Mathematical methods for students of physics and related fields~tqw~ darksiderg kho tài liệu bách khoa

Mathematical Methods


Sadri Hassani

Mathematical Methods
For Students of Physics and Related Fields

123


Sadri Hassani
IIlinois State University
Normal, IL
USA


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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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 Variations (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 generalization of—the transformation properties of covariant and contravariant
vectors. In light of these transformation properties, the Kronecker delta, introduced 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 Einstein’s general relativity.
The Fourier series treatment alone does not do justice to the many applications 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 thermodynamics. 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 connection 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 mathematics will, in the next life, be birds.
—Plato, Timaeos

This book is intended to help bridge the wide gap separating the level of mathematical sophistication expected of students of introductory physics from that

expected of students of advanced courses of undergraduate physics and engineering. 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 readiness 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 mathematical 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 language 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 textbooks, 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 physical situations in mechanics, electromagnetism, and, to a lesser extent, quantum 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 mathematical 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 student. 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
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 product. 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 companion volume that uses Mathematica R .
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 manipulations on the computers. Instructors using other symbolic algebra programs
such as Maple R and Macsyma R may generate their own examples or translate the Mathematica R 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.

xi


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 explain 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 material you learn in this book will become handy in the rest of your academic
training. Eventually, you are going to take courses in mechanics, electromagnetic 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 understanding of these subjects.
As you strive to grasp the (sometimes) difficult concepts, glance at the historical notes to appreciate the efforts of the past mathematicians and physicists 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 references, as a general companion, I frequently refer to my more advanced book,
Mathematical Physics: A Modern Introduction to Its Foundations, SpringerVerlag, 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 parallelism 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

I

xiii

Coordinates and Calculus

1 Coordinate Systems and Vectors
1.1 Vectors in a Plane and in Space . . . . .
1.1.1 Dot Product . . . . . . . . . . .
1.1.2 Vector or Cross Product . . . . .
1.2 Coordinate Systems . . . . . . . . . . .
1.3 Vectors in Different Coordinate Systems
1.3.1 Fields and Potentials . . . . . . .
1.3.2 Cross Product . . . . . . . . . .
1.4 Relations Among Unit Vectors . . . . .
1.5 Problems . . . . . . . . . . . . . . . . .

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3
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2 Differentiation
2.1 The Derivative . . . . . . . . . . . . . . . . . . . . .
2.2 Partial Derivatives . . . . . . . . . . . . . . . . . . .
2.2.1 Definition, Notation, and Basic Properties . .
2.2.2 Differentials . . . . . . . . . . . . . . . . . . .
2.2.3 Chain Rule . . . . . . . . . . . . . . . . . . .
2.2.4 Homogeneous Functions . . . . . . . . . . . .
2.3 Elements of Length, Area, and Volume . . . . . . . .
2.3.1 Elements in a Cartesian Coordinate System .
2.3.2 Elements in a Spherical Coordinate System .
2.3.3 Elements in a Cylindrical Coordinate System
2.4 Problems . . . . . . . . . . . . . . . . . . . . . . . .

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xvi

CONTENTS
3 Integration: Formalism

3.1 “ ” Means “ um” . . . . . . . . . . . . . . . . . . .
3.2 Properties of Integral . . . . . . . . . . . . . . . . . .
3.2.1 Change of Dummy Variable . . . . . . . . . .
3.2.2 Linearity . . . . . . . . . . . . . . . . . . . .
3.2.3 Interchange of Limits . . . . . . . . . . . . .
3.2.4 Partition of Range of Integration . . . . . . .
3.2.5 Transformation of Integration Variable . . . .
3.2.6 Small Region of Integration . . . . . . . . . .
3.2.7 Integral and Absolute Value . . . . . . . . . .
3.2.8 Symmetric Range of Integration . . . . . . .
3.2.9 Differentiating an Integral . . . . . . . . . . .
3.2.10 Fundamental Theorem of Calculus . . . . . .
3.3 Guidelines for Calculating Integrals . . . . . . . . . .
3.3.1 Reduction to Single Integrals . . . . . . . . .
3.3.2 Components of Integrals of Vector Functions
3.4 Problems . . . . . . . . . . . . . . . . . . . . . . . .

