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Sadri Hassani
Mathematical Methods
For Students of Physics and Related Fields
1 3

Sadri Hassani
IIlinois State University
Normal, IL
USA
hassani@entropy.phy.ilstu.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.
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 Mathematica
R
, a relatively self-contained com-
panion 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 manipu-
lations on the computers. Instructors using other symbolic algebra programs
such as Maple
R
and Macsyma
R
may generate their own examples or trans-
late 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.

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’Hôpital’s Rule . . . . . . . . . . . . 294

xviii CONTENTS
10.5 Multipole Expansion . . . . . . . . . . . . . . . . . . . . . . . . 297
10.6 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
10.7 Multivariable Taylor Series . . . . . . . . . . . . . . . . . . . . 305
10.8 Application to Diﬀerential Equations . . . . . . . . . . . . . . . 307
10.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
11 Integrals and Series as Functions 317
11.1 Integrals as Functions . . . . . . . . . . . . . . . . . . . . . . . 317
11.1.1 Gamma Function . . . . . . . . . . . . . . . . . . . . . . 318
11.1.2 The Beta Function . . . . . . . . . . . . . . . . . . . . . 320
11.1.3 The Error Function . . . . . . . . . . . . . . . . . . . . 322
11.1.4 Elliptic Functions . . . . . . . . . . . . . . . . . . . . . . 322
11.2 Power Series as Functions . . . . . . . . . . . . . . . . . . . . . 327
11.2.1 Hypergeometric Functions . . . . . . . . . . . . . . . . . 328
11.2.2 Conﬂuent Hypergeometric Functions . . . . . . . . . . . 332
11.2.3 Bessel Functions . . . . . . . . . . . . . . . . . . . . . . 333
11.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336
IV Analysis of Vectors 341
12 Vectors and Derivatives 343
12.1 Solid Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344
12.1.1 Ordinary Angle Revisited . . . . . . . . . . . . . . . . . 344
12.1.2 Solid Angle . . . . . . . . . . . . . . . . . . . . . . . . . 347
12.2 Time Derivative of Vectors . . . . . . . . . . . . . . . . . . . . 350
12.2.1 Equations of Motion in a Central Force Field . . . . . . 352
12.3 The Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
12.3.1 Gradient and Extremum Problems . . . . . . . . . . . . 359
12.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
13 Flux and Divergence 365
13.1 Flux of a Vector Field . . . . . . . . . . . . . . . . . . . . . . . 365
13.1.1 Flux Through an Arbitrary Surface . . . . . . . . . . . 370
13.2 Flux Density = Divergence . . . . . . . . . . . . . . . . . . . . 371
13.2.1 Flux Density . . . . . . . . . . . . . . . . . . . . . . . . 371
13.2.2 Divergence Theorem . . . . . . . . . . . . . . . . . . . . 374
13.2.3 Continuity Equation . . . . . . . . . . . . . . . . . . . . 378
13.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383
14 Line Integral and Curl 387
14.1 The Line Integral . . . . . . . . . . . . . . . . . . . . . . . . . . 387
14.2 Curl of a Vector Field and Stokes’ Theorem . . . . . . . . . . . 391
14.3 Conservative Vector Fields . . . . . . . . . . . . . . . . . . . . . 398
14.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404

CONTENTS xix
15 Applied Vector Analysis 407
15.1 Double Del Operations . . . . . . . . . . . . . . . . . . . . . . . 407
15.2 Magnetic Multipoles . . . . . . . . . . . . . . . . . . . . . . . . 409
15.3 Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411
15.3.1 A Primer of Fluid Dynamics . . . . . . . . . . . . . . . 413
15.4 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . 415
15.4.1 Maxwell’s Contribution . . . . . . . . . . . . . . . . . . 416
15.4.2 Electromagnetic Waves in Empty Space . . . . . . . . . 417
15.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420
16 Curvilinear Vector Analysis 423
16.1 Elements of Length . . . . . . . . . . . . . . . . . . . . . . . . . 423
16.2 The Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425
16.3 The Divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 427
16.4 The Curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431
16.4.1 The Laplacian . . . . . . . . . . . . . . . . . . . . . . . 435
16.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436
17 Tensor Analysis 439
17.1 Vectors and Indices . . . . . . . . . . . . . . . . . . . . . . . . . 439
17.1.1 Transformation Properties of Vectors . . . . . . . . . . . 441
17.1.2 Covariant and Contravariant Vectors . . . . . . . . . . . 445
17.2 From Vectors to Tensors . . . . . . . . . . . . . . . . . . . . . . 447
17.2.1 Algebraic Properties of Tensors . . . . . . . . . . . . . . 450
17.2.2 Numerical Tensors . . . . . . . . . . . . . . . . . . . . . 452
17.3 Metric Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454
17.3.1 Index Raising and Lowering . . . . . . . . . . . . . . . . 457
17.3.2 Tensors and Electrodynamics . . . . . . . . . . . . . . . 459
17.4 Differentiation of Tensors . . . . . . . . . . . . . . . . . . . . . 462
17.4.1 Covariant Differential and Affine Connection . . . . . . 462
17.4.2 Covariant Derivative . . . . . . . . . . . . . . . . . . . . 464
17.4.3 Metric Connection . . . . . . . . . . . . . . . . . . . . . 465
17.5 Riemann Curvature Tensor . . . . . . . . . . . . . . . . . . . . 468
17.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471
V Complex Analysis 475
18 Complex Arithmetic 477
18.1 Cartesian Form of Complex Numbers . . . . . . . . . . . . . . . 477
18.2 Polar Form of Complex Numbers . . . . . . . . . . . . . . . . . 482
18.3 Fourier Series Revisited . . . . . . . . . . . . . . . . . . . . . . 488
18.4 A Representation of Delta Function . . . . . . . . . . . . . . . 491
18.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493

xx CONTENTS
19 Complex Derivative and Integral 497
19.1 Complex Functions . . . . . . . . . . . . . . . . . . . . . . . . . 497
19.1.1 Derivatives of Complex Functions . . . . . . . . . . . . . 499
19.1.2 Integration of Complex Functions . . . . . . . . . . . . . 503
19.1.3 Cauchy Integral Formula . . . . . . . . . . . . . . . . . 508
19.1.4 Derivatives as Integrals . . . . . . . . . . . . . . . . . . 509
19.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 525
21.1 The Residue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525
21.2 Integrals of Rational Functions . . . . . . . . . . . . . . . . . . 529
21.3 Products of Rational and Trigonometric Functions . . . . . . . 532
21.4 Functions of Trigonometric Functions . . . . . . . . . . . . . . 534
21.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536
VI Differential Equations 539
22 From PDEs to ODEs 541
22.1 Separation of Variables . . . . . . . . . . . . . . . . . . . . . . . 542
22.2 Separation in Cartesian Coordinates . . . . . . . . . . . . . . . 544
22.3 Separation in Cylindrical Coordinates . . . . . . . . . . . . . . 547
22.4 Separation in Spherical Coordinates . . . . . . . . . . . . . . . 548
22.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550
23 First-Order Differential Equations 551
23.1 Normal Form of a FODE . . . . . . . . . . . . . . . . . . . . . 551
23.2 Integrating Factors . . . . . . . . . . . . . . . . . . . . . . . . . 553
23.3 First-Order Linear Differential Equations . . . . . . . . . . . . 556
23.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561
24 Second-Order Linear Differential Equations 563
24.1 Linearity, Superposition, and Uniqueness . . . . . . . . . . . . . 564
24.2 The Wronskian . . . . . . . . . . . . . . . . . . . . . . . . . . . 566
24.3 A Second Solution to the HSOLDE . . . . . . . . . . . . . . . . 567
24.4 The General Solution to an ISOLDE . . . . . . . . . . . . . . . 569
24.5 Sturm–Liouville Theory . . . . . . . . . . . . . . . . . . . . . . 570
24.5.1 Adjoint Differential Operators . . . . . . . . . . . . . . 571
24.5.2 Sturm–Liouville System . . . . . . . . . . . . . . . . . . 574
24.6 SOLDEs with Constant Coefficients . . . . . . . . . . . . . . . 575
24.6.1 The Homogeneous Case . . . . . . . . . . . . . . . . . . 576
24.6.2 Central Force Problem . . . . . . . . . . . . . . . . . . . 579

