The Very Basics of Tensors

Author: Nils K. Oeijord

Publisher: iUniverse

ISBN: 059535694X

Category: Mathematics

Page: 137

View: 2289


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.

TensorFlow für Dummies

Author: Matthew Scarpino

Publisher: John Wiley & Sons

ISBN: 3527818960

Category: Computers

Page: 324

View: 6851


TensorFlow ist Googles herausragendes Werkzeug für das maschinelle Lernen, und dieses Buch macht es zugänglich, selbst wenn Sie bisher wenig über neuronale Netze und Deep Learning wissen. Sie erfahren, auf welchen Prinzipien TensorFlow basiert und wie Sie mit TensorFlow Anwendungen schreiben. Gleichzeitig lernen Sie die Konzepte des maschinellen Lernens kennen. Wenn Sie Softwareentwickler sind und TensorFlow in Zukunft einsetzen möchten, dann ist dieses Buch der richtige Einstieg für Sie. Greifen Sie auch zu, wenn Sie einfach mehr über das maschinelle Lernen erfahren wollen.

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: 8394


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.

Tensor Calculus for Engineers and Physicists

Author: Emil de Souza Sánchez Filho

Publisher: Springer

ISBN: 331931520X

Category: Technology & Engineering

Page: 345

View: 8816


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.

The General Genetic Catastrophe

On the Discovery and the Discoverer

Author: Nils K. Oeijord

Publisher: iUniverse

ISBN: 9781450297653

Category: Health & Fitness

Page: 244

View: 7520


Nils K. Oeijord’s research since 1999 shows that we have a worldwide general genetic catastrophe (GGC) due to general local and global manmade mutagenic pollution. The GGC began in the 1700s, increased in the 1800s, and exploded in the 1900s. The HIGH and INCREASING prevalence and the HIGH and INCREASING incidence of gene damage and genetic diseases all over the world logically prove the existence of the GGC. Nils K. Oeijord is a science writer, a former researcher (plant production), a former assistant professor (mathematics), and a former science and mathematics lecturer (high school). He is the discoverer of the general genetic catastrophe, and has earned a place in Who’s Who in the World (28th Edition), in Great Minds of the 21st Century (5th Edition), and in 2000 Outstanding Intellectuals of the 21st Century (2011 Edition).

Why Minus Times Minus Is Plus

The Very Basic Mathematics of Real and Complex Numbers

Author: Nils K. Oeijord

Publisher: N.A

ISBN: 9781450240635

Category: Mathematics

Page: 136

View: 8135


MATHEMATICS / ALGEBRA This book is written for a very broad audience. There are no particular prerequisites for reading this book. We hope students of High Schools, Colleges, and Universities, as well as hobby mathematicians, will like and benefit from this book. The book is rigorous and self-contained. All results are proved (or the proofs are optional exercises) and stated as theorems. Important points are covered by examples and optional exercises. Additionally there are also two sections called "More optional exercises (with answers)." Modern technology uses complex numbers for just about everything. Actually, there is no way one can formulate quantum mechanics without resorting to complex numbers. Leonard Euler (1707-1786) considered it natural to introduce students to complex numbers much earlier than we do today. Even in his elementary algebra textbook he uses complex numbers throughout the book. Nils K. Oeijord is a science writer and a former assistant professor of mathematics at Tromsoe College, Norway. He is the author of The Very Basics of Tensors, and several other books in English and Norwegian. Nils K. Oeijord is the discoverer of the general genetic catastrophe (GGC).

Vectors, Tensors and the Basic Equations of Fluid Mechanics

Author: Rutherford Aris

Publisher: Courier Corporation

ISBN: 048613489X

Category: Mathematics

Page: 320

View: 3083


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.

