Mathematical methods for physics and related fields

Mathematical Methods

Sadri Hassani

Mathematical Methods

For Students of Physics and Related Fields

123

Sadri Hassani

IIlinois State University

Normal, IL

USA

[email protected]

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

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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 Gr¨ubl 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.

S.H.

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. If concepts (e.g., infinite series) do not succumb

easily to physical interpretation, then I immediately subjugate the physical

x Preface

situation to the mathematical concepts (e.g., 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.e., 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.g., 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 MathematicaR , a relatively self-contained com￾panion volume that uses MathematicaR .

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 MathematicaR in the accompanying companion volume is only a

reflection of my limited familiarity with the broader field of symbolic manipu￾lations on the computers. Instructors using other symbolic algebra programs

such as MapleR and MacsymaR may generate their own examples or trans￾late the MathematicaR commands of the companion volume into their favorite

language.

Acknowledgments

I would like to thank all my PHY 217 students at Illinois State University

who gave me a considerable amount of feedback. I am grateful to Thomas

von Foerster, Executive Editor of Mathematics, Physics and Engineering at

Springer-Verlag New York, Inc., for being very patient and supportive of the

project as soon as he took over its editorship. Finally, I thank my wife,

Sarah, my son, Dane, and my daughter, Daisy, for their understanding and

support.

Unless otherwise indicated, all biographical sketches have been taken from

the following sources:

Kline, M. Mathematical Thought: From Ancient to Modern Times, Vols. 1–3,

Oxford University Press, New York, 1972.

xii Preface

History of Mathematics archive at www-groups.dcs.st-and.ac.uk:80.

Simmons, G. Calculus Gems, McGraw-Hill, New York, 1992.

Gamow, G. The Great Physicists: From Galileo to Einstein, Dover, New York,

1961.

Although extreme care was taken to correct all the misprints, it is very

unlikely that I have been able to catch all of them. I shall be most grateful to

those readers kind enough to bring to my attention any remaining mistakes,

typographical or otherwise. Please feel free to contact me.

Sadri Hassani

Department of Physics, Illinois State University, Normal, Illinois

Note to the Reader

“Why,” said the Dodo, “the best way to ex￾plain it is to do it.”

—Lewis Carroll

Probably the best advice I can give you is, if you want to learn mathematics

and physics, “Just do it!” As a first step, read the material in a chapter

carefully, tracing the logical steps leading to important results. As a (very

important) second step, make sure you can reproduce these logical steps, as

well as all the relevant examples in the chapter, with the book closed. No

amount of following other people’s logic—whether in a book or in a lecture—

can help you learn as much as a single logical step that you have taken yourself.

Finally, do as many problems at the end of each chapter as your devotion and

dedication to this subject allows!

Whether you are a physics or an engineering student, almost all the ma￾terial you learn in this book will become handy in the rest of your academic

training. Eventually, you are going to take courses in mechanics, electro￾magnetic theory, strength of materials, heat and thermodynamics, quantum

mechanics, etc. A solid background of the mathematical methods at the level

of presentation of this book will go a long way toward your deeper under￾standing of these subjects.

As you strive to grasp the (sometimes) difficult concepts, glance at the his￾torical notes to appreciate the efforts of the past mathematicians and physi￾cists as they struggled through a maze of uncharted territories in search of

the correct “path,” a path that demands courage, perseverance, self-sacrifice,

and devotion.

At the end of most chapters, you will find a short list of references that you

may want to consult for further reading. In addition to these specific refer￾ences, as a general companion, I frequently refer to my more advanced book,

Mathematical Physics: A Modern Introduction to Its Foundations, Springer￾Verlag, 1999, which is abbreviated as [Has 99]. There are many other excellent

books on the market; however, my own ignorance of their content and the par￾allelism in the pedagogy of my two books are the only reasons for singling out

[Has 99].

Contents

Preface to Second Edition vii

Preface ix

Note to the Reader xiii

I Coordinates and Calculus 1

1 Coordinate Systems and Vectors 3

1.1 Vectors in a Plane and in Space . . . . . . . . . . . . . . . . . . 3

1.1.1 Dot Product . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.2 Vector or Cross Product . . . . . . . . . . . . . . . . . . 7

1.2 Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . 11

1.3 Vectors in Different Coordinate Systems . . . . . . . . . . . . . 16

1.3.1 Fields and Potentials . . . . . . . . . . . . . . . . . . . . 21

1.3.2 Cross Product . . . . . . . . . . . . . . . . . . . . . . . 28

1.4 Relations Among Unit Vectors . . . . . . . . . . . . . . . . . . 31

1.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2 Differentiation 43

2.1 The Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.2 Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.2.1 Definition, Notation, and Basic Properties . . . . . . . . 47

2.2.2 Differentials . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.2.3 Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . 55

2.2.4 Homogeneous Functions . . . . . . . . . . . . . . . . . . 57

2.3 Elements of Length, Area, and Volume . . . . . . . . . . . . . . 59

2.3.1 Elements in a Cartesian Coordinate System . . . . . . . 60

2.3.2 Elements in a Spherical Coordinate System . . . . . . . 62

2.3.3 Elements in a Cylindrical Coordinate System . . . . . . 65

2.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

xvi CONTENTS

3 Integration: Formalism 77

3.1 “

” Means “

um” . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.2 Properties of Integral . . . . . . . . . . . . . . . . . . . . . . . . 81