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 607
26.1 Frobenius Method . . . . . . . . . . . . . . . . . . . . . . . . . 608
26.2 Legendre Polynomials . . . . . . . . . . . . . . . . . . . . . . . 610
26.3 Second Solution of the Legendre DE . . . . . . . . . . . . . . . 617
26.4 Complete Solution . . . . . . . . . . . . . . . . . . . . . . . . . 619
26.5 Properties of Legendre Polynomials . . . . . . . . . . . . . . . . 622
26.5.1 Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622
26.5.2 Recurrence Relation . . . . . . . . . . . . . . . . . . . . 622
26.5.3 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . 624
26.5.4 Rodrigues Formula . . . . . . . . . . . . . . . . . . . . . 626
26.6 Expansions in Legendre Polynomials . . . . . . . . . . . . . . . 628
26.7 Physical Examples . . . . . . . . . . . . . . . . . . . . . . . . . 631
26.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635
27 Laplace’s Equation: Cylindrical Coordinates 639
27.1 The ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639
27.2 Solutions of the Bessel DE . . . . . . . . . . . . . . . . . . . . . 642
27.3 Second Solution of the Bessel DE . . . . . . . . . . . . . . . . . 645
27.4 Properties of the Bessel Functions . . . . . . . . . . . . . . . . 646
27.4.1 Negative Integer Order . . . . . . . . . . . . . . . . . . . 646
27.4.2 Recurrence Relations . . . . . . . . . . . . . . . . . . . . 646
27.4.3 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . 647
27.4.4 Generating Function . . . . . . . . . . . . . . . . . . . . 649
27.5 Expansions in Bessel Functions . . . . . . . . . . . . . . . . . . 653
27.6 Physical Examples . . . . . . . . . . . . . . . . . . . . . . . . . 654
27.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657
28 Other PDEs of Mathematical Physics 661
28.1 The Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . . 661
28.1.1 Heat-Conducting Rod . . . . . . . . . . . . . . . . . . . 662
28.1.2 Heat Conduction in a Rectangular Plate . . . . . . . . . 663
28.1.3 Heat Conduction in a Circular Plate . . . . . . . . . . . 664
28.2 The Schrödinger Equation . . . . . . . . . . . . . . . . . . . . . 666
28.2.1 Quantum Harmonic Oscillator . . . . . . . . . . . . . . 667
28.2.2 Quantum Particle in a Box . . . . . . . . . . . . . . . . 675
28.2.3 Hydrogen Atom . . . . . . . . . . . . . . . . . . . . . . 677
28.3 The Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . 680
28.3.1 Guided Waves . . . . . . . . . . . . . . . . . . . . . . . 682

xxii CONTENTS
28.3.2 Vibrating Membrane . . . . . . . . . . . . . . . . . . . . 686
28.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687
VII Special Topics 691
29 Integral Transforms 693
29.1 The Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . 693
29.1.1 Properties of Fourier Transform . . . . . . . . . . . . . . 696
29.1.2 Sine and Cosine Transforms . . . . . . . . . . . . . . . . 697
29.1.3 Examples of Fourier Transform . . . . . . . . . . . . . . 698
29.1.4 Application to Diﬀerential Equations . . . . . . . . . . . 702
29.2 Fourier Transform and Green’s Functions . . . . . . . . . . . . 705
29.2.1 Green’s Function for the Laplacian . . . . . . . . . . . . 708
29.2.2 Green’s Function for the Heat Equation . . . . . . . . . 709
29.2.3 Green’s Function for the Wave Equation . . . . . . . . . 711
29.3 The Laplace Transform . . . . . . . . . . . . . . . . . . . . . . 712
29.3.1 Properties of Laplace Transform . . . . . . . . . . . . . 713
29.3.2 Derivative and Integral of the Laplace Transform . . . . 717
29.3.3 Laplace Transform and Diﬀerential Equations . . . . . . 718
29.3.4 Inverse of Laplace Transform . . . . . . . . . . . . . . . 721
29.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723
30 Calculus of Variations 727
30.1 Variational Problem . . . . . . . . . . . . . . . . . . . . . . . . 728
30.1.1 Euler-Lagrange Equation . . . . . . . . . . . . . . . . . 729
30.1.2 Beltrami identity . . . . . . . . . . . . . . . . . . . . . . 731
30.1.3 Several Dependent Variables . . . . . . . . . . . . . . . 734
30.1.4 Several Independent Variables . . . . . . . . . . . . . . . 734
30.1.5 Second Variation . . . . . . . . . . . . . . . . . . . . . . 735
30.1.6 Variational Problems with Constraints . . . . . . . . . . 738
30.2 Lagrangian Dynamics . . . . . . . . . . . . . . . . . . . . . . . 740
30.2.1 From Newton to Lagrange . . . . . . . . . . . . . . . . . 740
30.2.2 Lagrangian Densities . . . . . . . . . . . . . . . . . . . . 744
30.3 Hamiltonian Dynamics . . . . . . . . . . . . . . . . . . . . . . . 747
30.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 750
31 Nonlinear Dynamics and Chaos 753
31.1 Systems Obeying Iterated Maps . . . . . . . . . . . . . . . . . . 754
31.1.1 Stable and Unstable Fixed Points . . . . . . . . . . . . . 755
31.1.2 Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . 757
31.1.3 Onset of Chaos . . . . . . . . . . . . . . . . . . . . . . . 761
31.2 Systems Obeying DEs . . . . . . . . . . . . . . . . . . . . . . . 763
31.2.1 The Phase Space . . . . . . . . . . . . . . . . . . . . . . 764
31.2.2 Autonomous Systems . . . . . . . . . . . . . . . . . . . 766
31.2.3 Onset of Chaos . . . . . . . . . . . . . . . . . . . . . . . 770

CONTENTS xxiii
31.3 Universality of Chaos . . . . . . . . . . . . . . . . . . . . . . . . 773
31.3.1 Feigenbaum Numbers . . . . . . . . . . . . . . . . . . . 773
31.3.2 Fractal Dimension . . . . . . . . . . . . . . . . . . . . . 775
31.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 778
32 Probability Theory 781
32.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . 781
32.1.1 A Set Theory Primer . . . . . . . . . . . . . . . . . . . . 782
32.1.2 Sample Space and Probability . . . . . . . . . . . . . . . 784
32.1.3 Conditional and Marginal Probabilities . . . . . . . . . 786
32.1.4 Average and Standard Deviation . . . . . . . . . . . . . 789
32.1.5 Counting: Permutations and Combinations . . . . . . . 791
32.2 Binomial Probability Distribution . . . . . . . . . . . . . . . . . 792
32.3 Poisson Distribution . . . . . . . . . . . . . . . . . . . . . . . . 797
32.4 Continuous Random Variable . . . . . . . . . . . . . . . . . . . 801
32.4.1 Transformation of Variables . . . . . . . . . . . . . . . . 804
32.4.2 Normal Distribution . . . . . . . . . . . . . . . . . . . . 806
32.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 physi-
cal problems. So, it is beneficial to start our study with these two concepts.
Both vectors and coordinates have generalizations that cover a wide vari-
ety 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 general properties
of vectors
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
1Vectors will be denoted by Roman letters printed in boldface type.

4 Coordinate Systems and Vectors
a
b
a
a
b
b
b + a
a + b
Δ
r
1
Δr2
ΔR
=Δr1
+ Δr 2
(a) (b)
A
C
B
Figure 1.1: Illustration of the commutative law of addition of vectors.
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
operations on
vectors 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 PQ 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
vector form of the
parametric
equation of a line
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.
2We 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
O
P
Q
X
x
y
z
p
q
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 use of unit vectors
by its length, one gets a unit vector in the direction of the original vector.
Unit vectors are generally denoted by the symbol ê with a subscript which
designates its direction. Thus, if we divided the vector a by its length |a| we
get the unit vector êa in the direction of a. Turning this definition around,
we have
Box 1.1.1. If we know the magnitude |a| of a vector quantity as well as
its direction êa, we can construct the vector: a = |a|êa.
This construction will be used often in the sequel.
The most commonly used unit vectors are those in the direction of coor- unit vectors along
the x-, y-, and
z-axes
dinate axes. Thus êx, êy, and êz are the unit vectors pointing in the positive
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 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 dot product
defined
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.
3These 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.

N
v
F
Figure 1.3: No work is done by a force orthogonal to displacement. If such a work
were not zero, it would have to be positive or negative; but no consistent rule exists to
assign a sign to the work.
The prime example is work which is defined as the scalar product of force and
displacement. The presence of cos θ ensures the requirement that the work
done by a force perpendicular to the displacement is zero. If this requirement
were not met, we would have the precarious situation of Figure 1.3 in which
the two vertical forces add up to zero but the total work done by them is
not zero! This is because it would be impossible to assign a “sign” to the
work done by forces being displaced perpendicular to themselves, and make
the rule of such an assignment in such a way that the work of F in the figure
cancels that of N. (The reader is urged to try to come up with a rule—e.g.,
assigning a positive sign to the work if the velocity points to the right of the
observer and a negative sign if it points to the observer’s left—and see that it
will not work, no matter how elaborate it may be!) The only logical definition
of work is that which includes a cos θ factor.
The dot product is clearly commutative, a · b = b · a. Moreover, it dis-
properties of dot
product tributes over vector addition
(a + b) · c = a · c + b · c.
To see this, note that Equation (1.1) can be interpreted as the product of the
length of a with the projection of b along a. Now Figure 1.4 demonstrates4
that the projection of a + b along c is the sum of the projections of a and b
along c (see Problem 1.2 for details). The third property of the inner product
is that a · a is always a positive number unless a is the zero vector in which
case a · a = 0. In mathematics, the collection of these three properties—
properties defining
the dot (inner)
product
commutativity, positivity, and distribution over addition—defines a dot (or
inner) product on a vector space.
The definition of the dot product leads directly to a · a = |a|2
or
|a| =
√
a · a, (1.2)
which is useful in calculating the length of sums or differences of vectors.
4Figure 1.4 appears to prove the distributive property only for vectors lying in the same
plane. However, the argument will be valid even if the three vectors are not coplanar.
Instead of dropping perpendicular lines from the tips of a and b, one drops perpendicular
planes.

