Mathematical Methods Sadri Hassani Mathematical Methods For Students of Physics and Related Fields 123 Sadri Hassani IIlinois State University Normal, IL USA hassani@entropy.edu ISBN: 978-0-387-09503-5 e-ISBN: 978-0-387-09504-2 Library of Congress Control Number: 2008935523 c Springer Science+Business Media, LLC 2009 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.
Printed on acid-free paper springer.com To my wife, Sarah, and to my children, Dane Arash and Daisy Bita Preface to the Second Edition In this new edition, which is a substantially revised version of the old one, I have added five new chapters: Vectors in Relativity (Chapter 8), Tensor Analysis (Chapter 17), Integral Transforms (Chapter 29), Calculus of Varia- tions (Chapter 30), and Probability Theory (Chapter 32). The discussion of vectors in Part II, especially the introduction of the inner product, offered the opportunity to present the special theory of relativity, which unfortunately, in most undergraduate physics curricula receives little attention. While the main motivation for this chapter was vectors, I grabbed the opportunity to develop the Lorentz transformation and Minkowski distance, the bedrocks of the special theory of relativity, from first principles. The short section, Vectors and Indices, at the end of Chapter 8 of the first edition, was too short to demonstrate the importance of what the indices are really used for, tensors.
So, I expanded that short section into a somewhat comprehensive discussion of tensors. Chapter 17, Tensor Analysis, takes a fresh look at vector transformations introduced in the earlier discussion of vectors, and shows the necessity of classifying them into the covariant and contravariant categories. It then introduces tensors based on—and as a gen- eralization of—the transformation properties of covariant and contravariant vectors. In light of these transformation properties, the Kronecker delta, in- troduced earlier in the book, takes on a new look, and a natural and extremely useful generalization of it is introduced leading to the Levi-Civita symbol.
A discussion of connections and metrics motivates a four-dimensional treatment of Maxwell’s equations and a manifest unification of electric and magnetic fields. The chapter ends with Riemann curvature tensor and its place in Ein- stein’s general relativity. The Fourier series treatment alone does not do justice to the many appli- cations in which aperiodic functions are to be represented. Fourier transform is a powerful tool to represent functions in such a way that the solution to many (partial) differential equations can be obtained elegantly and succinctly.
Chapter 29, Integral Transforms, shows the power of Fourier transform in many illustrations including the calculation of Green’s functions for Laplace, heat, and wave differential operators. Laplace transforms, which are useful in solving initial-value problems, are also included. viii Preface to Second Edition The Dirac delta function, about which there is a comprehensive discussion in the book, allows a very smooth transition from multivariable calculus to the Calculus of Variations, the subject of Chapter 30. This chapter takes an intuitive approach to the subject: replace the sum by an integral and the Kronecker delta by the Dirac delta function, and you get from multivariable calculus to the calculus of variations! Well, the transition may not be as simple as this, but the heart of the intuitive approach is.
Once the transition is made and the master Euler-Lagrange equation is derived, many examples, including some with constraint (which use the Lagrange multiplier technique), and some from electromagnetism and mechanics are presented. Probability Theory is essential for quantum mechanics and thermody- namics. This is the subject of Chapter 32. Starting with the basic notion of the probability space, whose prerequisite is an understanding of elementary set theory, which is also included, the notion of random variables and its con- nection to probability is introduced, average and variance are defined, and binomial, Poisson, and normal distributions are discussed in some detail.
Aside from the above major changes, I have also incorporated some other important changes including the rearrangement of some chapters, adding new sections and subsections to some existing chapters (for instance, the dynamics of fluids in Chapter 15), correcting all the mistakes, both typographic and conceptual, to which I have been directed by many readers of the first edition, and adding more problems at the end of each chapter. Stylistically, I thought splitting the sometimes very long chapters into smaller ones and collecting the related chapters into Parts make the reading of the text smoother. I hope I was not wrong! I would like to thank the many instructors, students, and general readers who communicated to me comments, suggestions, and errors they found in the book. Among those, I especially thank Dan Holland for the many discussions we have had about the book, Rafael Benguria and Gebhard Grübl for pointing out some important historical and conceptual mistakes, and Ali Erdem and Thomas Ferguson for reading multiple chapters of the book, catching many mistakes, and suggesting ways to improve the presentation of the material.
Jerome Brozek meticulously and diligently read most of the book and found numerous errors. Although a lawyer by profession, Mr. Brozek, as a hobby, has a keen interest in mathematical physics. I thank him for this interest and for putting it to use on my book.