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77

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4 Integration: Applications
4.1 Single Integrals . . . . . . . . . . . . . . . . . . .
4.1.1 An Example from Mechanics . . . . . . .
4.1.2 Examples from Electrostatics and Gravity
4.1.3 Examples from Magnetostatics . . . . . .
4.2 Applications: Double Integrals . . . . . . . . . .
4.2.1 Cartesian Coordinates . . . . . . . . . . .
4.2.2 Cylindrical Coordinates . . . . . . . . . .
4.2.3 Spherical Coordinates . . . . . . . . . . .
4.3 Applications: Triple Integrals . . . . . . . . . . .
4.4 Problems . . . . . . . . . . . . . . . . . . . . . .

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5 Dirac Delta Function
5.1 One-Variable Case . . . . . . . . . . . .
5.1.1 Linear Densities of Points . . . .
5.1.2 Properties of the Delta Function
5.1.3 The Step Function . . . . . . . .
5.2 Two-Variable Case . . . . . . . . . . . .
5.3 Three-Variable Case . . . . . . . . . . .
5.4 Problems . . . . . . . . . . . . . . . . .

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II

Algebra of Vectors

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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

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215

216
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8 Vectors in Relativity
8.1 Proper and Coordinate Time . . .
8.2 Spacetime Distance . . . . . . . . .
8.3 Lorentz Transformation . . . . . .
8.4 Four-Velocity and Four-Momentum
8.4.1 Relativistic Collisions . . .
8.4.2 Second Law of Motion . . .
8.5 Problems . . . . . . . . . . . . . .

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250
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6.2

6.3
6.4
6.5

6.1.3 Orthogonal Transformation
Vectors in Space . . . . . . . . . .
6.2.1 Transformation of Vectors .
6.2.2 Inner Product . . . . . . . .
Determinant . . . . . . . . . . . . .
The Jacobian . . . . . . . . . . . .
Problems . . . . . . . . . . . . . .

7 Finite-Dimensional Vector Spaces
7.1 Linear Transformations . . . . .
7.2 Inner Product . . . . . . . . . . .
7.3 The Determinant . . . . . . . . .
7.4 Eigenvectors and Eigenvalues . .
7.5 Orthogonal Polynomials . . . . .
7.6 Systems of Linear Equations . . .
7.7 Problems . . . . . . . . . . . . .

III

Infinite Series

257

9 Infinite Series
9.1 Infinite Sequences . . . . . . . . . . . . . . . . . .
9.2 Summations . . . . . . . . . . . . . . . . . . . . .
9.2.1 Mathematical Induction . . . . . . . . . .
9.3 Infinite Series . . . . . . . . . . . . . . . . . . . .
9.3.1 Tests for Convergence . . . . . . . . . . .
9.3.2 Operations on Series . . . . . . . . . . . .
9.4 Sequences and Series of Functions . . . . . . . .
9.4.1 Properties of Uniformly Convergent Series
9.5 Problems . . . . . . . . . . . . . . . . . . . . . .

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279

10 Application of Common Series
10.1 Power Series . . . . . . . . . . . . . .
10.1.1 Taylor Series . . . . . . . . .
10.2 Series for Some Familiar Functions .
10.3 Helmholtz Coil . . . . . . . . . . . .
10.4 Indeterminate Forms and L’Hˆ
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xviii

CONTENTS
10.5
10.6
10.7
10.8
10.9

Multipole Expansion . . . . . . . . .
Fourier Series . . . . . . . . . . . . .

Multivariable Taylor Series . . . . .
Application to Differential Equations
Problems . . . . . . . . . . . . . . .

11 Integrals and Series as Functions
11.1 Integrals as Functions . . . . . .
11.1.1 Gamma Function . . . . .
11.1.2 The Beta Function . . . .
11.1.3 The Error Function . . .
11.1.4 Elliptic Functions . . . . .
11.2 Power Series as Functions . . . .
11.2.1 Hypergeometric Functions
11.2.2 Confluent Hypergeometric
11.2.3 Bessel Functions . . . . .
11.3 Problems . . . . . . . . . . . . .

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311

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318
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322
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327
328
332
333
336

Analysis of Vectors

341

12 Vectors and Derivatives
12.1 Solid Angle . . . . . . . . . . . . . . . . . . . . . . .
12.1.1 Ordinary Angle Revisited . . . . . . . . . . .
12.1.2 Solid Angle . . . . . . . . . . . . . . . . . . .
12.2 Time Derivative of Vectors . . . . . . . . . . . . . .
12.2.1 Equations of Motion in a Central Force Field
12.3 The Gradient . . . . . . . . . . . . . . . . . . . . . .
12.3.1 Gradient and Extremum Problems . . . . . .