A
B
O
a
b
a+b
c
Proj. of a
Proj. of b
Figure 1.4: The distributive property of the dot product is clearly demonstrated if we
interpret the dot product as the length of one vector times the projection of the other
vector on the first.
One can use the distributive property of the dot product to show that
if (ax, ay, az) and (bx, by, bz) represent the components of a and b along the
axes x, y, and z, then dot product in
terms of
components
a · b = axbx + ayby + azbz. (1.3)
From the definition of the dot product, we can draw an important conclu-
sion. If we divide both sides of a · b = |a| |b| cos θ by |a|, we get
a · b
|a|
= |b| cosθ or

a
|a|

· b = |b| cos θ ⇒ êa · b = |b| cos θ.
Noting that |b| cos θ is simply the projection of b along a, we conclude
a useful relation to
be used frequently
in the sequel
Box 1.1.2. To find the perpendicular projection of a vector b along
another vector a, take the dot product of b with êa, the unit vector along a.
Sometimes “component” is used for perpendicular projection. This is not
entirely correct. For any set of three mutually perpendicular unit vectors in
space, Box 1.1.2 can be used to find the components of a vector along the
three unit vectors. Only if the unit vectors are mutually perpendicular do
components and projections coincide.
1.1.2 Vector or Cross Product
Given two space vectors, a and b, we can find a third space vector c, called
the cross product of a and b, and denoted by c = a × b. The magnitude cross product of
two space vectors
of c is defined by |c| = |a| |b| sin θ where θ is the angle between a and b.
The direction of c is given by the right-hand rule: If a is turned to b (note
right-hand rule
explained
the order in which a and b appear here) through the angle between a and b,

a (right-handed) screw that is perpendicular to a and b will advance in the
direction of a × b. This definition implies that
a × b = −b × a.
This property is described by saying that the cross product is antisymmet-
cross product is
antisymmetric ric. The definition also implies that
a · (a × b) = b · (a × b) = 0.
That is, a × b is perpendicular to both a and b.5
The vector product has the following properties:
a × (αb) = (αa) × b = α(a × b), a × b = −b × a,
a × (b + c) = a × b + a × c, a × a = 0. (1.4)
Using these properties, we can write the vector product of two vectors in terms
of their components. We are interested in a more general result valid in other
coordinate systems as well. So, rather than using x, y, and z as subscripts for
unit vectors, we use the numbers 1, 2, and 3. In that case, our results can
cross product in
terms of
components
also be used for spherical and cylindrical coordinates which we shall discuss
shortly.
a × b = (α1ê1 + α2ê2 + α3ê3) × (β1ê1 + β2ê2 + β3ê3)
= α1β1ê1 × ê1 + α1β2ê1 × ê2 + α1β3ê1 × ê3
+ α2β1ê2 × ê1 + α2β2ê2 × ê2 + α2β3ê2 × ê3
+ α3β1ê3 × ê1 + α3β2ê3 × ê2 + α3β3ê3 × ê3.
But, by the last property of Equation (1.4), we have
ê1 × ê1 = ê2 × ê2 = ê3 × ê3 = 0.
Also, if we assume that ê1, ê2, and ê3 form a so-called right-handed set,
i.e., if
right-handed set
of unit vectors
ê1 × ê2 = −ê2 × ê1 = ê3,
ê1 × ê3 = −ê3 × ê1 = −ê2, (1.5)
ê2 × ê3 = −ê3 × ê2 = ê1,
then we obtain
a × b = (α2β3 − α3β2)ê1 + (α3β1 − α1β3)ê2 + (α1β2 − α2β1)ê3
5This fact makes it clear why a × b is not defined in the plane. Although it is possible
to define a × b for vectors a and b lying in a plane, a × b will not lie in that plane (it
will be perpendicular to that plane). For the vector product, a and b (although lying in a
plane) must be considered as space vectors.

e1
e2 e3
α1 α2 α3
β1 β2 β3
det
e1 e2 e3
α1 α2 α3
β1 β2 β3
e1 e2 e3
α1 α2
α3
β1 β2 β3
=
Figure 1.5: A 3 × 3 determinant is obtained by writing the entries twice as shown,
multiplying all terms on each slanted line and adding the results. The lines from upper
left to lower right bear a positive sign, and those from upper right to lower left a negative
sign.
which can be nicely written in a determinant form6
cross product in
terms of the
determinant of
components
a × b = det
⎛
⎝
ê1 ê2 ê3
α1 α2 α3
β1 β2 β3
⎞
⎠ . (1.6)
Figure 1.5 explains the rule for “expanding” a determinant.
Example 1.1.2. From the definition of the vector product and Figure 1.6(a),
we note that area of a
parallelogram in
terms of cross
product of its two
sides
|a × b| = area of the parallelogram defined by a and b.
So we can use Equation (1.6) to find the area of a parallelogram defined by two
vectors directly in terms of their components. For instance, the area defined by
a = (1, 1, −2) and b = (2, 0, 3) can be found by calculating their vector product
a × b = det
⎛
⎝
ê1 ê2 ê3
1 1 −2
2 0 3
⎞
⎠ = 3ê1 − 7ê2 − 2ê3,
and then computing its length
|a × b| = 32 + (−7)2 + (−2)2 =
√
62.
a
b
c
θ θ
|a| cos θ
a
b
|a| sin θ
θ
b × c
(a) (b)
Figure 1.6: (a) The area of a parallelogram is the absolute value of the cross product of
the two vectors describing its sides. (b) The volume of a parallelepiped can be obtained
by mixing the dot and the cross products.
6No knowledge of determinants is necessary at this point. The reader may consider (1.6)
to be a mnemonic device useful for remembering the components of a × b.

Example 1.1.3. The volume of a parallelepiped defined by three non-coplanar
vectors, a, b, and c, is given by |a · (b × c)|. This can be seen from Figure 1.6(b),
where it is clear that
volume of a
parallelepiped as a
combination of
dot and cross
products
volume = (area of base)(altitude) = |b × c|(|a| cos θ) = |(b × c) · a|.
The absolute value is taken to ensure the positivity of the area. In terms of compo-
nents we have
volume = |(b × c)1α1 + (b × c)2α2 + (b × c)3α3|
= |(β2γ3 − β3γ2)α1 + (β3γ1 − β1γ3)α2 + (β1γ2 − β2γ1)α3|,
which can be written in determinant form as
volume of a
parallelepiped as
the determinant of
the components of
its side vectors
volume = |a · (b × c)| = det
⎛
⎝
α1 α2 α3
β1 β2 β3
γ1 γ2 γ3
⎞
⎠ .
Note how we have put the absolute value sign around the determinant of the matrix,
so that the area comes out positive.
Historical Notes
The concept of vectors as directed line segments that could represent velocities,
forces, or accelerations has a very long history. Aristotle knew that the effect of two
forces acting on an object could be described by a single force using what is now
called the parallelogram law. However, the real development of the concept took an
unexpected turn in the nineteenth century.
With the advent of complex numbers and the realization by Gauss, Wessel, and
especially Argand, that they could be represented by points in a plane, mathemati-
cians discovered that complex numbers could be used to study vectors in a plane.
A complex number is represented by a pair7
of real numbers—called the real and
imaginary parts of the complex number—which could be considered as the two
components of a planar vector.
This connection between vectors in a plane and complex numbers was well es-
tablished by 1830. Vectors are, however, useful only if they are treated as objects
in space. After all, velocities, forces, and accelerations are mostly three-dimensional
objects. So, the two-dimensional complex numbers had to be generalized to three
dimensions. This meant inventing ways of adding, subtracting, multiplying, and
dividing objects such as (x, y, z).
The invention of a spatial analogue of the planar complex numbers is due to
William R. Hamilton. Next to Newton, Hamilton is the greatest of all English
William R.
Hamilton
1805–1865
mathematicians, and like Newton he was even greater as a physicist than as a
mathematician. At the age of five Hamilton could read Latin, Greek, and Hebrew.
At eight he added Italian and French; at ten he could read Arabic and Sanskrit,
and at fourteen, Persian. A contact with a lightning calculator inspired him to
study mathematics. In 1822 at the age of seventeen and a year before he entered
Trinity College in Dublin, he prepared a paper on caustics which was read before the
Royal Irish Academy in 1824 but not published. Hamilton was advised to rework
and expand it. In 1827 he submitted to the Academy a revision which initiated the
science of geometrical optics and introduced new techniques in analytical mechanics.
7See Chapter 18.