Spezielle Relativitätstheorie für Studienanfänger

Author: Jürgen Freund

Publisher: vdf Hochschulverlag AG

ISBN: 3825228843

Category: Science

Page: 252

View: 8921


Die Spezielle Relativitätstheorie ist nichts Schwieriges und Geheimnisvolles! Studienanfänger der Natur- und Technikwissenschaften können sie mit ihren Kenntnissen der Mathematik und der Physik in vollem Umfang und bis ins Detail verstehen, wenn sie sich die Mühe machen, sich durch dieses Lehrbuch durchzuarbeiten. Zahlreiche, z.T. auch kompliziertere Problemstellungen, bis hin zum Energie-Impuls-Vierervektor und zum elektromagnetischen Feld, werden ausführlich behandelt und zeigen dem Anfänger, wie relativistische Rechnungen anzupacken sind. Auch die bekannten Paradoxa und Paradoxien werden umfassend erklärt. Für den Wissenschaftler, der mit diesem Gebiet schon vertraut ist, ist dieses Lehrbuch wegen seiner umfangreichen Formelauflistungen nicht minder wertvoll. Kurz nach dem einhundertsten Jahrestag der Speziellen Relativitätstheorie liegt hiermit ein Kompendium vor, das dieses Teilgebiet der Physik entzaubert und in die elementare Grundlagenphysik einreiht.

Tensor Calculus

Author: J. L. Synge,A. Schild

Publisher: Courier Corporation

ISBN: 048614139X

Category: Mathematics

Page: 336

View: 8885


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.

Yetter-Drinfel'd Hopf Algebras Over Groups of Prime Order

Author: Yorck Sommerhauser,Yorck Sommerhäuser

Publisher: Springer Science & Business Media

ISBN: 9783540437994

Category: Mathematics

Page: 157

View: 3230


Being the first monograph devoted to this subject, the book addresses the classification problem for semisimple Hopf algebras, a field that has attracted considerable attention in the last years. The special approach to this problem taken here is via semidirect product decompositions into Yetter-Drinfel'd Hopf algebras and group rings of cyclic groups of prime order. One of the main features of the book is a complete treatment of the structure theory for such Yetter-Drinfel'd Hopf algebras.


Author: Klaus-Jürgen Bathe

Publisher: DrMaster Publications

ISBN: 9783540668060

Category: Technology & Engineering

Page: 1253

View: 9642


Dieses Lehr- und Handbuch behandelt sowohl die elementaren Konzepte als auch die fortgeschrittenen und zukunftsweisenden linearen und nichtlinearen FE-Methoden in Statik, Dynamik, Festkörper- und Fluidmechanik. Es wird sowohl der physikalische als auch der mathematische Hintergrund der Prozeduren ausführlich und verständlich beschrieben. Das Werk enthält eine Vielzahl von ausgearbeiteten Beispielen, Rechnerübungen und Programmlisten. Als Übersetzung eines erfolgreichen amerikanischen Lehrbuchs hat es sich in zwei Auflagen auch bei den deutschsprachigen Ingenieuren etabliert. Die umfangreichen Änderungen gegenüber der Vorauflage innerhalb aller Kapitel - vor allem aber der fortgeschrittenen - spiegeln die rasche Entwicklung innerhalb des letzten Jahrzehnts auf diesem Gebiet wieder. TOC:Eine Einführung in den Gebrauch von Finite-Elemente-Verfahren.-Vektoren, Matrizen und Tensoren.-Einige Grundbegriffe ingenieurwissenschaftlicher Berechnungen.-Formulierung der Methode der finiten Elemente.-Formulierung und Berechnung von isoparametrischen Finite-Elemente-Matrizen.-Nichtlineare Finite-Elemente-Berechnungen in der Festkörper- und Strukturmechanik.-Finite-Elemente-Berechnungen von Wärmeübertragungs- und Feldproblemen.-Lösung von Gleichgewichtsbeziehungen in statischen Berechnungen.-Lösung von Bewegungsgleichungen in kinetischen Berechnungen.-Vorbemerkungen zur Lösung von Eigenproblemen.-Lösungsverfahren für Eigenprobleme.-Implementierung der Finite-Elemente-Methode.

Tensors, Differential Forms, and Variational Principles

Author: David Lovelock,Hanno Rund

Publisher: Courier Corporation

ISBN: 048613198X

Category: Mathematics

Page: 400

View: 9509


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.