12.4 Problems . . . . . . . . . . . . . . . . . . . . . . . .

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343
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347

350
352
355
359
362

13 Flux and Divergence
13.1 Flux of a Vector Field . . . . . . . . . . . .
13.1.1 Flux Through an Arbitrary Surface
13.2 Flux Density = Divergence . . . . . . . . .
13.2.1 Flux Density . . . . . . . . . . . . .
13.2.2 Divergence Theorem . . . . . . . . .
13.2.3 Continuity Equation . . . . . . . . .
13.3 Problems . . . . . . . . . . . . . . . . . . .

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. 404

14 Line Integral and Curl
14.1 The Line Integral . . . . . . . . . . . . . . .
14.2 Curl of a Vector Field and Stokes’ Theorem
14.3 Conservative Vector Fields . . . . . . . . . .
14.4 Problems . . . . . . . . . . . . . . . . . . .

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CONTENTS

xix

15 Applied Vector Analysis

15.1 Double Del Operations . . . . . . . . . .
15.2 Magnetic Multipoles . . . . . . . . . . .
15.3 Laplacian . . . . . . . . . . . . . . . . .
15.3.1 A Primer of Fluid Dynamics . .
15.4 Maxwell’s Equations . . . . . . . . . . .
15.4.1 Maxwell’s Contribution . . . . .
15.4.2 Electromagnetic Waves in Empty
15.5 Problems . . . . . . . . . . . . . . . . .
16 Curvilinear Vector Analysis
16.1 Elements of Length . . . . .
16.2 The Gradient . . . . . . . .
16.3 The Divergence . . . . . . .
16.4 The Curl . . . . . . . . . .
16.4.1 The Laplacian . . .
16.5 Problems . . . . . . . . . .

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17 Tensor Analysis
17.1 Vectors and Indices . . . . . . . . . . . . . . . . . . .
17.1.1 Transformation Properties of Vectors . . . . .
17.1.2 Covariant and Contravariant Vectors . . . . .
17.2 From Vectors to Tensors . . . . . . . . . . . . . . . .
17.2.1 Algebraic Properties of Tensors . . . . . . . .
17.2.2 Numerical Tensors . . . . . . . . . . . . . . .
17.3 Metric Tensor . . . . . . . . . . . . . . . . . . . . . .
17.3.1 Index Raising and Lowering . . . . . . . . . .
17.3.2 Tensors and Electrodynamics . . . . . . . . .
17.4 Differentiation of Tensors . . . . . . . . . . . . . . .
17.4.1 Covariant Differential and Affine Connection
17.4.2 Covariant Derivative . . . . . . . . . . . . . .
17.4.3 Metric Connection . . . . . . . . . . . . . . .
17.5 Riemann Curvature Tensor . . . . . . . . . . . . . .
17.6 Problems . . . . . . . . . . . . . . . . . . . . . . . .

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Complex Analysis

18 Complex Arithmetic
18.1 Cartesian Form of Complex Numbers .
18.2 Polar Form of Complex Numbers . . .
18.3 Fourier Series Revisited . . . . . . . .

18.4 A Representation of Delta Function .
18.5 Problems . . . . . . . . . . . . . . . .

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417
420

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477
477
482
488
491
493


xx

CONTENTS
19 Complex Derivative and Integral
19.1 Complex Functions . . . . . . . . . . . . .

19.1.1 Derivatives of Complex Functions .
19.1.2 Integration of Complex Functions .
19.1.3 Cauchy Integral Formula . . . . .
19.1.4 Derivatives as Integrals . . . . . .
19.2 Problems . . . . . . . . . . . . . . . . . .

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497
497
499
503
508
509
511

20 Complex Series
515
20.1 Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516
20.2 Taylor and Laurent Series . . . . . . . . . . . . . . . . . . . . . 518
20.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522
21 Calculus of Residues
21.1 The Residue . . . . . . . . . . . . . . . . . . . . . .
21.2 Integrals of Rational Functions . . . . . . . . . . .
21.3 Products of Rational and Trigonometric Functions
21.4 Functions of Trigonometric Functions . . . . . . .
21.5 Problems . . . . . . . . . . . . . . . . . . . . . . .

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Differential Equations

22 From PDEs to ODEs
22.1 Separation of Variables . . . . . . . . .
22.2 Separation in Cartesian Coordinates .
22.3 Separation in Cylindrical Coordinates
22.4 Separation in Spherical Coordinates .
22.5 Problems . . . . . . . . . . . . . . . .