1.2 Coordinate Systems 11
In 1827, while still an undergraduate, he was appointed Professor of Astronomy
at Trinity College in which capacity he had to manage the astronomical observations
and teach science. He did not do much of the former, but he was a fine lecturer.
Hamilton had very good intuition, and knew how to use analogy to reason from
the known to the unknown. Although he lacked great flashes of insight, he worked
very hard and very long on special problems to see what generalizations they would
lead to. He was patient and systematic in working on specific problems and was
willing to go through detailed and laborious calculations to check or prove a point.
After mastering and clarifying the concept of complex numbers and their relation
to planar vectors (see Problem 18.11 for the connection between complex multiplica-
tion on the one hand, and dot and cross products on the other), Hamilton was able
to think more clearly about the three-dimensional generalization. His efforts led
unfortunately to frustration because the vectors (a) required four components, and
(b) defied commutativity! Both features were revolutionary and set the standard
for algebra. He called these new numbers quaternions.
In retrospect, one can see that the new three-dimensional complex numbers had
to contain four components. Each “number,” when acting on a vector, rotates the
latter about an axis and stretches (or contracts) it. Two angles are required to
specify the axis of rotation, one angle to specify the amount of rotation, and a
fourth number to specify the amount of stretch (or contraction).
Hamilton announced the invention of quaternions in 1843 at a meeting of the
Royal Irish Academy, and spent the rest of his life developing the subject.
1.2 Coordinate Systems
Coordinates are “functions” that specify points of a space. The smallest
number of these functions necessary to specify a point is called the dimension
of that space. For instance, a point of a plane is specified by two numbers, and
as the point moves in the plane the two numbers change, i.e., the coordinates
are functions of the position of the point. If we designate the point as P, we
may write the coordinate functions of P as (f(P), g(P)).8
Each pair of such coordinate
systems as
functions.
functions is called a coordinate system.
There are two coordinate systems used for a plane, Cartesian, denoted
by (x(P), y(P)), and polar, denoted by (r(P), θ(P)). As shown in Figure 1.7,
P
y(P)
x(P)
O
P
O
θ(P)
r(P)
Figure 1.7: Cartesian and polar coordinates of a point P in two dimensions.
8Think of f (or g) as a rule by which a unique number is assigned to each point P .

the “function” x is defined as giving the distance from P to the vertical axis,
while θ is the function which gives the angle that the line OP makes with a
given fiducial (usually horizontal) line. The origin O and the fiducial line are
completely arbitrary. Similarly, the functions r and y give distances from the
origin and to the horizontal axis, respectively.
Box 1.2.1. In practice, one drops the argument P and writes (x, y) and
(r, θ).
We can generalize the above concepts to three dimensions. There are three
coordinate functions now. So for a point P in space we write
the three common
coordinate
systems:
Cartesian,
cylindrical and
spherical
(f(P), g(P), h(P)),
where f, g, and h are functions on the three-dimensional space. There are
three widely used coordinate systems, Cartesian (x(P), y(P), z(P)), cylin-
drical (ρ(P), ϕ(P), z(P)), and spherical (r(P), θ(P), ϕ(P)). ϕ(P) is called
the azimuth or the azimuthal angle of P, while θ(P) is called its polar
angle. To find the spherical coordinates of P, one chooses an arbitrary point
as the origin O and an arbitrary line through O called the polar axis. One
measures OP and calls it r(P); θ(P) is the angle between OP and the polar
axis. To find the third coordinate, we construct the plane through O and per-
pendicular to the polar axis, drop a projection from P to the plane meeting
the latter at H, draw an arbitrary fiducial line through O in this plane, and
measure the angle between this line and OH. This angle is ϕ(P). Cartesian
and cylindrical coordinate systems can be described similarly. The three co-
ordinate systems are shown in Figure 1.8. As indicated in the figure, the polar
axis is usually taken to be the z-axis, and the fiducial line from which ϕ(P)
is measured is chosen to be the x-axis. Although there are other coordinate
systems, the three mentioned above are by far the most widely used.
x
y
z
x(P)
y(P)
z(P)
P
(a) (b)
x
y
z
P
z(P)
(P) H
ρ (P)
ϕ
(c)
x
y
z
P
H
(P)
r (P)
θ (P)
ϕ
Figure 1.8: (a) Cartesian, (b) cylindrical, and (c) spherical coordinates of a point P in
three dimensions.

Which one of the three systems of coordinates to use in a given physi-
cal problem is dictated mainly by the geometry of that problem. As a rule,
spherical coordinates are best suited for spheres and spherically symmetric
problems. Spherical symmetry describes situations in which quantities of in-
terest are functions only of the distance from a ﬁxed point and not on the
orientation of that distance. Similarly, cylindrical coordinates ease calcula-
tions when cylinders or cylindrical symmetries are involved. Finally, Cartesian
coordinates are used in rectangular geometries.
Of the three coordinate systems, Cartesian is the most complete in the
following sense: A point in space can have only one triplet as its coordinates.
This property is not shared by the other two systems. For example, a point limitations of
non-Cartesian
coordinates
P located on the z-axis of a cylindrical coordinate system does not have a
well-deﬁned ϕ(P). In practice, such imperfections are not of dire consequence
and we shall ignore them.
Once we have three coordinate systems to work with, we need to know
how to translate from one to another. First we give the transformation rule
from spherical to cylindrical. It is clear from Figure 1.9 that transformation
from spherical to
cylindrical
coordinates
ρ = r sin θ, ϕcyl = ϕsph, z = r cos θ. (1.7)
Thus, given (r, θ, ϕ) of a point P, we can obtain (ρ, ϕ, z) of the same point by
substituting in the RHS.
Next we give the transformation rule from cylindrical to Cartesian. Again transformation
from cylindrical to
Cartesian
coordinates
Figure 1.9 gives the result:
x = ρ cosϕ, y = ρ sin ϕ, zcar = zcyl. (1.8)
We can combine (1.7) and (1.8) to connect Cartesian and spherical coordi- transformation
from spherical to
Cartesian
coordinates
nates:
x = r sin θ cos ϕ, y = r sin θ sin ϕ, z = r cos θ. (1.9)
x
y
z
P
θ
r
ρ
ρ
ϕ
Figure 1.9: The relation between the cylindrical and spherical coordinates of a point
P can be obtained using this diagram.

Box 1.2.2. Equations (1.7)–(1.9) are extremely important and worth be-
ing committed to memory. The reader is advised to study Figure 1.9
carefully and learn to reproduce (1.7)–(1.9) from the figure!
The transformations given are in their standard form. We can turn them
around and give the inverse transformations. For instance, squaring the first
and third equations of (1.7) and adding gives ρ2
+ z2
= r2
or r = ρ2 + z2.
Similarly, dividing the first and third equation yields tan θ = ρ/z, which
implies that θ = tan−1
(ρ/z), or equivalently,
z
r
= cos θ ⇒ θ = cos−1 z
r
= cos−1 z
ρ2 + z2

.
Thus, the inverse of (1.7) is
transformation
from cylindrical to
spherical
coordinates
r = ρ2 + z2, θ = tan−1 ρ
z
= cos−1 z
ρ2 + z2

, ϕsph = ϕcyl.
(1.10)
Similarly, the inverse of (1.8) is
ρ = x2 + y2,
ϕ = tan−1 y
x
= cos−1 x
x2 + y2

= sin−1 y
x2 + y2

, (1.11)
zcyl = zcar,
and that of (1.9) is
transformation
from Cartesian to
spherical
coordinates
r = x2 + y2 + z2,
θ = tan−1 x2 + y2
z

= cos−1 z
x2 + y2 + z2

= sin−1 x2 + y2
x2 + y2 + z2

, (1.12)
ϕ = tan−1 y
x
= cos−1 x
x2 + y2
= sin−1 y
x2 + y2
.
An important question concerns the range of these quantities. In other
words: In what range should we allow these quantities to vary in order to cover
the whole space? For Cartesian coordinates all three variables vary between
−∞ and +∞. Thus,
range of
coordinate
variables −∞ x +∞, −∞ y +∞, −∞ z +∞.
The ranges of cylindrical coordinates are
0 ≤ ρ ∞, 0 ≤ ϕ ≤ 2π, −∞ z ∞.

Note that ρ, being a distance, cannot have negative values.9
Similarly, the
ranges of spherical coordinates are
0 ≤ r ∞, 0 ≤ θ ≤ π, 0 ≤ ϕ ≤ 2π.
Again, r is never negative for similar reasons as above. Also note that the
range of θ excludes values larger than π. This is because the range of ϕ takes
care of points where θ “appears” to have been increased by π.
Historical Notes
One of the greatest achievements in the development of mathematics since Euclid
was the introduction of coordinates. Two men take credit for this development: Fer-
mat and Descartes. These two great French mathematicians were interested in the
unification of geometry and algebra, which resulted in the creation of a most fruitful
branch of mathematics now called analytic geometry. Fermat and Descartes who
were heavily involved in physics, were keenly aware of both the need for quantitative
methods and the capacity of algebra to deliver that method.
Fermat’s interest in the unification of geometry and algebra arose because of his
involvement in optics. His interest in the attainment of maxima and minima—thus
Pierre de Fermat
1601–1665
his contribution to calculus—stemmed from the investigation of the passage of light
rays through media of different indices of refraction, which resulted in Fermat’s
principle in optics and the law of refraction. With the introduction of coordinates,
Fermat was able to quantify the study of optics and set a trend to which all physicists
of posterity would adhere. It is safe to say that without analytic geometry the
progress of science, and in particular physics, would have been next to impossible.
Born into a family of tradespeople, Pierre de Fermat was trained as a lawyer
and made his living in this profession becoming a councillor of the parliament of
the city of Toulouse. Although mathematics was but a hobby for him and he could
devote only spare time to it, he made great contributions to number theory, to
calculus, and, together with Pascal, initiated work on probability theory.
The coordinate system introduced by Fermat was not a convenient one. For one
thing, the coordinate axes were not at right angles to one another. Furthermore,
the use of negative coordinates was not considered. Nevertheless, he was able to
translate geometric curves into algebraic equations.
René Descartes was a great philosopher, a founder of modern biology, and a
superb physicist and mathematician. His interest in mathematics stemmed from his
desire to understand nature. He wrote:
. . . I have resolved to quit only abstract geometry, that is to say, the
consideration of questions which serve only to exercise the mind, and
this, in order to study another kind of geometry, which has for its object
the explanation of the phenomena of nature.
His father, a relatively wealthy lawyer, sent him to a Jesuit school at the age
René Descartes
1596–1650
of eight where, due to his delicate health, he was allowed to spend the mornings in
bed, during which time he worked. He followed this habit during his entire life. At
twenty he graduated from the University of Poitier as a lawyer and went to Paris
where he studied mathematics with a Jesuit priest. After one year he decided to
9In some calculus books ρ is allowed to have negative values to account for points on the
opposite side of the origin. However, in physics literature ρ is assumed to be positive.To go
to “the other side” of the origin along ρ, we change ϕ by π, keeping ρ positive at all times.