Transport Phenomena in Newtonian Fluids - A Concise Primer

Author: Per Olsson

Publisher: Springer Science & Business Media

ISBN: 3319013092

Category: Science

Page: 94

View: 364


This short primer provides a concise and tutorial-style introduction to transport phenomena in Newtonian fluids , in particular the transport of mass, energy and momentum. The reader will find detailed derivations of the transport equations for these phenomena, as well as selected analytical solutions to the transport equations in some simple geometries. After a brief introduction to the basic mathematics used in the text, Chapter 2, which deals with momentum transport, presents a derivation of the Navier-Stokes-Duhem equation describing the basic flow in a Newtonian fluid. Also provided at this stage are the derivations of the Bernoulli equation, the pressure equation and the wave equation for sound waves. The boundary layer, turbulent flow and flow separation are briefly reviewed. Chapter 3, which addresses energy transport caused by thermal conduction and convection, examines a derivation of the heat transport equation. Finally, Chapter 4, which focuses on mass transport caused by diffusion and convection, discusses a derivation of the mass transport equation.

Introduction to Diffusion Tensor Imaging

And Higher Order Models

Author: Susumu Mori,J-Donald Tournier

Publisher: Academic Press

ISBN: 0123984076

Category: Medical

Page: 140

View: 4121


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

Introduction to Vectors and Tensors

Author: Ray M. Bowen,Chao-cheng Wang

Publisher: Courier Corporation

ISBN: 048646914X

Category: Mathematics

Page: 520

View: 1310


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.

Tensor Analysis with Applications in Mechanics

Author: L. P. Lebedev

Publisher: World Scientific

ISBN: 9814313998

Category: Mathematics

Page: 380

View: 9310


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.

Basic Optical Properties of Materials

Summaries of Papers Presented at the Topical Conference on Basic Optical Properties of Materials, Held at the National Bureau of Standards, Gaithersburg, Maryland, May 5-7, 1980

Author: Albert Feldman

Publisher: N.A


Category: Optical materials

Page: 241

View: 1775



Author: Peter C. Kendall

Publisher: Springer-Verlag

ISBN: 332294056X

Category: Technology & Engineering

Page: 260

View: 2745


Bücher über Vektoranalysis beginnen üblicherweise mit der Definition eines Vektors als Äquivalenzklasse gerichteter Strecken - oder weniger genau, als Größe, die sowohl eine Richtung als auch eine Länge hat. Diese Einführung ist wegen ihres einfach erscheinenden Konzeptes einprägsam, aber sie führt zu logischen Schwierigkeiten, die nur durch sorgfältiges Vorgehen gelöst werden können. Folgerichtig haben Studenten oft Probleme, die Anfänge der Vektoranalysis vollständig zu verstehen und verlieren schnell an Vertrauen. Eine andere Unzulänglichkeit ist es, daß bei der weiteren Entwicklung häufig auf die geometrische Anschauung zurückgegriffen wird und viel Sorgfalt nötig ist, um analytische Zusammenhänge nicht zu verwischen oder zu übersehen. So wird z. B. selten klar, daß bei der Definition des Gradienten eines Skalarfeldes, der Divergenz oder der Rotation eines Vektorfeldes vorausgesetzt werden muß, daß die Felder stetig differenzierbar sind und daß die bloße Existenz der partiellen Ableitungen erster Ordnung unzureichend ist. Der Einstieg in die Vektoranalysis, der in diesem Band gewählt wurde, basiert auf der Definition eines Vektors mit Hilfe rechtwinkliger kartesischer Komponenten, die bei einer Änderung der Achsen vorgegebene Transformationsgesetze erfüllen. Dieser Einstieg wurde seit 10 Jahren erfolgreich in Anfängervorlesungen für Mathematiker und andere Naturwissenschaftler benutzt und bietet einige Vorteile. Regeln zur Addition und Subtraktion von Vektoren, zur Berechnung des Skalar- und Vektor produktes und zum Differenzieren sind schnell greifbar und die Möglichkeit, Vektoren so einfach zu handhaben, gibt den Studenten unmittelbares Zutrauen. Der spätere Einstieg in die Theorie der Vektorfelder erscheint natürlich, da Gradient, Divergenz und Rotation in ihrer Koordinatenform definiert sind.