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534
536

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548
550

23 First-Order Differential Equations
23.1 Normal Form of a FODE . . . . . . . . .
23.2 Integrating Factors . . . . . . . . . . . . .
23.3 First-Order Linear Differential Equations
23.4 Problems . . . . . . . . . . . . . . . . . .

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24 Second-Order Linear Differential Equations
24.1 Linearity, Superposition, and Uniqueness . . .

24.2 The Wronskian . . . . . . . . . . . . . . . . .
24.3 A Second Solution to the HSOLDE . . . . . .
24.4 The General Solution to an ISOLDE . . . . .
24.5 Sturm–Liouville Theory . . . . . . . . . . . .
24.5.1 Adjoint Differential Operators . . . .
24.5.2 Sturm–Liouville System . . . . . . . .
24.6 SOLDEs with Constant Coefficients . . . . .
24.6.1 The Homogeneous Case . . . . . . . .
24.6.2 Central Force Problem . . . . . . . . .

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CONTENTS

xxi

24.6.3 The Inhomogeneous Case . . . . . . . . . . . . . . . . . 583
24.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587
25 Laplace’s Equation: Cartesian Coordinates
591
25.1 Uniqueness of Solutions . . . . . . . . . . . . . . . . . . . . . . 592
25.2 Cartesian Coordinates . . . . . . . . . . . . . . . . . . . . . . . 594
25.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603

26 Laplace’s Equation: Spherical Coordinates
26.1 Frobenius Method . . . . . . . . . . . . . .
26.2 Legendre Polynomials . . . . . . . . . . . .
26.3 Second Solution of the Legendre DE . . . .
26.4 Complete Solution . . . . . . . . . . . . . .
26.5 Properties of Legendre Polynomials . . . . .
26.5.1 Parity . . . . . . . . . . . . . . . . .
26.5.2 Recurrence Relation . . . . . . . . .
26.5.3 Orthogonality . . . . . . . . . . . . .
26.5.4 Rodrigues Formula . . . . . . . . . .
26.6 Expansions in Legendre Polynomials . . . .
26.7 Physical Examples . . . . . . . . . . . . . .
26.8 Problems . . . . . . . . . . . . . . . . . . .

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27 Laplace’s Equation: Cylindrical Coordinates
27.1 The ODEs . . . . . . . . . . . . . . . . . . . .
27.2 Solutions of the Bessel DE . . . . . . . . . . .

27.3 Second Solution of the Bessel DE . . . . . . .
27.4 Properties of the Bessel Functions . . . . . .
27.4.1 Negative Integer Order . . . . . . . . .
27.4.2 Recurrence Relations . . . . . . . . . .
27.4.3 Orthogonality . . . . . . . . . . . . . .
27.4.4 Generating Function . . . . . . . . . .
27.5 Expansions in Bessel Functions . . . . . . . .
27.6 Physical Examples . . . . . . . . . . . . . . .
27.7 Problems . . . . . . . . . . . . . . . . . . . .

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. 657

28 Other PDEs of Mathematical Physics
28.1 The Heat Equation . . . . . . . . . . . . . . . .
28.1.1 Heat-Conducting Rod . . . . . . . . . .
28.1.2 Heat Conduction in a Rectangular Plate
28.1.3 Heat Conduction in a Circular Plate . .
28.2 The Schr¨
odinger Equation . . . . . . . . . . . .
28.2.1 Quantum Harmonic Oscillator . . . . .
28.2.2 Quantum Particle in a Box . . . . . . .
28.2.3 Hydrogen Atom . . . . . . . . . . . . .
28.3 The Wave Equation . . . . . . . . . . . . . . .
28.3.1 Guided Waves . . . . . . . . . . . . . .

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661
661
662

663
664
666
667
675
677
680
682


xxii

CONTENTS
28.3.2 Vibrating Membrane . . . . . . . . . . . . . . . . . . . . 686
28.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687

VII

Special Topics

691

29 Integral Transforms
29.1 The Fourier Transform . . . . . . . . . . . . . . . . . . .
29.1.1 Properties of Fourier Transform . . . . . . . . . .
29.1.2 Sine and Cosine Transforms . . . . . . . . . . . .
29.1.3 Examples of Fourier Transform . . . . . . . . . .
29.1.4 Application to Differential Equations . . . . . . .
29.2 Fourier Transform and Green’s Functions . . . . . . . .
29.2.1 Green’s Function for the Laplacian . . . . . . . .