join the army of Prince Maurice of Orange in 1617. During the next nine years he
vacillated between various armies while studying mathematics.
He eventually returned to Paris, where he devoted his efforts to the study of
optical instruments motivated by the newly discovered power of the telescope. In
1628 he moved to Holland to a quieter and freer intellectual environment. There he
lived for the next twenty years and wrote his famous works. In 1649 Queen Christina
of Sweden persuaded Descartes to go to Stockholm as her private tutor. However
the Queen had an uncompromising desire to draw curves and tangents at 5 a.m.,
causing Descartes to break the lifelong habit of getting up at 11 o’clock! After only
a few months in the cold northern climate, walking to the palace for the 5 o’clock
appointment with the queen, he died of pneumonia in 1650.
Descartes described his algebraic approach to geometry in his monumental work
La Géométrie. It is in this work that he solves geometrical problems using algebra
by introducing coordinates. These coordinates, as in Fermat’s case, were not lengths
along perpendicular axes. Nevertheless they paved the way for the later generations
of scientists such as Newton to build on Descartes’ and Fermat’s ideas and improve
on them.
Throughout the seventeenth century, mathematicians used one axis with the y
values drawn at an oblique or right angle onto that axis. Newton, however, in a book
Newton uses polar
coordinates for the
first time
called The Method of Fluxions and Infinite Series written in 1671, and translated
much later into English in 1736, describes a coordinate system in which points are
located in reference to a fixed point and a fixed line through that point. This was
the first introduction of essentially the polar coordinates we use today.
1.3 Vectors in Different Coordinate Systems
Many physical situations require the study of vectors in different coordinate
systems. For example, the study of the solar system is best done in spherical
coordinates because of the nature of the gravitational force. Similarly calcu-
lation of electromagnetic fields in a cylindrical cavity will be easier if we use
cylindrical coordinates. This requires not only writing functions in terms of
these coordinate variables, but also expressing vectors in terms of unit vectors
suitable for these coordinate systems. It turns out that, for the three coordi-
nate systems described above, the most natural construction of such vectors
renders them mutually perpendicular.
Any set of three (two) mutually perpendicular unit vectors in space (in the
plane) is called an orthonormal basis.10
Basis vectors have the property
orthonormal basis
that any vector can be written in terms of them.
Let us start with the plane in which the coordinate system could be Carte-
sian or polar. In general, we construct an orthonormal basis at a point and
note that
10The word “orthonormal” comes from orthogonal meaning “perpendicular,” and normal
meaning “of unit length.”

1.3 Vectors in Different Coordinate Systems 17
P
Q
ex
^
ey
^
ex
^
ey
^
(a)
P
Q
er
^
er
^
eθ
^
(b)
eθ
^
Figure 1.10: The unit vectors in (a) Cartesian coordinates and (b) polar coordinates.
The unit vectors at P and Q are the same for Cartesian coordinates, but different in
polar coordinates.
Box 1.3.1. The orthonormal basis, generally speaking, depends on the
point at which it is constructed.
The vectors of a basis are constructed as follows. To find the unit vector
corresponding to a coordinate at a point P, hold the other coordinate fixed
and increase the coordinate in question. The initial direction of motion of P
is the direction of the unit vector sought. Thus, we obtain the Cartesian unit
vectors at point P of Figure 1.10(a): êx is obtained by holding y fixed and
letting x vary in the increasing direction; and êy is obtained by holding x fixed
at P and letting y increase. In each case, the unit vectors show the initial
direction of the motion of P. It should be clear that one obtains the same set general rule for
constructing a
basis at a point
of unit vectors regardless of the location of P. However, the reader should
take note that this is true only for coordinates that are defined in terms of
axes whose directions are fixed, such as Cartesian coordinates.
If we use polar coordinates for P, then holding θ fixed at P gives the
direction of êr as shown in Figure 1.10(b), because for fixed θ, that is the
direction of increase for r. Similarly, if r is fixed at P, the initial direction
of motion of P when θ is increased is that of êθ shown in the figure. If we
choose another point such as Q shown in the figure, then a new set of unit
vectors will be obtained which are different form those of P. This is because
polar coordinates are not defined in terms of any fixed axes.
Since {êx, êy} and {êr, êθ} form a basis in the plane, any vector a in the
plane can be expressed in terms of either basis as shown in Figure 1.11. Thus,
we can write
a = axP êxP + ayP êyP = arP êrP + aθP êθP = arQ êrQ + aθQ êθQ , (1.13)
where the coordinates are subscripted to emphasize their dependence on the
points at which the unit vectors are erected. In the case of Cartesian coor-
dinates, this, of course, is not necessary because the unit vectors happen to
be independent of the point. In the case of polar coordinates, although this

P
Q
ex
^
ey
^
ex
^
ey
^
a
a
P
Q
er
^
er
^
eθ
^
eθ
^
a
a
(a) (b)
Figure 1.11: (a) The vector a has the same components along unit vectors at P and Q
in Cartesian coordinates. (b) The vector a has different components along unit vectors
at different points for a polar coordinate system.
dependence exists, we normally do not write the points as subscripts, being
aware of this dependence every time we use polar coordinates.
So far we have used parentheses to designate the (components of) a vector.
angle brackets
denote vector
components
Since, parentheses—as a universal notation—are used for coordinates of points,
we shall write components of a vector in angle brackets. So Equation (1.13)
can also be written as
a = ax, ayP = ar, aθP = ar, aθQ,
where again the subscript indicating the point at which the unit vectors are
defined is normally deleted. However, we need to keep in mind that although
ax, ay is independent of the point in question, ar, aθ is very much point-
dependent. Caution should be exercised when using this notation as to the
location of the unit vectors.
The unit vectors in the coordinate systems of space are defined the same
way. We follow the rule given before:
Box 1.3.2. (Rule for Finding Coordinate Unit Vectors). To find
the unit vector corresponding to a coordinate at a point P, hold the other
coordinates fixed and increase the coordinate in question. The initial di-
rection of motion of P is the direction of the unit vector sought.
It should be clear that the Cartesian basis {êx, êy, êz} is the same for all
points, and usually they are drawn at the origin along the three axes. An
arbitrary vector a can be written as
a = axêx + ayêy + azêz or a = ax, ay, az, (1.14)
where we used angle brackets to denote components of the vector, reserving
the parentheses for coordinates of points in space.

x
y
z
O
P(ρ, ϕ, z)
ϕ
eϕ
^
eρ
^
ez
^
z
ρ
Figure 1.12: Unit vectors of cylindrical coordinates.
The unit vectors at a point P in the other coordinate systems are obtained
similarly. In cylindrical coordinates, êρ lies along and points in the direction
of increasing ρ at P; êϕ is perpendicular to the plane formed by P and the
z-axis and points in the direction of increasing ϕ; êz points in the direction of
positive z (see Figure 1.12). We note that only êz is independent of the point
at which the unit vectors are defined because z is a fixed axis in cylindrical
coordinates. Given any vector a, we can write it as
a = aρêρ + aϕêϕ + azêz or a = aρ, aϕ, az. (1.15)
The unit vectors in spherical coordinates are defined similarly: êr is taken
along r and points in the direction of increasing r; this direction is called radial direction
radial; êθ is taken to lie in the plane formed by P and the z-axis, is per-
pendicular to r, and points in the direction of increasing θ; êϕ is as in the
cylindrical case (Figure 1.13). An arbitrary vector in space can be expressed
in terms of the spherical unit vectors at P:
a = arêr + aθêθ + aϕêϕ or a = ar, aθ, aϕ. (1.16)
It should be emphasized that
Box 1.3.3. The cylindrical and spherical unit vectors êρ, êr, êθ, and êϕ
are dependent on the position of P.
Once an origin O is designated, every point P in space will define a vector,
called a position vector and denoted by r. This is simply the vector drawn position vector
from O to P. In Cartesian coordinates this vector has components x, y, z,
thus one can write
r = xêx + yêy + zêz. (1.17)

er
^
eϕ
^
eθ
^
x
y
z
O
r
θ
P(r, θ, ϕ)
ϕ
Figure 1.13: Unit vectors of spherical coordinates. Note that the intersection of the
shaded plane with the xy-plane is a line along the cylindrical coordinate ρ.
But (x, y, z) are also the coordinates of the point P. This can be a source of
diﬀerence between
coordinates and
components
explained
confusion when other coordinate systems are used. For example, in spherical
coordinates, the components of the vector r at P are r, 0, 0 because r has
only a component along êr and none along êθ or êϕ. One writes11
r = rêr. (1.18)
However, the coordinates of P are still (r, θ, ϕ)! Similarly, the coordinates of
P are (ρ, ϕ, z) in a cylindrical system, while
r = ρ êρ + zêz, (1.19)
because r lies in the ρz-plane and has no component along êϕ. Therefore,
Box 1.3.4. Make a clear distinction between the components of the
vector r and the coordinates of the point P.
A common symptom of confusing components with coordinates is as fol-
lows. Point P1 has position vector r1 with spherical components r1, 0, 0
at P1. The position vector of a second point P2 is r2 with spherical compo-
nents r2, 0, 0 at P2. It is easy to fall into the trap of thinking that r1 − r2
has spherical components r1 − r2, 0, 0! This is, of course, not true, because
the spherical unit vectors at P1 are completely diﬀerent from those at P2,
and, therefore, contrary to the Cartesian case, we cannot simply subtract
components.
11We should really label everything with P . But, as usual, we assume this labeling to be
implied.

