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Thinking About Equations THINKING ABOUT EQUATIONS THINKING ABOUT EQUATIONS A Practical Guide for Developing Mathematical Intuition in the Physical Sciences and Engineering MATT A. BERNSTEIN, PhD WILLIAM A. FRIEDMAN, PhD A JOHN WILEY & SONS, INC., PUBLICATION Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6088, or online at http://www.wiley.com/go/permission. Limit of Liability/Disclaimer of Warranty: While the publishers and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifi cally disclaim any implied warranties of merchantability or fi tness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profi t or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002. Wiley also publishes is books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com. Library of Congress Cataloging-in-Publication Data: Bernstein, Matt A., 1958– Thinking about equations : a practical guide for developing mathematical intuition in the physical sciences and engineering / Matt A. Bernstein, William A. Friedman. p. cm. Includes bibliographical references and index. ISBN 978-0-470-18620-6 (pbk.) 1. Mathematical analysis. 2. Physical sciences–Mathematics. 3. Engineering– Mathematics. I. Friedman, William A. (William Albert) II. Title. QA300.B496 2009 515—dc22 2009001787 Printed in the United States of America. 10 9 8 7 6 5 4 3 2 1 CONTENTS Preface vii Acknowledgments xi List of Worked-Out Example Problems xiii 1 Equations Representing Physical Quantities 1 1.1 Systems of Units, 2 1.2 Conversion of Units, 4 1.3 Dimensional Checks and the Use of Symbolic Parameters, 5 1.4 Arguments of Transcendental Functions, 7 1.5 Dimensional Checks to Generalize Equations, 9 1.6 Other Types of Units, 10 1.7 Simplifying Intermediate Calculations, 11 Exercises, 15 2 A Few Pitfalls and a Few Useful Tricks 18 2.1 A Few Instructive Pitfalls, 18 2.2 A Few Useful Tricks, 31 2.3 A Few “Advanced” Tricks, 37 Exercises, 42 3 Limiting and Special Cases 47 3.1 Special Cases to Simplify and Check Algebra, 47 3.2 Special Cases and Heuristic Arguments, 49 3.3 Limiting Cases of a Differential Equation, 57 v vi CONTENTS 3.4 Transition Points, 62 Exercises 76 4 Diagrams, Graphs, and Symmetry 83 4.1 Introduction, 83 4.2 Diagrams for Equations, 84 4.3 Graphical Solutions, 88 4.4 Symmetry to Simplify Equations, 99 Exercises, 100 5 Estimation and Approximation 115 5.1 Powers of Two for Estimation, 116 5.2 Fermi Questions, 118 5.3 Estimates Based on Simple Physics, 121 5.4 Approximating Defi nite Integrals, 125 5.5 Perturbation Analysis, 136 5.6 Isolating Important Variables, 145 Exercises, 150 6 Introduction to Dimensional Analysis and Scaling 155 6.1 Dimensional Analysis: An Introduction, 155 6.2 Dimensional Analysis: A Systematic Approach, 158 6.3 Introduction to Scaling, 170 Exercises, 183 7 Generalizing Equations 188 7.1 Binomial Expressions, 189 7.2 Motivating a General Expression, 190 7.3 Recurring Themes, 192 7.4 General yet Simple: Euler’s Identity, 203 7.5 When to Try to Generalize, 207 Exercises, 208 8 Several Instructive Examples 214 8.1 Choice of Coordinate System, 214 8.2 Solution Has Unexpected Properties, 223 8.3 Solutions in Search of Problems, 231 8.4 Learning from Remarkable Results, 249 Exercises, 249 Index 253 PREFACE Equations play a central role in day - to - day problem solving for the physical sciences, engineering, and related fi elds. Those of us who pursue these technical careers must learn to understand and to solve equations. It is often a challenge to fi gure out what an equation means and then to form a strategy to solve it. One aim of this book is to illustrate that by investing a little effort into think- ing about equations, your task can become easier and your solutions can become more reliable. On a deeper level, studying the topics presented here could provide you with greater insight into your own fi eld of study, and you may even derive some intellectual pleasure in the process. Today, a staggering amount of knowl- edge is readily available through the Internet. With this abundance of informa- tion comes an ever - greater premium on understanding the interconnections and themes embedded within this information. This book can help you gain that intuitive understanding while you acquire some practical mathematical tools. We hope that our selection of topics and approaches will open your eyes to this useful and rewarding process. One might ask why we wrote a book about equations, i.e., focusing on analytic methods, when today so many problems are solved using numerical methods. In our own scientifi c research, we rely heavily on numerical computa- tion. Equations, however, are the basis for almost all numerical solutions and simulations. A deeper understanding of the underlying equations often can improve the strategy for a numerical approach, and importantly, it helps one to check whether the resulting numerical answers are reasonable. With the advent and widespread availability of powerful computational packages such as MATLAB ®, MAPLE ™ , IDL ®, and others, some students have spent less time studying and thinking about equations and have not yet developed an intuitive “ feel ” for this topic that experience brings. This book vii viii PREFACE can serve as a resource and workbook for students who wish to improve those skills in an effi cient manner. We believe that the powerful graphics capabilities of those computational packages can complement the material presented here, and users of symbolic manipulation software such as Mathematica ® will also fi nd this book relevant to their work. The core of our intended audience is undergraduate students majoring in any discipline within engineering or any of the physical sciences including physics, geology, astronomy, and chemistry, as well as related fi elds such as applied mathematics and biophysics. Even students in peripherally related fi elds, such as computational fi nance, will fi nd the book useful, particularly when they apply methods that originated in the physical sciences and engi- neering to their own work. Graduate students with an undergraduate back- ground in a fi eld like biology who are transitioning into a more quantitative area such as biomedical engineering could also fi nd the techniques described in this book quite useful. We have tried to target the level of the book so it is accessible to under- graduate students who have a background in basic calculus and some familiar- ity with differential equations. An undergraduate course in introductory physics is also recommended. Some of the worked example problems draw on concepts and results from linear algebra and complex analysis, but this back- ground is not essential to follow most of the material. We are not mathemati- cians, and there are no formal mathematical proofs in the book. Instead, the derivations, where provided, are presented at a level of rigor that is typical of work in the physical sciences. A few of the sections and example problems are labeled with an asterisk ( * ), because either they assume a higher level of mathematical background or else the material contained in them is not central to the rest of the book. These starred sections and problems can be skipped without loss of continuity. This book is not a comprehensive “ how - to ” manual of mathematical methods. Instead, selected concepts are introduced by a short discussion in each section of each chapter. Then, the concepts are illustrated and further developed with example problems followed by detailed solutions. These worked example problems form the backbone of this book. Our own backgrounds are in physics, with specialization in magnetic reso- nance imaging, medical physics, and biomedical engineering (MAB) and theo- retical nuclear physics (WAF). While we have chosen example problems drawn from subjects that are familiar to us, we have tried to avoid problems that require highly specialized knowledge. We also have tried to provide expla- nations of terms that might be unfamiliar and to avoid the use of jargon. We hope you will read and work through example problems that are outside of your immediate area of interest. If you can recognize concepts that can be applied to your own work, then you will have made great progress toward our ultimate goal.
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