29.2.2 Green’s Function for the Heat Equation . . . . .
29.2.3 Green’s Function for the Wave Equation . . . . .
29.3 The Laplace Transform . . . . . . . . . . . . . . . . . .
29.3.1 Properties of Laplace Transform . . . . . . . . .
29.3.2 Derivative and Integral of the Laplace Transform
29.3.3 Laplace Transform and Differential Equations . .
29.3.4 Inverse of Laplace Transform . . . . . . . . . . .
29.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . .

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693
693
696
697
698
702
705
708
709
711
712
713
717
718
721
723

30 Calculus of Variations
30.1 Variational Problem . . . . . . . . . . . . . .
30.1.1 Euler-Lagrange Equation . . . . . . .
30.1.2 Beltrami identity . . . . . . . . . . . .
30.1.3 Several Dependent Variables . . . . .
30.1.4 Several Independent Variables . . . . .
30.1.5 Second Variation . . . . . . . . . . . .
30.1.6 Variational Problems with Constraints
30.2 Lagrangian Dynamics . . . . . . . . . . . . .
30.2.1 From Newton to Lagrange . . . . . . .
30.2.2 Lagrangian Densities . . . . . . . . . .
30.3 Hamiltonian Dynamics . . . . . . . . . . . . .
30.4 Problems . . . . . . . . . . . . . . . . . . . .

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727
728
729
731
734
734

735
738
740
740
744
747
750

31 Nonlinear Dynamics and Chaos
31.1 Systems Obeying Iterated Maps . . . . . .
31.1.1 Stable and Unstable Fixed Points .
31.1.2 Bifurcation . . . . . . . . . . . . .
31.1.3 Onset of Chaos . . . . . . . . . . .
31.2 Systems Obeying DEs . . . . . . . . . . .
31.2.1 The Phase Space . . . . . . . . . .
31.2.2 Autonomous Systems . . . . . . .
31.2.3 Onset of Chaos . . . . . . . . . . .

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753
754
755
757
761
763
764
766
770

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CONTENTS
31.3 Universality of Chaos . . . . .
31.3.1 Feigenbaum Numbers
31.3.2 Fractal Dimension . .
31.4 Problems . . . . . . . . . . .

xxiii
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773

773
775
778

32 Probability Theory
32.1 Basic Concepts . . . . . . . . . . . . . . . . . . . .
32.1.1 A Set Theory Primer . . . . . . . . . . . . .
32.1.2 Sample Space and Probability . . . . . . . .
32.1.3 Conditional and Marginal Probabilities . .
32.1.4 Average and Standard Deviation . . . . . .
32.1.5 Counting: Permutations and Combinations
32.2 Binomial Probability Distribution . . . . . . . . . .
32.3 Poisson Distribution . . . . . . . . . . . . . . . . .
32.4 Continuous Random Variable . . . . . . . . . . . .
32.4.1 Transformation of Variables . . . . . . . . .
32.4.2 Normal Distribution . . . . . . . . . . . . .
32.5 Problems . . . . . . . . . . . . . . . . . . . . . . .

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781
781
782
784
786
789
791
792
797
801
804

806
809

Bibliography

815

Index

817


Part I

Coordinates and Calculus


Chapter 1

Coordinate Systems
and Vectors
Coordinates and vectors—in one form or another—are two of the most
fundamental concepts in any discussion of mathematics as applied to physical problems. So, it is beneficial to start our study with these two concepts.
Both vectors and coordinates have generalizations that cover a wide variety of physical situations including not only ordinary three-dimensional space
with its ordinary vectors, but also the four-dimensional spacetime of relativity
with its so-called four vectors, and even the infinite-dimensional spaces used
in quantum physics with their vectors of infinite components. Our aim in this
chapter is to review the ordinary space and how it is used to describe physical
phenomena. To facilitate this discussion, we first give an outline of some of
the properties of vectors.