One of the great advantages of vectors is their ability to express results Physical laws
ought to be
coordinate
independent!
independent of any specific coordinate systems. Physical laws are always
coordinate-independent. For example, when we write F = ma both F and a
could be expressed in terms of Cartesian, spherical, cylindrical, or any other
convenient coordinate system. This independence allows us the freedom to
choose the coordinate systems most convenient for the problem at hand. For
example, it is extremely difficult to solve the planetary motions in Cartesian
coordinates, while the use of spherical coordinates facilitates the solution of
the problem tremendously.
Example 1.3.1. We can express the coordinates of the center of mass (CM) of center of mass
a collection of particles in terms of their position vectors.12
Thus, if r denotes the
position vector of the CM of the collection of N mass points, m1, m2, . . . , mN with
respective position vectors r1, r2, . . . , rN relative to an origin O, then13
r =
m1r1 + m2r2 + · · · + mN rN
m1 + m2 + · · · + mN
=
N
k=1 mkrk
M
, (1.20)
where M =
N
k=1 mk is the total mass of the system. One can also think of Equation
(1.20) as a vector equation. To find the component equations in a coordinate system,
one needs to pick a fixed point (say the origin), a set of unit vectors at that point
(usually the unit vectors along the axes of some coordinate system), and substitute
the components of rk along those unit vectors to find the components of r along the
unit vectors.
1.3.1 Fields and Potentials
The distributive property of the dot product and the fact that the unit vectors
of the bases in all coordinate systems are mutually perpendicular can be used
to derive the following: dot product in
terms of
components in the
three coordinate
systems
a · b = axbx + ayby + azbz (Cartesian),
a · b = aρbρ + aϕbϕ + azbz (cylindrical), (1.21)
a · b = arbr + aθbθ + aϕbϕ (spherical).
The first of these equations is the same as (1.3 ).
It is important to keep in mind that the components are to be expressed
in the same set of unit vectors. This typically means setting up mutually per-
pendicular unit vectors (an orthonormal basis) at a single point and resolving
all vectors along those unit vectors.
The dot product, in various forms and guises, has many applications in
physics. As pointed out earlier, it was introduced in the definition of work,
but soon spread to many other concepts of physics. One of the simplest—and
most important—applications is its use in writing the laws of physics in a
coordinate-independent way.
12This implies that the equation is most useful only when Cartesian coordinates are
used, because only for these coordinates do the components of the position vector of a
point coincide with the coordinates of that point.
13We assume that the reader is familiar with the symbol

simply as a summation
symbol. We shall discuss its properties and ways of manipulating it in Chapter 9.

q
q'
er
^
r
(x, y, z)
x
y
z
Figure 1.14: The diagram illustrating the electrical force when one charge is at the
origin.
Example 1.3.2. A point charge q is situated at the origin. A second charge q
is
located at (x, y, z) as shown in Figure 1.14. We want to express the electric force
on q
in Cartesian, spherical, and cylindrical coordinate systems.
We know that the electric force, as given by Coulomb’s law, lies along the line
joining the two charges and is either attractive or repulsive according to the signs
of q and q
. All of this information can be summarized in the formula
Coulomb’s law
Fq =
keqq
r2
êr (1.22)
where ke = 1/(4π0) ≈ 9 × 109
in SI units. Note that if q and q
are unlike, qq
0
and Fq is opposite to êr, i.e., it is attractive. On the other hand, if q and q
are of
the same sign, qq
0 and Fq is in the same direction as êr, i.e., repulsive.
Equation (1.22) expresses Fq in spherical coordinates. Thus, its components in
terms of unit vectors at q
are

keqq
/r2
, 0, 0

. To get the components in the other
coordinate systems, we rewrite (1.22). Noting that êr = r/r, we write
Fq =
keqq
r2
r
r
=
keqq
r3
r. (1.23)
For Cartesian coordinates we use (1.12) to obtain r3
= (x2
+y2
+z2
)3/2
. Substituting
this and (1.17) in (1.23) yields
Fq =
keqq
(x2 + y2 + z2)3/2
(xêx + yêy + zêz).
Therefore, the components of Fq in Cartesian coordinates are

keqq
x
(x2 + y2 + z2)3/2
,
keqq
y
(x2 + y2 + z2)3/2
,
keqq
z
(x2 + y2 + z2)3/2

.
Finally, using (1.10) and (1.19) in (1.23), we obtain
Fq =
keqq
(ρ2 + z2)3/2
(ρ êρ + zêz).

x
y
z P1
P2
r1
r2 −r1
r2
Figure 1.15: The displacement vector between P1 and P2 is the diﬀerence between
their position vectors.
Thus the components of Fq along the cylindrical unit vectors constructed at the
location of q
are
keqq
ρ
(ρ2 + z2)3/2
, 0,
keqq
z
(ρ2 + z2)3/2

.

Since r gives the position of a point in space, one can use it to write
the distance between two points P1 and P2 with position vectors r1 and r2.
Figure 1.15 shows that r2 − r1 is the displacement vector from P1 to P2. The
importance of this vector stems from the fact that many physical quantities
are functions of distances between point particles, and r2 −r1 is a concise way
of expressing this distance. The following example illustrates this.
Historical Notes
During the second half of the eighteenth century many physicists were engaged in a
quantitative study of electricity and magnetism. Charles Augustin de Coulomb,
who developed the so-called torsion balance for measuring weak forces, is credited
with the discovery of the law governing the force between electrical charges.
Coulomb was an army engineer in the West Indies. After spending nine years
there, due to his poor health, he returned to France about the same time that the
French Revolution began, at which time he retired to the country to do scientiﬁc
Charles Coulomb
1736–1806
research.
Beside his experiments on electricity, Coulomb worked on applied mechanics,
structural analysis, the fracture of beams and columns, the thrust of arches, and the
thrust of the soil.
At about the same time that Coulomb discovered the law of electricity, there
lived in England a very reclusive character named Henry Cavendish. He was
born into the nobility, had no close friends, was afraid of women, and disinterested
in music or arts of any kind. His life revolved around experiments in physics and
chemistry that he carried out in a private laboratory located in his large mansion.
During his long life he published only a handful of relatively unimportant pa-
pers. But after his death about one million pounds sterling were found in his bank
Henry Cavendish
1731–1810
account and twenty bundles of notes in his laboratory. These notes remained in
the possession of his relatives for a long time, but when they were published one

hundred years later, it became clear that Henry Cavendish was one of the greatest
experimental physicists ever. He discovered all the laws of electric and magnetic
interactions at the same time as Coulomb, and his work in chemistry matches that
of Lavoisier. Furthermore, he used a torsion balance to measure the universal grav-
itational constant for the first time, and as a result was able to arrive at the exact
mass of the Earth.
Example 1.3.3. Coulomb’s law for two arbitrary charges
Suppose there are point charges q1 at P1 and q2 at P2. Let us write the force exerted
on q2 by q1. The magnitude of the force is
F21 =
keq1q2
d2
,
where d = P1P2 is the distance between the two charges. We use d because the
usual notation r has special meaning for us: it is one of the coordinates in spherical
systems. If we multiply this magnitude by the unit vector describing the direction
of the force, we obtain the full force vector (see Box 1.1.1). But, assuming repulsion
for the moment, this unit vector is
r2 − r1
|r2 − r1|
≡ ê21.
Also, since d = |r2 − r1|, we have
F21 =
keq1q2
d2
ê21 =
keq1q2
|r2 − r1|2
r2 − r1
|r2 − r1|
or
Coulomb’s law
when charges are
arbitrarily located
F21 =
keq1q2
|r2 − r1|3
(r2 − r1). (1.24)
Although we assumed repulsion, we see that (1.24) includes attraction as well. In-
deed, if q1q2 0, F21 is opposite to r2 −r1, i.e., F21 is directed from P2 to P1. Since
F21 is the force on q2 by q1, this is an attraction. We also note that Newton’s third
law is included in (1.24):
F12 =
keq2q1
|r1 − r2|3
(r1 − r2) = −F21
because r2 − r1 = −(r1 − r2) and |r2 − r1| = |r1 − r2|.
We can also write the gravitational force immediately
vector form of
gravitational force
F21 = −
Gm1m2
|r2 − r1|3
(r2 − r1), (1.25)
where m1 and m2 are point masses and the minus sign is introduced to ensure
attraction.
Now that we have expressions for electric and gravitational forces, we can
obtain the electric field of a point charge and the gravitational field of a point
mass. First recall that the electric field at a point P is defined to be the
force on a test charge q located at P divided by q. Thus if we have a charge
q1, at P1 with position vector r1 and we are interested in its fields at P with

