Mathtensor: A System for Doing Tensor Analysis by Computer - Softcover

Parker, Leonard; Christensen, Steven M.

Softcover

ISBN 10: 0201569906 ISBN 13: 9780201569902

Publisher: Addison-Wesley Professional, 1994

Synopsis

An introduction to MathTensor, software that can be used with Mathematica to do tensor analysis with a computer. Software is available separately from publisher. DLC: MathTensor.

About this title

From the Inside Flap

This book should be of interest to practitioners, researchers, and students in the sciences, engineering, and mathematics who wish to make efficient and accurate use of tensors and differential forms in solving problems. MathTensor is a computer program that works with Mathematica by extending its capabilities to include tensor analysis and differential forms. These pages are intended to serve, first of all, as a comprehensive introduction to all aspects of MathTensor. In addition, they contain pedagogical introductions to tensors and differential forms. The mathematical techniques for dealing with tensors have been developed by mathematicians and scientists over many years. Tensors are applied, among other things, to describe electromagnetic fields, energy and momentum, elementary particles, gravitational fields and spacetime, elastic stresses and strains in deformed bodies, fluid dynamics, the geometry of curved surfaces and spaces, and the curvature of the cornea of the eye. These applications range from physics, elasticity, and differential geometry to aerodynamics, meteorology, and medicine. Whether you are doing extensive tensor analysis or are only starting to learn about tensors and differential forms, you will find this book worth reading. Pedagogical introductions to tensors and differential forms make this book an excellent learning tool, suitable for self-study or as a textbook in a computer-based college or introductory graduate level course.

Following a brief chapter of short examples showing what you can do with MathTensor, the book is divided into three main parts. The first part begins with a chapter giving a pedagogical introduction to tensors. This introduction to tensors is followed by two comprehensive chapters discussing the MathTensor commands for dealing with tensors carrying symbolic or concrete indices. Included are short examples from physics, engineering, and mathematics.

The second part consists of two chapters on differential forms. The first chapter introduces differential forms and describes and illustrates the various differential form operations in MathTensor. The second chapter shows examples of how you may work with differential forms (as well as tensor-valued differential forms) in MathTensor. The examples include calculations involving explicit functions in a coordinate basis, as well as the Cartan structure equations with torsion in an arbitrary basis.

The third part gives longer examples and applications of MathTensor, including chapters on electromagnetism and special relativity, nonlinear elasticity in engineering mechanics, and general relativity, including variational methods. You are not expected to know all these areas, but you will nevertheless find in these chapters helpful techniques for using MathTensor. There is also an Appendix summarizing the usage of the various MathTensor commands and objects and a detailed index.

Chapter 1 through Chapter 8, except for Section 3.6 were written by Leonard Parker. Chapter 9 and Section 3.6 were written by Steven M. Christensen.

Reading this book will benefit anyone who is using or planning to use techniques of tensor analysis and differential forms. Even if you do not yet use Mathematica MathTensor, you will gain an understanding of how they can be of help to you. The intended audience includes not only researchers and practitioners, but also students. MathTensor can be a great help in learning about tensors and subjects involving tensors. If you are already using Mathematica, this book will introduce you to new and powerful methods by means of which you can explore new areas. If you are using or expecting to use MathTensor, this book is essential reading.

Depending on your background, you will want to study some chapters more thoroughly than others. All readers will want to read the chapters in Part A because they serve as an introduction to tensor analysis and to MathTensor. If you are already familiar with tensors, you can skim through the introduction to tensors, but you should carefully read the chapter on how to use MathTensor.

Part B should be read if you wish to learn about or use differential forms. Differential forms are elegant objects related to antisymmetric tensors. Reading Part B of this book in conjunction with one of the many textbooks on differential forms is an effective way to teach yourself the subject. MathTensor will also help you carry out complicated differential form computations in your research and development work.

The chapters of Part C are independent of one another. You do not have to be a specialist in the subject of each chapter to find useful techniques that you may be able to apply to your own problems. Thus if you have the time you should look through all chapters of Part C. Engineers should certainly read the chapter on applications of MathTensor to nonlinear elasticity, as well as detailed examples. The chapter on electromagnetism and special relativity should be read by anyone interested in this fundamental subject, which forms the basis for much of twentieth-century physics. The chapter on general relativity is worth reading if you are interested in gravitation or other areas involving the curvature of surfaces or spaces. The part of this chapter that uses variational techniques will be of interest to many physicists and mathematicians. Most of the basic equations of physics can be obtained through variation of a suitable integral called the action. Such variational methods occur in optics, gravitation, mechanics, electromagnetism, and many other areas.

The convention for numbering equations is illustrated by the following example. Equation 10 of Chapter 2 is referenced as Eq. (10) in Chapter 2 and as Eq. (2.10) in other chapters.

The definitive reference on Mathematica is Stephen Wolfram, Mathematica, A System for Doing Mathematica by Computer (Addison-Wesley, 1991). You can find discussions of MathTensor and some of its applications in Richard Crandall, Mathematica for the Sciences (Addison-Wesley, 1991), and Thomas B. Bahder, Mathematica for Scientists and Engineers (Addison-Wesley, 1994).

MathTensor is in use at leading universities, governmental laboratories, research institutes, and corporations both here and abroad. MathTensor will run on any microcomputer, workstation, server, or mainframe that can run Mathematica. You can obtain more information about MathTensor, or order it, by contacting MathSolutions, Inc., 3049 N. Lake Drive, Milwaukee, WI 53211, USA, or MathSolutions, Inc., P.O. Box 16175, Chapel Hill, NC 27516, USATelephone numbers: (414) 964-6284 and (919) 967-9853Fax numbers: (414) 964-6284 and (919) 967-9853Electronic mail address: mathtensor@wriAn information requests form is included on the last page of this volume.

The authors thank Stephen Wolfram, Roman Maeder, Kevin McIsaac, Paul Abbott, Jamie Petersen, Tom Sherlock, Ben Friedman, Joe Kaiping, Maury Kendell, and the entire staff of Wolfram Research, Inc., for their extremely valuable assistance over the past several years.

We are also grateful to Sun Microsystems for the donation of three workstations early in this project.

We thank numerous colleagues for helpful suggestions. These include Stephen Fulling, David Boulware, Ben Chow, Gautam Dasgupta, Timothy Gallivan, David Hobill, Richard Isaacson, Yefim Ivshin, Thomas Pence, Ulrich Jentschura, and all of the beta testers and users from many institutions.

Leonard Parker and Steven M. Christensen 0201569906P04062001

From the Back Cover

This book is an introduction to the authors' MathTensor software, which provides a computer program that works with Mathematica by extending its capabilities to include tensor analysis. Included is an introduction to MathTensor's commands and functions along with information on how to apply MathTensor to specific problems, among them problems in electromagnetism and relativity. For students and researchers in science, engineering and mathematics who use tensors in solving problems.

0201569906B04062001

About this edition

• Publisher: Addison-Wesley Professional
• Publication date: 1994
• ISBN 10: 0201569906
• ISBN 13: 9780201569902
• Binding: Paperback
• Edition number: 1
• Number of pages: 400