and {l:;J. This device enables the reader to skip various topics without disturbing the main flow of the text. Some of these provide additional background material intended for completeness, to minimize the necessity of consulting too many outside references. We treat finite and infinite-dimensional manifolds simultaneously. This is partly for efficiency of exposition. Without advanced applications, using manifolds of mappings, the study of infinite-dimensional manifolds can be hard to motivate.
Release

Tensor Analysis with Applications in Mechanics

Author: L. P. Lebedev

Publisher: World Scientific

ISBN: 9814313998

Category: Mathematics

Page: 380

View: 1009

The tensorial nature of a quantity permits us to formulate transformation rules for its components under a change of basis. These rules are relatively simple and easily grasped by any engineering student familiar with matrix operators in linear algebra. More complex problems arise when one considers the tensor fields that describe continuum bodies. In this case general curvilinear coordinates become necessary. The principal basis of a curvilinear system is constructed as a set of vectors tangent to the coordinate lines. Another basis, called the dual basis, is also constructed in a special manner. The existence of these two bases is responsible for the mysterious covariant and contravariant terminology encountered in tensor discussions. A tensor field is a tensor-valued function of position in space. The use of tensor fields allows us to present physical laws in a clear, compact form. A byproduct is a set of simple and clear rules for the representation of vector differential operators such as gradient, divergence, and Laplacian in curvilinear coordinate systems. This book is a clear, concise, and self-contained treatment of tensors, tensor fields, and their applications. The book contains practically all the material on tensors needed for applications. It shows how this material is applied in mechanics, covering the foundations of the linear theories of elasticity and elastic shells. The main results are all presented in the first four chapters. The remainder of the book shows how one can apply these results to differential geometry and the study of various types of objects in continuum mechanics such as elastic bodies, plates, and shells. Each chapter of this new edition is supplied with exercises and problems most with solutions, hints, or answers to help the reader progress. An extended appendix serves as a handbook-style summary of all important formulas contained in the book.
Release

Vectors and Tensors in Crystallography

Author: Donald E. Sands

Publisher: Courier Corporation

ISBN: 9780486495163

Category: Science

Page: 228

View: 4606

Ample instruction on vector and tensor manipulations in general coordinate systems, plus specific examples and applications. Although the text emphasizes crystallographic applications, the methods developed are essential in any problems pertaining to nonorthogonal systems. "Heartily recommended to every crystallographer, students of crystallography and other solid-state scientists." —Acta Crystallographica. 1982 edition.
Release

Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers

Author: Hung Nguyen-Schäfer,Jan-Philip Schmidt

Publisher: Springer

ISBN: 3662484978

Category: Technology & Engineering

Page: 376

View: 8216

This book presents tensors and differential geometry in a comprehensive and approachable manner, providing a bridge from the place where physics and engineering mathematics end, and the place where tensor analysis begins. Among the topics examined are tensor analysis, elementary differential geometry of moving surfaces, and k-differential forms. The book includes numerous examples with solutions and concrete calculations, which guide readers through these complex topics step by step. Mindful of the practical needs of engineers and physicists, book favors simplicity over a more rigorous, formal approach. The book shows readers how to work with tensors and differential geometry and how to apply them to modeling the physical and engineering world. The authors provide chapter-length treatment of topics at the intersection of advanced mathematics, and physics and engineering: • General Basis and Bra-Ket Notation • Tensor Analysis • Elementary Differential Geometry • Differential Forms • Applications of Tensors and Differential Geometry • Tensors and Bra-Ket Notation in Quantum Mechanics The text reviews methods and applications in computational fluid dynamics; continuum mechanics; electrodynamics in special relativity; cosmology in the Minkowski four-dimensional space time; and relativistic and non-relativistic quantum mechanics. Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers benefits research scientists and practicing engineers in a variety of fields, who use tensor analysis and differential geometry in the context of applied physics, and electrical and mechanical engineering. It will also interest graduate students in applied physics and engineering.
Release

Matrix-Tensor Methods in Continuum Mechanics

(Revised Second Printing)

Author: S F Borg

Publisher: World Scientific Publishing Company

ISBN: 9813103671

Category: Technology & Engineering

Page: 356

View: 1505

The purposes of the text are: To introduce the engineer to the very important discipline in applied mathematics-tensor methods as well as to show the fundamental unity of the different fields in continuum mechanics-with the unifying material formed by the matrix-tensor theory and to present to the engineer modern engineering problems. Request Inspection Copy
Release

Continuum Mechanics

Author: Fridtjov Irgens

Publisher: Springer Science & Business Media

ISBN: 9783540742982

Category: Technology & Engineering

Page: 661

View: 5418

This book presents an introduction into the entire science of Continuum Mechanics in three parts. The presentation is modern and comprehensive. Its introduction into tensors is very gentle. The book contains many examples and exercises, and is intended for scientists, practitioners and students of mechanics.
Release