position vector r, we introduce a charge q at r and calculate the force on q
from Equation (1.24):
Fq =
keq1q
|r − r1|3
(r − r1).
Dividing by q gives electric field of a
point charge
E1 =
keq1
|r − r1|3
(r − r1), (1.26)
where we have given the field the same index as the charge producing it.
The calculation of the gravitational field follows similarly. The result is
g1 = −
Gm1
|r − r1|3
(r − r1). (1.27)
In (1.26) and (1.27), P is called the field point and P1 the source point. field point and
source point
Note that in both expressions, the field position vector comes first.
If there are several point charges (or masses) producing an electric (gravita-
tional) field, we simply add the contributions from each source. The principle superposition
principle explained
behind this procedure is called the superposition principle. It is a princi-
ple that “seems” intuitively obvious, but upon further reflection its validity
becomes surprising. Suppose a charge q1 produces a field E1 around itself.
Now we introduce a second charge q2 which, far away and isolated from any
other charges, produced a field E2 around itself. It is not at all obvious that
once we move these charges together, the individual fields should not change.
After all, this is not what happens to human beings! We act completely dif-
ferently when we are alone than when we are in the company of others. The
presence of others drastically changes our individual behaviors. Nevertheless,
charges and masses, unfettered by any social chains, retain their individuality
and produce fields as if no other charges were present.
It is important to keep in mind that the superposition principle applies
only to point sources. For example, a charged conducting sphere will not
produce the same field when another charge is introduced nearby, because the
presence of the new charge alters the charge distribution of the sphere and
indeed does change the sphere’s field. However each individual point charge
(electron) on the sphere, whatever location on the sphere it happens to end
up in, will retain its individual electric field.14
Going back to the electric field, we can write
E = E1 + E2 + · · · + En
for n point charges q1, q2, . . . , qn (see Figure 1.16). Substituting from (1.26),
with appropriate indices, we obtain
E =
keq1
|r − r1|3
(r − r1) +
keq2
|r − r2|3
(r − r2) + · · · +
keqn
|r − rn|3
(r − rn)
or, using the summation symbol, we obtain
14The superposition principle, which in the case of electrostatics and gravity is needed
to calculate the fields of large sources consisting of many point sources, becomes a vital
pillar upon which quantum theory is built and by which many of the strange phenomena
of quantum physics are explained.

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openly assailed,—until at last the republic was constrained to
take up arms in their defence.
“Such are these two great wars in which these two chiefs bore
such part. Washington fought for national independence and
triumphed, making his country an example to mankind. Lincoln
drew a reluctant sword to save those great ideas, essential to
the life and character of the republic. * * *
“Rejoice as you point to this child of the people, who was lifted
so high that republican institutions became manifest in him! * *
* Above all, see to it that his constant vows are fulfilled, and
that the promises of the fathers are maintained, so that no
person in the upright form of man can be shut out from their
protection. Then will the unity of the republic be fixed on a
foundation that cannot fail, and other nations will enjoy its
security. The cornerstone of national independence is already in
its place, and on it is inscribed the name of George Washington.
There is another stone which must have its place at the corner
also. This is the Declaration of Independence, with all its
promises fulfilled. On this stone we will gratefully inscribe the
name of Abraham Lincoln.”

Emancipation Statue of Lincoln—Washington, D. C.
Carlyle says that “sincerity, a deep, great, genuine sincerity, is the
first characteristic of all men in any way heroic. All great men have
this as the primary material in them.” This is why the so-called “art
for art’s sake” never can be great. It is sincerity for merely formal
success, and not for the spirit of “life more abundantly.” Formal
efficiency is achieved only in the complicated training of an extended

education, but social efficiency of immeasurably greater value is the
simplicity of knowledge. It is the source and explanation of all
interests, and in that learning, Lincoln had no superior. He never
achieved any good that he did not at once want to share it with
others. As a boy he never learned anything good that he did not
want to express it to others. In this process of receiving and giving is
the fundamental means of building character and mind. In teaching
others, he taught himself, and thus in losing his life he found it. In
being able to tell his observations and interpretations to his
comrades, he was training to be the schoolmaster of the world.
Lincoln’s earnest sincerity relating to himself, his associates, his
community, his country, and for all mankind, may be illustrated in a
few quotations:
“The man who will not investigate both sides of a question is
dishonest.”
“After all, the one meaning of life is simply to be kind.”
“I have not done much, but this I have done—wherever I have
found a thistle growing, I have tried to pluck it up, and in its place to
plant a flower.”
“I have been too familiar with disappointment, to be very much
chagrined by defeat.”
“Without the assistance of that Divine Being I cannot succeed, and
with that assistance I cannot fail.”
“If destruction be our lot, we must ourselves be its author and
finisher. As a nation of freemen we must live through all time, or die
by suicide.”
“A majority held in restraint by constitutional checks and limitations,
and always changing easily with deliberate changes of popular
opinions and sentiments, is the only true sovereign of a free people.”

“Twenty-five years ago I was a hired laborer. The hired laborer of
yesterday may labor on his own account today, and hire others to
labor for him to-morrow. Advancement and improvement in
conditions is the order of things in a society of equals,—in a
democracy.”
In a speech at Columbus, Ohio, September 16, 1859, he said, “I
believe there is a genuine popular sovereignty. I think a definition of
genuine popular sovereignty, in the abstract, would be about this:
That each man shall do precisely as he pleases with himself, and
with all those things which exclusively concern him. Applied to
government this principle would be, that a general government shall
do all those things which pertain to it, and all the local governments
shall do precisely as they please in respect to those matters which
exclusively concern them. I understand that this government of the
United States, under which we live, is based upon that principle; and
I am misunderstood if it is supposed that I have any war to make
upon that principle.”
But, there is a patriotic masterpiece of Lincoln’s thought, which, with
the reinforcement of occasion and place, such as the field of
Gettysburg was, contains all the unmeasurable and priceless
meaning of Lincoln for American patriotism and the manhood of
America. It is his address of dedication on the battlefield of
Gettysburg. In effect on the human mind, it probably can never be
surpassed as a message of political freedom for the rights of man.
II. A MASTERPIECE OF MEANING FOR
AMERICA
The battle of Gettysburg is regarded by historians as one of the
decisive battles of the world. It was fought July 2, 3 and 4, 1863. On
the first anniversary, a great national meeting was held there to
dedicate the ground as a government burial place for the soldiers
who had died there.

Mr. Seward, Secretary of State, on the eve of the dedication, in the
course of an address, said, “I thank my God for the hope that this is
the last fratricidal war which will fall upon this country, vouchsafed
us from heaven, as the richest, the broadest, the most beautiful and
capable of a great destiny, that has ever been given to any part of
the human race.”
At the opening of the ceremonies, before a vast concourse of
people, from all the Northern states, convened on the open
battlefield, Rev. T. H. Stockton said in the course of his dedicatory
prayer, “In behalf of all humanity, whose ideal is divine, whose first
memory is Thine image lost, and whose last hope is Thine image
restored, and especially of our own nation, whose history has been
so favored, whose position is so peerless, whose mission is so
sublime, and whose future so attractive, we thank Thee for the
unspeakable patience of Thy compassion, and the exceeding
greatness of Thy loving kindness.... By this Altar of Sacrifice, on this
Field of Deliverance, on this Mount of Salvation, within the fiery and
bloody line of these ‘munitions of rocks,’ looking back to the dark
days of fear and trembling, and to the rapture of relief that came
after, we multiply our thanksgivings and confess our obligations....
Our enemies ... prepared to cast the chain of Slavery around the
form of Freedom, binding life and death together forever.... But,
behind these hills was heard the feeble march of a smaller, but still
pursuing host. Onward they hurried, day and night, for God and
their country. Footsore, wayworn, hungry, thirsty, faint,—but not in
heart,—they came to dare all, to bear all, and to do all that is
possible to heroes.... Baffled, bruised, broken, their enemies
recoiled, retired and disappeared.... But oh, the slain!... From the
Coasts beneath the Eastern Star, from the shores of Northern lakes
and rivers, from the flowers of Western prairies, and from the homes
of the Midway and Border, they came here to die for us and for
mankind.... As the trees are not dead, though their foliage is gone,
so our heroes are not dead, though their forms have fallen.... The
spirit of their example is here. And, so long as time lasts, the

pilgrims of our own land, and from all lands, will thrill with its
inspiration.”
Edward Everett, as the orator of the day, said in the course of his
scholarly address, “As my eye ranges over the fields whose sod was
so recently moistened by the blood of gallant and loyal men, I feel,
as never before, how truly it was said of old, ‘it is sweet and
becoming to die for one’s country.’ I feel, as never before, how justly
from the dawn of history to the present time, men have paid the
homage of their gratitude and admiration to the memory of those
who nobly sacrificed their lives, that their fellowmen may live in
safety and honor.... I do not believe there is in all history, the record
of a Civil War of such gigantic dimensions where so little has been
done in the spirit of vindictiveness as in this war.... There is no
bitterness in the hearts of the masses.... The bonds that unite us as
one People,—a substantial community of origin, language, belief and
law; common, national and political interests ... these bonds of
union are of perennial force and energy, while the causes of
alienation are imaginary, factitious and transient. The heart of the
People, North and South, is for the Union.... The weary masses of
the people are yearning to see the dear old flag floating over their
capitols, and they sigh for the return of peace, prosperity and
happiness, which they enjoyed under a government whose power
was felt only in its blessings.... You feel, though the occasion is
mournful, that it is good to be here! God bless the Union! It is
dearer to us for the blood of brave men which has been shed in its
defense.... ‘The whole earth,’ said Pericles, as he stood over the
remains of his fellow citizens, who had fallen in the first year of the
Peloponnesian War, ‘the whole earth is the sepulchre of illustrious
men.’ All time, he might have added, is the millennium of their
glory.”
The place and the occasion were supremely inspiring to patriotism,
not only for the triumph of moral principle in one’s country, but for
its meaning to all humanity. The great battlefield spread out before

