MATHEMATICAL METHODS IN THE PHYSICAL SCIENCES Third Edition MARY L. BOAS DePaul University www.com MATHEMATICAL METHODS IN THE PHYSICAL SCIENCES www.com MATHEMATICAL METHODS IN THE PHYSICAL SCIENCES Third Edition MARY L. BOAS DePaul University www.com PUBLISHER Kaye Pace SENIOR ACQUISITIONS Editor Stuart Johnson PRODUCTION MANAGER Pam Kennedy PRODUCTION EDITOR Sarah Wolfman-Robichaud MARKETING MANAGER Amanda Wygal SENIOR DESIGNER Dawn Stanley EDITORIAL ASSISTANT Krista Jarmas/Alyson Rentrop PRODUCTION MANAGER Jan Fisher/Publication Services This book was set in 10/12 Computer Modern by Publication Services and printed and bound by R. The cover was printed by Lehigh Press.
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Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030-5774, (201)748-6011, fax (201)748-6008, or online at http://www.com/go/permissions. To order books or for customer service please, call 1-800-CALL WILEY (225-5945). ISBN 0-471-19826-9 ISBN-13 978-0-471-19826-0 ISBN-WIE 0-471-36580-7 ISBN-WIE-13 978-0-471-36580-8 Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 www.com To the memory of RPB www.com PREFACE This book is particularly intended for the student with a year (or a year and a half) of calculus who wants to develop, in a short time, a basic competence in each of the many areas of mathematics needed in junior to senior-graduate courses in physics, chemistry, and engineering. Thus it is intended to be accessible to sophomores (or freshmen with AP calculus from high school).
It may also be used effectively by a more advanced student to review half-forgotten topics or learn new ones, either by independent study or in a class. Although the book was written especially for students of the physical sciences, students in any field (say mathematics or mathematics for teaching) may find it useful to survey many topics or to obtain some knowledge of areas they do not have time to study in depth. Since theorems are stated carefully, such students should not need to unlearn anything in their later work. The question of proper mathematical training for students in the physical sci- ences is of concern to both mathematicians and those who use mathematics in appli- cations.
Some instructors may feel that if students are going to study mathematics at all, they should study it in careful and thorough detail. For the undergradu- ate physics, chemistry, or engineering student, this means either (1) learning more mathematics than a mathematics major or (2) learning a few areas of mathematics thoroughly and the others only from snatches in science courses. The second alter- native is often advocated; let me say why I think it is unsatisfactory. It is certainly true that motivation is increased by the immediate application of a mathematical technique, but there are a number of disadvantages: 1.
The discussion of the mathematics is apt to be sketchy since that is not the primary concern. Students are faced simultaneously with learning a new mathematical method and applying it to an area of science that is also new to them. Frequently the vii www.com viii Preface difficulty in comprehending the new scientific area lies more in the distraction caused by poorly understood mathematics than it does in the new scientific ideas. Students may meet what is actually the same mathematical principle in two different science courses without recognizing the connection, or even learn ap- parently contradictory theorems in the two courses! For example, in thermody- namics students learn that the integral of an exact differential around a closed 2π path is always zero.
In electricity or hydrodynamics, they run into 0 dθ, which is certainly the integral of an exact differential around a closed path but is not equal to zero! Now it would be fine if every science student could take the separate mathematics courses in differential equations (ordinary and partial), advanced calculus, linear algebra, vector and tensor analysis, complex variables, Fourier series, probability, calculus of variations, special functions, and so on. However, most science students have neither the time nor the inclination to study that much mathematics, yet they are constantly hampered in their science courses for lack of the basic techniques of these subjects. It is the intent of this book to give these students enough background in each of the needed areas so that they can cope successfully with junior, senior, and beginning graduate courses in the physical sciences. I hope, also, that some students will be sufficiently intrigued by one or more of the fields of mathematics to pursue it futher.
It is clear that something must be omitted if so many topics are to be compressed into one course. I believe that two things can be left out without serious harm at this stage of a student’s work: generality, and detailed proofs. Stating and proving a theorem in its most general form is important to the mathematician and to the advanced student, but it is often unnecessary and may be confusing to the more elementary student. This is not in the least to say that science students have no use for careful mathematics.
Scientists, even more than pure mathematicians, need careful statements of the limits of applicability of mathematical processes so that they can use them with confidence without having to supply proof of their validity. Consequently I have endeavored to give accurate statements of the needed theorems, although often for special cases or without proof. Interested students can easily find more detail in textbooks in the special fields. Mathematical physics texts at the senior-graduate level are able to assume a degree of mathematical sophistication and knowledge of advanced physics not yet attained by students at the sophomore level.