3.2.1 Change of Dummy Variable . . . . . . . . . . . . . . . . 82

3.2.2 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.2.3 Interchange of Limits . . . . . . . . . . . . . . . . . . . 82

3.2.4 Partition of Range of Integration . . . . . . . . . . . . . 82

3.2.5 Transformation of Integration Variable . . . . . . . . . . 83

3.2.6 Small Region of Integration . . . . . . . . . . . . . . . . 83

3.2.7 Integral and Absolute Value . . . . . . . . . . . . . . . . 84

3.2.8 Symmetric Range of Integration . . . . . . . . . . . . . 84

3.2.9 Differentiating an Integral . . . . . . . . . . . . . . . . . 85

3.2.10 Fundamental Theorem of Calculus . . . . . . . . . . . . 87

3.3 Guidelines for Calculating Integrals . . . . . . . . . . . . . . . . 91

3.3.1 Reduction to Single Integrals . . . . . . . . . . . . . . . 92

3.3.2 Components of Integrals of Vector Functions . . . . . . 95

3.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4 Integration: Applications 101

4.1 Single Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.1.1 An Example from Mechanics . . . . . . . . . . . . . . . 101

4.1.2 Examples from Electrostatics and Gravity . . . . . . . . 104

4.1.3 Examples from Magnetostatics . . . . . . . . . . . . . . 109

4.2 Applications: Double Integrals . . . . . . . . . . . . . . . . . . 115

4.2.1 Cartesian Coordinates . . . . . . . . . . . . . . . . . . . 115

4.2.2 Cylindrical Coordinates . . . . . . . . . . . . . . . . . . 118

4.2.3 Spherical Coordinates . . . . . . . . . . . . . . . . . . . 120

4.3 Applications: Triple Integrals . . . . . . . . . . . . . . . . . . . 122

4.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5 Dirac Delta Function 139

5.1 One-Variable Case . . . . . . . . . . . . . . . . . . . . . . . . . 139

5.1.1 Linear Densities of Points . . . . . . . . . . . . . . . . . 143

5.1.2 Properties of the Delta Function . . . . . . . . . . . . . 145

5.1.3 The Step Function . . . . . . . . . . . . . . . . . . . . . 152

5.2 Two-Variable Case . . . . . . . . . . . . . . . . . . . . . . . . . 154

5.3 Three-Variable Case . . . . . . . . . . . . . . . . . . . . . . . . 159

5.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

II Algebra of Vectors 171

6 Planar and Spatial Vectors 173

6.1 Vectors in a Plane Revisited . . . . . . . . . . . . . . . . . . . . 174

6.1.1 Transformation of Components . . . . . . . . . . . . . . 176

6.1.2 Inner Product . . . . . . . . . . . . . . . . . . . . . . . . 182

CONTENTS xvii

6.1.3 Orthogonal Transformation . . . . . . . . . . . . . . . . 190

6.2 Vectors in Space . . . . . . . . . . . . . . . . . . . . . . . . . . 192

6.2.1 Transformation of Vectors . . . . . . . . . . . . . . . . . 194

6.2.2 Inner Product . . . . . . . . . . . . . . . . . . . . . . . . 198

6.3 Determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

6.4 The Jacobian . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

6.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

7 Finite-Dimensional Vector Spaces 215

7.1 Linear Transformations . . . . . . . . . . . . . . . . . . . . . . 216

7.2 Inner Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

7.3 The Determinant . . . . . . . . . . . . . . . . . . . . . . . . . . 222

7.4 Eigenvectors and Eigenvalues . . . . . . . . . . . . . . . . . . . 224

7.5 Orthogonal Polynomials . . . . . . . . . . . . . . . . . . . . . . 227

7.6 Systems of Linear Equations . . . . . . . . . . . . . . . . . . . . 230

7.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

8 Vectors in Relativity 237

8.1 Proper and Coordinate Time . . . . . . . . . . . . . . . . . . . 239

8.2 Spacetime Distance . . . . . . . . . . . . . . . . . . . . . . . . . 240

8.3 Lorentz Transformation . . . . . . . . . . . . . . . . . . . . . . 243

8.4 Four-Velocity and Four-Momentum . . . . . . . . . . . . . . . . 247

8.4.1 Relativistic Collisions . . . . . . . . . . . . . . . . . . . 250

8.4.2 Second Law of Motion . . . . . . . . . . . . . . . . . . . 253

8.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

III Infinite Series 257

9 Infinite Series 259

9.1 Infinite Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . 259

9.2 Summations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

9.2.1 Mathematical Induction . . . . . . . . . . . . . . . . . . 265

9.3 Infinite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

9.3.1 Tests for Convergence . . . . . . . . . . . . . . . . . . . 267

9.3.2 Operations on Series . . . . . . . . . . . . . . . . . . . . 273

9.4 Sequences and Series of Functions . . . . . . . . . . . . . . . . 274

9.4.1 Properties of Uniformly Convergent Series . . . . . . . . 277

9.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

10 Application of Common Series 283

10.1 Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

10.1.1 Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . 286

10.2 Series for Some Familiar Functions . . . . . . . . . . . . . . . . 287

10.3 Helmholtz Coil . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

10.4 Indeterminate Forms and L’Hˆopital’s Rule . . . . . . . . . . . . 294