the eyes of the vast concourse gathered there from all the states,
and the spirit of the heroic scenes animated every mind.
Edward Everett, then regarded as the greatest orator in America,
had delivered the dedicatory oration through a long strain of
attention, during the weary and fatiguing hours. The President was
then called on to close the dedication with whatever he might feel
desirable to say. He did so in a few words, but these few words are
cherished as among the greatest contributions to the meaning of
civilization. To one of the decisive battles for freedom in the world, it
gave a starry crown from “the voice of the people” as “the voice of
God.”
The War Department appropriated five thousand dollars to cast this
speech in bronze and set it up on the battlefield of Gettysburg. It is
regarded as a masterpiece of dedication in the literature of the
world.
“Fourscore and seven years ago our fathers brought forth on this
continent a new nation, conceived in liberty, and dedicated to the
proposition that all men are created equal.
“Now we are engaged in a great civil war testing whether that
nation, or any nation so conceived and so dedicated, can long
endure. We are met on a great battlefield of that war. We have come
to dedicate a portion of that field as a final resting place for those
who here gave their lives that that nation might live. It is altogether
fitting and proper that we should do this.
“But, in a larger sense, we cannot dedicate, we cannot consecrate,
we cannot hallow this ground. The brave men, living and dead, who
struggled here, have consecrated it far above our poor power to add
or detract. The world will little note, nor long remember, what we
say here, but it can never forget what they did here.

“It is for us, the living, rather to be dedicated here to the unfinished
work which they who fought here have thus far so nobly advanced.
It is rather for us to be here dedicated to the great task remaining
before us: that from the same honored dead we take increased
devotion to that cause for which they gave the last full measure of
devotion; that we here highly resolve that these dead should not
have died in vain; that this nation, under God, shall have a new birth
of freedom, and that government of the people, by the people, for
the people, shall not perish from the earth.”
III. THE MISSION OF AMERICA
The understanding person who becomes conscious of a meaning for
his life, realizes a most important responsibility to work for the
betterment of his mind and the material conditions that are to
become as his future self. The moral person, who becomes
conscious of a meaning for human life, works for this betterment as
his contribution to the progress of posterity. This means that a moral
individual coincides with a social humanity. Anything not thus
harmonizing morally for the world as it is, in order to promote a
world as it ought to be, is an enemy of both self and society.
Lincoln admonishes us to remember that “The struggle of today is
not altogether for today,—it is for a vast future also.” We learned
rapidly, when the true situation came into our view, that, as
Professor Phelps voiced it long ago, “To save America we must save
the world.” American patriotism is clearly world-patriotism, and it has
become synonymous with humanity. This old truth was discovered
by the Revolutionary Fathers, and it is the mission of America to
make it the truth of the World.
The International Teachers’ Congress representing eighteen nations,
which met at Liege in 1905, adopted five definite ideas of
International Peace, that should be promoted through all available

ways, in all the schools of civilized nations. Briefly stated, those
fundamental ideas were as follows:
1. The morality of individuals is the same for people and
nations.
2. The ideal of brotherly love has no limit.
3. All life must be duly respected.
4. Human rights are the same for one and all.
5. Love of country coincides with love of humanity.
Such principles and such a definition of patriotism were upheld by
the makers and preservers of America, at the greatest cost of
treasure and life, and they are the life-interest of every one worthy
of the name American. It moved Bishop J. P. Newman to say of
Lincoln in his anniversary oration of 1894, “Lincoln’s mission was as
large as his country, vast as humanity, enduring as time. No greater
thought can ever enter the human mind than obedience to law and
freedom for all.... Time has vindicated the character of his
statesmanship, that to preserve the Union was to save this great
nation for human liberty.”
American faith has at last come to the conditions when it can realize
itself in fulfilling the moral work of the world. That vision came into
full view during the Great European War.
President Wilson, in his address to Congress, April 2, 1917, said:
“We are at the beginning of an age in which it will be insisted that
the same standards of conduct and of responsibility for wrong shall
be observed among nations and their Governments that are
observed among the individual citizens of civilized states.”
Congress acted upon this reaffirmation of the responsibility of
Americans and the mission of America. Concerning the monstrous

invasion of humanity and ruthless denial of international law, he
said:
“Neutrality is no longer feasible or desirable where the peace of the
world is involved and the freedom of its peoples and the menace to
that peace and freedom lies in the existence of autocratic
Governments backed by organized force which is controlled wholly
by their will, not by the will of their people. We have seen the last of
neutrality in such circumstances.”
The Way of Peace, as the morality of democracies, he clearly
defined, so that even the worst prejudice could not becloud the
issue with irrelevant or contradictory assertions.
“A steadfast concert for peace can never be maintained except by a
partnership of democratic nations. No autocratic Government could
be trusted to keep faith within it or observe its covenants. It must be
a league of honor, a partnership of opinion. Intrigue would eat its
vitals away; the plotters of inner circles who could plan what they
would and render account to no one would be a corruption seated at
its very heart. Only free peoples can hold their purpose and their
honor steady to a common end and prefer the interests of mankind
to any narrow interest of their own.”
Washington was charged with the heroic task of making the thirteen
colonies safe for “Life, Liberty, and the Pursuit of Happiness;”
Lincoln’s patriotic mission was to unchain this Ideal for all America:
and Wilson’s sublime conception was to make the world “safe for
democracy,” that its peace might be planted on “the trusted
foundations of liberty.”
A mind-union upon human meaning as an ideal is necessary for the
patriotism of America. The right to life means that the making of
right life has a right way. Those who deny the meaning of America
divest themselves of all claims in reason upon the rights of life
defined in American history. The American kingdom of right is
perfecting itself as rapidly as minds can be mobilized for its sublime

task. The war-message extending the definition of American
freedom says:
“We have no selfish ends to serve. We desire no conquest, no
dominion. We seek no indemnities for ourselves, no material
compensation for the sacrifices we shall freely make. We are but one
of the champions of the rights of mankind. We shall be satisfied
when those rights have been made as secure as the faith and the
freedom of the nations can make them.”
And, finally, the duty of every American, worthy of America, enters
the third epoch of American history, as did the patriot duty of
Washington and Lincoln in their time. The message concludes in
these measured terms:
“It is a fearful thing to lead this great, peaceful people into war—into
the most terrible and disastrous of all wars, civilization itself seeming
to be in the balance.
“But the right is more precious than peace, and we shall fight for the
things which we have always carried nearest our hearts—for
democracy, for the right of those who submit to authority to have a
voice in their own Governments, for the rights and liberties of small
nations, for a universal dominion of right by such a concert of free
peoples as shall bring peace and safety to all nations and make the
world itself at last free.
“To such a task we can dedicate our lives and our fortunes,
everything that we are and everything that we have, with the pride
of those who know that the day has come when America is
privileged to spend her blood and her might for the principles that
gave her birth and happiness and the peace which she has
treasured. God helping her, she can do no other.”
The world in its social evolution has come on through its immemorial
struggle to the crisis in its history, where civilization, as liberty in
moral law, can progress further only as the forces of humanity are
organized “to make the world safe for democracy.” The final truth is

that the world will be made safe for democracy when democracy is
made safe for the individual. All political creeds, religious interests
and moral ideals, must have this democracy in which to work, before
they can become free to develop their own truth.
Autocratic egotism, whether framed in national or personal will,
among many or few, must perish from the earth, with all its spoils
and masteries, before there can be any possible “government of the
people, for the people and by the people.” As “a house divided
against itself cannot stand,” so, a civilization cannot stand whose
humanity is divided into the three special interests known to us as
individuals, the nation and an alien world.
The human task of conscience and reason, made clear in the
progress of experience, finds the humanity of child, mother and man
in all its relations and interests, or it has not found God or the
meaning of the Universe.
Human peace and salvation are gained, not only through persuasion,
education and regeneration, but also that the composing conditions
of “peace on earth” shall be made materially safe for “life, liberty
and the pursuit of happiness.”
Physically, as well as spiritually, the faith that is “without works is
dead.” The righteousness that allows its right to be defeated is not
righteous, and the conscience that permits the crimes of inhumanity
is no less unlawful before man and God. In such conditions, the
prophet cried out, “Cursed be he that doeth the work of the Lord
negligently, and cursed be he that keepeth back his sword from
blood.”
The American democracy of Washington and Lincoln, with their
hosts of devoted associates, means individual righteousness and
responsibility making safe the free-born mind for a moral world.
What is an American and why so is the patriotic and religious
interest developed through ages of sacrifice and suffering. Only
those who are willing “to give the last full measure of devotion” to

that divine work are heirs to the humanity of Washington and
Lincoln, and who are thus entitled to be named Americans, or are
worthy to share the heritage of America.

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