Last but not least, I want to thank my family, especially my wife Sarah for her unwavering support. Normal, IL January, 2008 Preface Innocent light-minded men, who think that astronomy can be learnt by looking at the stars without knowledge of math- ematics will, in the next life, be birds. —Plato, Timaeos This book is intended to help bridge the wide gap separating the level of math- ematical sophistication expected of students of introductory physics from that expected of students of advanced courses of undergraduate physics and engi- neering. While nothing beyond simple calculus is required for introductory physics courses taken by physics, engineering, and chemistry majors, the next level of courses—both in physics and engineering—already demands a readi- ness for such intricate and sophisticated concepts as divergence, curl, and Stokes’ theorem.
It is the aim of this book to make the transition between these two levels of exposure as smooth as possible. Level and Pedagogy I believe that the best pedagogy to teach mathematics to beginning students of physics and engineering (even mathematics, although some of my mathe- matical colleagues may disagree with me) is to introduce and use the concepts in a multitude of applied settings. This method is not unlike teaching a lan- guage to a child: it is by repeated usage—by the parents or the teacher—of the same word in different circumstances that a child learns the meaning of the word, and by repeated active (and sometimes wrong) usage of words that the child learns to use them in a sentence. And what better place to use the language of mathematics than in Nature itself in the context of physics.
I start with the familiar notion of, say, a derivative or an integral, but interpret it entirely in terms of physical ideas. Thus, a derivative is a means by which one obtains velocity from position vectors or acceleration from velocity vectors, and integral is a means by which one obtains the gravitational or electric field of a large number of charged or massive particles., infinite series) do not succumb easily to physical interpretation, then I immediately subjugate the physical x Preface situation to the mathematical concepts (e., multipole expansion of electric potential). Because of my belief in this pedagogy, I have kept formalism to a bare minimum. After all, a child needs no knowledge of the formalism of his or her language (i., grammar) to be able to read and write.
Similarly, a novice in physics or engineering needs to see a lot of examples in which mathematics is used to be able to “speak the language.” And I have spared no effort to provide these examples throughout the book. Of course, formalism, at some stage, becomes important. Just as grammar is taught at a higher stage of a child’s education (say, in high school), mathematical formalism is to be taught at a higher stage of education of physics and engineering students (possibly in advanced undergraduate or graduate classes). Features The unique features of this book, which set it apart from the existing text- books, are • the inseparable treatments of physical and mathematical concepts, • the large number of original illustrative examples, • the accessibility of the book to sophomores and juniors in physics and engineering programs, and • the large number of historical notes on people and ideas.
All mathematical concepts in the book are either introduced as a natural tool for expressing some physical concept or, upon their introduction, immediately used in a physical setting. Thus, for example, differential equations are not treated as some mathematical equalities seeking solutions, but rather as a statement about the laws of Nature (e., the second law of motion) whose solutions describe the behavior of a physical system. Almost all examples and problems in this book come directly from physi- cal situations in mechanics, electromagnetism, and, to a lesser extent, quan- tum mechanics and thermodynamics. Although the examples are drawn from physics, they are conceptually at such an introductory level that students of engineering and chemistry will have no difficulty benefiting from the mathe- matical discussion involved in them.
Most mathematical-methods books are written for readers with a higher level of sophistication than a sophomore or junior physics or engineering stu- dent. This book is directly and precisely targeted at sophomores and juniors, and seven years of teaching it to such an audience have proved both the need for such a book and the adequacy of its level. My experience with sophomores and juniors has shown that peppering the mathematical topics with a bit of history makes the subject more enticing. It also gives a little boost to the motivation of many students, which at times can Preface xi run very low.
The history of ideas removes the myth that all mathematical concepts are clear cut, and come into being as a finished and polished prod- uct. It reveals to the students that ideas, just like artistic masterpieces, are molded into perfection in the hands of many generations of mathematicians and physicists. Use of Computer Algebra As soon as one applies the mathematical concepts to real-world situations, one encounters the impossibility of finding a solution in “closed form.” One is thus forced to use approximations and numerical methods of calculation. Computer algebra is especially suited for many of the examples and problems in this book.
Because of the variety of the computer algebra softwares available on the market, and the diversity in the preference of one software over another among instructors, I have left any discussion of computers out of this book. Instead, all computer and numerical chapters, examples, and problems are collected in Mathematical Methods Using Mathematica R , a relatively self-contained com- R panion volume that uses Mathematica. By separating the computer-intensive topics from the text, I have made it possible for the instructor to use his or her judgment in deciding how much and in what format the use of computers should enter his or her pedagogy. The usage of Mathematica R in the accompanying companion volume is only a reflection of my limited familiarity with the broader field of symbolic manipu- lations on the computers.