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. 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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 P REFACE E quations play a central role in day- t o- d ay 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. O n 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- g reater 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. O ne 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. W ith 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 “ f eel” 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. T he 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. W e 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. T his book is not a comprehensive “ h ow- t o” 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. O ur own backgrounds are in physics, with specialization in magnetic reso- nance imaging, m edical 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. At the end of each chapter, there are a number of exercises designed to test and further develop your understanding of the material. We also encourage you to devise and to solve some of your own exercises, perhaps related to your own specifi c fi eld of interest. PREFACE ix C hapter 1 deals with units and dimensions of physical quantities. Although this is a very basic topic, it has some surprisingly profound implications that are developed more fully in Chapter 6 , which deals with dimensional analysis and scaling. Chapter 2 illustrates several common pitfalls to avoid when dealing with equations, along with a few handy techniques and tricks that are used in later chapters. Chapter 3 discusses special cases and their use to check equations and to guess at the solution of diffi cult problems. Chapter 4 focuses on pictorial and graphical methods, and provides a basic introduction to the concept of symmetry and its use for simplifying equations. Chapter 5 discusses a variety of estimation and approximation techniques. Chapter 7 discusses how and when to generalize equations. Finally, Chapter 8 provides a few more instructive examples that illustrate several problem -s olving techniques while reinforcing some of the concepts introduced in earlier chapters. T here are many interrelations and cross- r eferences among the chapters. Generalization, which is discussed in Chapter 7 , is closely related to the subject of special cases introduced in Chapter 3 . Similarly, dimensional analysis and scaling (Chapter 6 ) are closely related to the study of units and dimensions (Chapter 1 ) . Notice that these linked chapters are spaced apart in the book. This placement is intended to give the reader time to think about the material before it is revisited. Despite the interconnections among them, the chapters are, for the most part, self- c ontained and need not necessarily be read in numerical sequence. The book can be used to construct modules for a quick study of a more focused topic. Two such possible modules are: ( 1) d imensional analysis and scaling: Chapters 1 and 6 ; ( 2) s pecial cases, approximation, and generalization: Chapters 3 , 5 , and 7 . W e have also tried to enliven the discussion by identifying the nationality and approximate era of some of the luminaries who made major contributions to the scientifi c and mathematical literature that we have drawn upon, and by providing a few historical anecdotes. While this material is not necessary for dealing with the equations themselves, we hope that you fi nd it entertaining and that it will provide some insight into how science often has progressed along an unexpected or a roundabout path. I f you have suggestions for additional material that could be included in future editions of this book, fi nd errors or descriptions that are unclear, or wish to provide feedback of any other type, we would be happy to hear from you. M att A. Bernstein Rochester, Minnesota A CKNOWLEDGEMENTS S everal individuals offered their time and effort to review the manuscript and provide comments. We especially want to thank Juliet Bernstein, who offered many insightful suggestions, which have greatly improved the clarity of the book. Kristi Welle reviewed several of the example problems, and Linda Greene proofread many of the pages. We thank Professor Ž e ljko Bajzer for showing us the curious integral that is discussed in Example 2.1. We thank them all for their outstanding efforts and emphasize that any remaining mis- takes and inaccuracies are solely our own. I ndividually, we would like to express our gratitude. M AB: I thank my friends and coworkers at the Mayo Clinic and also in the broader magnetic resonance imaging community. It has been a privilege to work with such outstanding people during such an exciting time. I am also indebted to my entire family and especially to my wife Rhoda Lichy for her unwavering support and patience during the writing of this book. W AF: I am grateful to the many students at all levels with whom I have had contact during my nearly 40 years of teaching, and to my late wife Cheryl who was by my side during that journey. xi LIST OF WORKED-OUT EXAMPLE PROBLEMS 1.1 Conversion of units: A standard procedure 4 1.2 Choice of units to simplify calculations: Compton scattering 12 2.1 Conditions of a theorem: A curious defi nite integral 19 2.2 A calculation trick: Mean and variance of an exponential distribution 32 2.3 Parametric representation: Solving a differential equation 35 2.4 Delta function to remove constraints: Random walk problem* 39 3.1 Special cases to guess the solution: Area of a square 50 3.2 Limiting case of a differential equation: Bessel and Struve functions 59 3.3 Number of real solutions: Ferromagnetism 66 3.4 Number of real solutions: van der Waals equation of state 68 3.5 Critical points: Landau model of second-order phase transition 72 3.6 Critical points: Landau model of fi rst-order phase transition 73 4.1 Diagrams to suggest equations: Balls and urns 85 4.2 Graphical solution: Jones plot 89 4.3 Unexpected graphical solution: Eigenvalues of a type of matrix 93 4.4 Symmetry to simplify equations: Network of resistors 99 4.5 Symmetry to simplify equations: Fourier series 104 5.1 Estimating geometric growth: The rule of 72 116 5.2 Fermi questions: Number of hair salons 119 5.3 Estimate based on simple physics: Maximal height of a mountain 121 5.4 Estimate based on simple physics: Bee’s compound eye 123 5.5 Approximating integrals: Stirling’s formula 129 xiii