Yet such students, if given simple and clear explanations, can readily master the techniques we cover in this text. (They not only can, but will have to in one way or another, if they are going to pass their junior and senior physics courses!) These students are not ready for detailed applications—these they will get in their science courses—but they do need and want to be given some idea of the use of the methods they are studying, and some simple applications. This I have tried to do for each new topic. For those of you familiar with the second edition, let me outline the changes for the third: 1.
Prompted by several requests for matrix diagonalization in Chapter 3, I have moved the first part of Chapter 10 to Chapter 3 and then have amplified the treatment of tensors in Chapter 10. I have also changed Chapter 3 to include more detail about linear vector spaces and then have continued the discussion of basis functions in Chapter 7 (Fourier series), Chapter 8 (Differential equations), www.com Preface ix Chapter 12 (Series solutions) and Chapter 13 (Partial differential equations). Again, prompted by several requests, I have moved Fourier integrals back to the Fourier series Chapter 7. Since this breaks up the integral transforms chapter (old Chapter 15), I decided to abandon that chapter and move the Laplace transform and Dirac delta function material back to the ordinary differential equations Chapter 8.
I have also amplified the treatment of the delta function. The Probability chapter (old Chapter 16) now becomes Chapter 15. Here I have changed the title to Probability and Statistics, and have revised the latter part of the chapter to emphasize its purpose, namely to clarify for students the theory behind the rules they learn for handling experimental data. The very rapid development of technological aids to computation poses a steady question for instructors as to their best use.
Without selecting any particular Computer Algebra System, I have simply tried for each topic to point out to students both the usefulness and the pitfalls of computer use. (Please see my comments at the end of ”To the Student” just ahead.) The material in the text is so arranged that students who study the chapters in order will have the necessary background at each stage. However, it is not always either necessary or desirable to follow the text order. Let me suggest some rearrangements I have found useful.
If students have previously studied the material in any of chapters 1, 3, 4, 5, 6, or 8 (in such courses as second-year calculus, differential equations, linear algebra), then the corresponding chapter(s) could be omitted, used for reference, or, preferably, be reviewed briefly with emphasis on problem solving. Students may know Taylor’s theorem, for example, but have little skill in using series approximations; they may know the theory of multiple integrals, but find it difficult to set up a double integral for the moment of inertia of a spherical shell; they may know existence theorems for differential equations, but have little skill in solving, say, y + y = x sin x. Problem solving is the essential core of a course on Mathematical Methods. This gives students an introduction to Partial Differential Equations but requires only the use of Fourier series expansions.
Later on, after studying Chapter 12, students can return to complete Chapter 13. Chapter 15 (Probability and Statistics) is almost independent of the rest of the text; I have covered this material anywhere from the beginning to the end of a one-year course. It has been gratifying to hear the enthusiastic responses to the first two editions, and I hope that this third edition will prove even more useful. I want to thank many readers for helpful suggestions and I will appreciate any further comments.
If you find misprints, please send them to me at MLBoas@aol. I also want to thank the University of Washington physics students who were my LATEX typists: Toshiko Asai, Jeff Sherman, and Jeffrey Frasca. And I especially want to thank my son, Harold P. Boas, both for mathematical consultations, and for his expert help with LATEX problems.
Instructors who have adopted the book for a class should consult the publisher about an Instructor’s Answer Book, and about a list correlating 2nd and 3rd edition problem numbers for problems which appear in both editions.com TO THE STUDENT As you start each topic in this book, you will no doubt wonder and ask “Just why should I study this subject and what use does it have in applications?” There is a story about a young mathematics instructor who asked an older professor “What do you say when students ask about the practical applications of some mathematical topic?” The experienced professor said “I tell them!” This text tries to follow that advice. However, you must on your part be reasonable in your request. It is not possible in one book or course to cover both the mathematical methods and very many detailed applications of them. You will have to be content with some information as to the areas of application of each topic and some of the simpler applications.
In your later courses, you will then use these techniques in more advanced applications. At that point you can concentrate on the physical application instead of being distracted by learning new mathematical methods. One point about your study of this material cannot be emphasized too strongly: To use mathematics effectively in applications, you need not just knowledge but skill. Skill can be obtained only through practice.
You can obtain a certain superficial knowledge of mathematics by listening to lectures, but you cannot obtain skill this way.