From Calculus to Chaos
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Dr D.J. Acheson Jesus College Oxford OX1 3DW Tel: 01865 279700 Fax: 01865 279687 Email: david.acheson@jesus.ox.ac.uk
What is calculus really for? This book is a highly readable introduction to applications of calculus, from Newton's time to the present day. These often involve questions of dynamics, i.e., of how--and why--things change with time. Problems of this kind lie at the heart of much of applied mathematics, physics, and engineering. From Calculus to Chaos takes a fresh approach to the subject as a whole, by moving from first steps to the frontiers, and by focusing on the many important and interesting ideas which can get lost amid a snowstorm of detail in conventional texts. The book is aimed at a wide readership, and assumes only some knowledge of elementary calculus. There are exercises (with full solutions) and simple but powerful computer programs which are suitable even for readers with no previous computing experience. David Acheson's book will inspire new students by providing a foretaste of more advanced mathematics and some of its liveliest applications.
1. Introduction 2. A brief review of calculus 3. Ordinary differential equations 4. Computer solution methods 5. Elementary oscillations 6. Planetary motion 7. Waves and diffusion 8. The best of all possible worlds? 9. Fluid flow 10. Instability and catastrophe 11. Nonlinear oscillations and chaos 12. The not-so-simple pendulum Further reading Appendix A. Elementary prgramming in QBASIC Appendix B. Ten programs for exploring dynamics Solutions to the exercises Index
"Despite public interest, it has been difficult to find a suitable introductory book on chaos for mathematics students. In 'From Calculus to Chaos', David Acheson manages to bridge the gap, tie the topic into the undergraduate curriculum, throw in some history and practical techniques, and tell readers about an experimental basis of dynamical systems theory--all this without being stuffy."--New Scientist "This is a thoroughly excellent little book and a most valuable addition to the literature on dynamics. Its approach is quite unique, bringing together a vast range of real physical phenomena and elucidating the essential dynamics by means of well chosen toy models in the form of differential equations. All the necessary analytical techniques are slipped in with the minimum of fuss, and numerical methods are employed throughout in such a way that the reader is encouraged to use the computer as an experimental tool. . . .the book deserves a place on the shelves of all serious students, teachers, and researchers." --UK Nonlinear News "The book is quite suitable as a text in dynamics and can serve as a refresher for those wishing to see how applied math courses they struggle through can be made simple."--Bulletin of the American Meteorological Society "This short introductory-level book illustrates several important ideas of contemporary research in nonlinear dynamics by the numerical solution of rigid solids and fluids examples. . . .[m]ost of the complex phenomena currently studied in nonlinear dynamics are modeled by equations whose properties can only be uncovered by numerical methods or by more oblique analytic attacks on their qualitative behavior. Students would likely be more successful if they have this library of physical examples with which to test ideas." --Applied Mechanics Reviews "Acheson presents an introduction to the calculus-based development of dynamics (continuous dynamical systems). The text is a beautiful historical review of physical mathematics from Newton and Leibniz to Lorenz in the late 20th century. It begins with a brief review of projectile and planetary motion and related topics in calculus and differential equations, including numerical (computer) solutions and the theory of oscillations. Later chapters discuss more advanced topics: the three-body problem, wave and diffusion equations, action and Hamilton's principle, calculus of variations, Lagrange's equations, fluid flow, theory of linear stability, bifurcation and catastrophic change, nonlinear oscillations, and the Lorenz equations. . . . an excellent overview of a broad body of material in a historically accurate setting. Chapter exercises; appendixes with solutions to exercises and an elementary introduction to programming in QBASIC. Undergraduates through professionals."--Choice "The book under review falls into the category of 'books I would have liked to have read in high school and first year in college, and then I would have been much better off'. Indeed, this is a very attractive introduction to a number of topics in dynamics, understood widely to include classical mechanics, wave motion, fluid flow, etc. The author has given us earlier another well-written and frequently cited textbook . . . , and here we find again the same laudable style of exposition."--Mathematical Reviews "Mathematics and Physics have always had close connections, it seems. In fact, the boundaries between the two subjects have been quite fluid over time. Not too long ago, Newtonian mechanics . . . was a part of mathematics, and number theory and algebraic geometry were the purest of pure mathematics. Things look different now. It also used to be the case that all mathematicians knew lots of physics, or at least theoretical physics. That, too, is no longer the case. Many mathematicians have never even had a course in theoretical mechanics. David Acheson's From Calculus to Chaos is a book that can remedy that lac
I enjoyed reading this book and learned quite a lot from it. I recommend it to anyone who - like myself - knows calculus better than chaos, and would like to begin rectifying the situation as painlessly as possible
'The author has been really successful in combining very intuitive mathematics with very intuitive physics in a highly readable book.....this is a most advisable book for first year courses on applied calculus, dynamics or introductory physics (or for a part of such courses). It will be alsovery useful to science teachers in schools and to general readers interested in science who wish to use their home computers to keep up with an important aspect of physical science.'Institute of Physics
The project as a whole succeeds well, and the book deserves to be on theshelves of people ranging from school science teachers to undergraduates inmathematics and physics....The last chapter deals with stability of invertedpendulums, and gives an excellent rounded account of how the theory,experiments, and simulations interact to explain this fascinating effect...Thematerial presented here is a fascinating and unpretentious sweep through thesubject, and it would make an ideal course text at undergraduate level, or anindividual study book for well-motivated 'A' level readers...The aim of givingthe reader insight into a wide array of dynamical problems using very elementarymathematics is achieved well, and the excellent selection of examples andhistorical asides adds depth to the topics covered.
'Using carefully selected examples, from the time of Newton to the presentday the author demonstrates in a highly readable form what we usually call themathematization of physical problems.'European Mathematical Society Newsletter, issue 27, March 1998
What is calculus really for? It helps tell us how - and why - things change with time. This is the purpose of dynamics, which lies at the heart of much of applied mathematics, science and engineering. From calculus to chaos takes a fresh approach to the subject as a whole. This is a highly readable and well-illustrated introduction to dynamics which aims to inspire the interest and enthusiasm of students embarking on a course of mathematical study. The writing is inviting, illuminating, and informative, not heavy or pedantic. Acheson explains why the mathematics which students learn is useful by exploring the most central and interesting ideas of calculus and dynamics from Newton to the present.
1. Introduction 2. A Brief Review of Calculus 3. Ordinary Differential Equations 4. Computer Solution Methods 5. Elementary Oscillations 6. Planetary Motion 7. Waves and diffusion 8. The Best of all Possible Worlds? 9. Fluid Flow 10. Instability and Catastrophe 11. Nonlinear Oscillations and Chaos 12. The Not-so-simple Pendulum Further reading Appendix A: Elementary programming in QBASIC Appendix B: Ten programs for exploring dynamics Solutions to the exercises Index
작가정보
저자(글) Acheson, D. J.
목차
Introduction A Brief Review of Calculus Ordinary Differential Equations Computer Solution Methods Elementary Oscillations Planetary Motion Waves and diffusion The Best of all Possible Worlds? Fluid Flow Instability and Catastrophe Nonlinear Oscillations and Chaos The Not-so-simple Pendulum Further reading Elementary programming in QBASIC Ten programs for exploring dynamics Solutions to the exercises Index Table of Contents provided by Publisher. All Rights Reserved.
기본정보
ISBN | 9780198500773 ( 0198500777 ) |
---|---|
발행(출시)일자 | 1998년 01월 08일 |
쪽수 | 280쪽 |
크기 |
156 * 232
* 15
mm
/ 476 g
|
총권수 | 1권 |
언어 | 영어 |
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