1.1

Vectors in a Plane and in Space

We start with the most common definition of a vector as a directed line
segment without regard to where the vector is located. In other words, a vector
is a directed line segment whose only important attributes are its direction
and its length. As long as we do not change these two attributes, the vector is
not affected. Thus, we are allowed to move a vector parallel to itself without
changing the vector. Examples of vectors1 are position r, displacement Δr,
velocity v, momentum p, electric field E, and magnetic field B. The vector
that has no length is called the zero vector and is denoted by 0.
Vectors would be useless unless we could perform some kind of operation
on them. The most basic operation is changing the length of a vector. This
is accomplished by multiplying the vector by a real positive number. For
example, 3.2r is a vector in the same direction as r but 3.2 times longer. We
1 Vectors

will be denoted by Roman letters printed in boldface type.

general properties
of vectors


4

Coordinate Systems and Vectors

Δr

1

A

a+b
b+a

C

b

r2

ΔR

=Δ

r1

b

a

a

Δr
2

B

+Δ

b

(a)

a

(b)

Figure 1.1: Illustration of the commutative law of addition of vectors.

operations on
vectors

vector form of the
parametric
equation of a line

can flip the direction of a vector by multiplying it by −1. That is, (−1) × r =
−r is a vector having the same length as r but pointing in the opposite
direction. We can combine these two operations and think of multiplying a
vector by any real (positive or negative) number. The result is another vector
lying along the same line as the original vector. Thus, −0.732r is a vector
that is 0.732 times as long as r and points in the opposite direction. The zero
vector is obtained every time one multiplies any vector by the number zero.
Another operation is the addition of two vectors. This operation, with
which we assume the reader to have some familiarity, is inspired by the obvious
addition law for displacements. In Figure 1.1(a), a displacement, Δr1 from
A to B is added to the displacement Δr2 from B to C to give ΔR their

resultant, or their sum, i.e., the displacement from A to C: Δr1 + Δr2 = ΔR.
Figure 1.1(b) shows that addition of vectors is commutative: a + b = b + a.
It is also associative, a + (b + c) = (a + b) + c, i.e., the order in which you
add vectors is irrelevant. It is clear that a + 0 = 0 + a = a for any vector a.
Example 1.1.1. The parametric equation of a line through two given points
can be obtained in vector form by noting that any point in space defines a vector
whose components are the coordinates of the given point.2 If the components of
the points P and Q in Figure 1.2 are, respectively, (px , py , pz ) and (qx , qy , qz ), then
we can define vectors p and q with those components. An arbitrary point X with
components (x, y, z) will lie on the line P Q if and only if the vector x = (x, y, z)
has its tip on that line. This will happen if and only if the vector joining P and X,
namely x − p, is proportional to the vector joining P and Q, namely q − p. Thus,
for some real number t, we must have
x − p = t(q − p)

or

x = t(q − p) + p.

This is the vector form of the equation of a line. We can write it in component
form by noting that the equality of vectors implies the equality of corresponding
components. Thus,
x = (qx − px )t + px ,
y = (qy − py )t + py ,
z = (qz − pz )t + pz ,
which is the usual parametric equation for a line.
2 We shall discuss components and coordinates in greater detail later in this chapter. For
now, the knowledge gained in calculus is sufficient for our discussion.

1.1 Vectors in a Plane and in Space

5

z
Q

X
x

P
q

p

O

y

x

Figure 1.2: The parametric equation of a line in space can be obtained easily using
vectors.

There are some special vectors that are extremely useful in describing
physical quantities. These are the unit vectors. If one divides a vector
by its length, one gets a unit vector in the direction of the original vector.
ˆ with a subscript which
Unit vectors are generally denoted by the symbol e
designates its direction. Thus, if we divided the vector a by its length |a| we

ˆa in the direction of a. Turning this definition around,
get the unit vector e
we have

use of unit vectors

Box 1.1.1. If we know the magnitude |a| of a vector quantity as well as
ˆa , we can construct the vector: a = |a|ˆ
its direction e
ea .
This construction will be used often in the sequel.
The most commonly used unit vectors are those in the direction of coorˆx , e
ˆy , and e
ˆz are the unit vectors pointing in the positive
dinate axes. Thus e
directions of the x-, y-, and z-axes, respectively.3 We shall introduce unit
vectors in other coordinate systems when we discuss those coordinate systems
later in this chapter.

1.1.1

unit vectors along
the x-, y-, and
z-axes

Dot Product

The reader is no doubt familiar with the concept of dot product whereby
two vectors are “multiplied” and the result is a number. The dot product of
a and b is defined by

a · b ≡ |a| |b| cos θ,
(1.1)
where |a| is the length of a, |b| is the length of b, and θ is the angle between
the two vectors. This definition is motivated by many physical situations.
3 These unit vectors are usually denoted by i, j, and k, a notation that can be confusing
when other non-Cartesian coordinates are used. We shall not use this notation, but adhere
to the more suggestive notation introduced above.

dot product
defined