This page intentionally left blank
MATHEMATICAL METHODS FOR OPTICAL
PHYSICS AND ENGINEERING
The first textbook on mathematical methods focusing on techniques for optical science and
engineering, this textbook is ideal for advanced undergraduates and graduate students in
optical physics.
Containing detailed sections on the basic theory, the textbook places strong emphasis
on connecting the abstract mathematical concepts to the optical systems to which they
are applied. It covers many topics which usually only appear in more specialized books,
such as Zernike polynomials, wavelet and fractional Fourier transforms, vector spherical
harmonics, the z-transform, and the angular spectrum representation.
Most chapters end by showing how the techniques covered can be used to solve an
optical problem. Essay problems in each chapter based on research publications, together
with numerous exercises, help to further strengthen the connection between the theory and
its application.
gregory j. gbur is anAssociate Professorof Physics and Optical Science at the Univer-
sity of North Carolina at Charlotte, where he has taught a graduate course on mathematical
methods for optics for the past five years and a course on advanced physical optics for two
years.

MATHEMATICAL METHODS FOR
OPTICAL PHYSICS AND
ENGINEERING
GREGORY J. GBUR
University of North Carolina
CAMBRIDGE UNIVERSITY PRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore,
São Paulo, Delhi, Dubai, Tokyo, Mexico City
Cambridge University Press
The Edinburgh Building, Cambridge CB2 8RU, UK

Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521516105
© G. J. Gbur 2011
This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without
the written permission of Cambridge University Press.
First published 2011
Printed in the United Kingdom at the University Press, Cambridge
A catalog record for this publication is available from the British Library
Library of Congress Cataloging in Publication data
Gbur, Greg.
Mathematical methods for optical physics and engineering / Greg Gbur.
p. cm.
Includes bibliographical references and index.
ISBN 978-0-521-51610-5 (hardback)
1. Optics–Mathematics. I. Title.
QC355.3.G38 2011
535.01

51–dc22 2010036195
ISBN 978-0-521-51610-5 Hardback
Cambridge University Press has no responsibility for the persistence or
accuracy of URLs for external or third-party internet websites referred to in
this publication, and does not guarantee that any content on such websites is,
or will remain, accurate or appropriate.
Dedicated to my wife Beth,
and my parents


Contents
Preface page xv
1 Vector algebra 1
1.1 Preliminaries 1
1.2 Coordinate system invariance 4
1.3 Vector multiplication 9
1.4 Useful products of vectors 12
1.5 Linear vector spaces 13
1.6 Focus: periodic media and reciprocal lattice vectors 17
1.7 Additional reading 24
1.8 Exercises 24
2 Vector calculus 28
2.1 Introduction 28
2.2 Vector integration 29
2.3 The gradient, ∇ 35
2.4 Divergence, ∇· 37
2.5 The curl, ∇× 41
2.6 Further applications of ∇ 43
2.7 Gauss’ theorem (divergence theorem) 45
2.8 Stokes’ theorem 47
2.9 Potential theory 48
2.10 Focus: Maxwell’s equations in integral and differential form 51
2.11 Focus: gauge freedom in Maxwell’s equations 57
2.12 Additional reading 60
2.13 Exercises 60
3 Vector calculus in curvilinear coordinate systems 64
3.1 Introduction: systems with different symmetries 64
3.2 General orthogonal coordinate systems 65
3.3 Vector operators in curvilinear coordinates 69
3.4 Cylindrical coordinates 73

vii
viii Contents
3.5 Spherical coordinates 76
3.6 Exercises 79
4 Matrices and linear algebra 83
4.1 Introduction: Polarization and Jones vectors 83
4.2 Matrix algebra 88
4.3 Systems of equations, determinants, and inverses 93
4.4 Orthogonal matrices 102
4.5 Hermitian matrices and unitary matrices 105
4.6 Diagonalization of matrices, eigenvectors, and eigenvalues 107
4.7 Gram–Schmidt orthonormalization 115
4.8 Orthonormal vectors and basis vectors 118
4.9 Functions of matrices 120
4.10 Focus: matrix methods for geometrical optics 120
4.11 Additional reading 133
4.12 Exercises 133
5 Advanced matrix techniques and tensors 139
5.1 Introduction: Foldy–Lax scattering theory 139
5.2 Advanced matrix terminology 142
5.3 Left–right eigenvalues and biorthogonality 143
5.4 Singular value decomposition 146
5.5 Other matrix manipulations 153
5.6 Tensors 159
5.7 Additional reading 174
5.8 Exercises 174
6 Distributions 177
6.1 Introduction: Gauss’ law and the Poisson equation 177
6.2 Introduction to delta functions 181
6.3 Calculus of delta functions 184

6.4 Other representations of the delta function 185
6.5 Heaviside step function 187
6.6 Delta functions of more than one variable 188
6.7 Additional reading 192
6.8 Exercises 192
7 Infinite series 195
7.1 Introduction: the Fabry–Perot interferometer 195
7.2 Sequences and series 198
7.3 Series convergence 201
7.4 Series of functions 210
7.5 Taylor series 213
7.6 Taylor series in more than one variable 218
7.7 Power series 220
7.8 Focus: convergence of the Born series 221
Contents ix
7.9 Additional reading 226
7.10 Exercises 226
8 Fourier series 230
8.1 Introduction: diffraction gratings 230
8.2 Real-valued Fourier series 233
8.3 Examples 236
8.4 Integration range of the Fourier series 239
8.5 Complex-valued Fourier series 239
8.6 Properties of Fourier series 240
8.7 Gibbs phenomenon and convergence in the mean 243
8.8 Focus: X-ray diffraction from crystals 246
8.9 Additional reading 249
8.10 Exercises 249
9 Complex analysis 252
9.1 Introduction: electric potential in an infinite cylinder 252

9.2 Complex algebra 254
9.3 Functions of a complex variable 258
9.4 Complex derivatives and analyticity 261
9.5 Complex integration and Cauchy’s integral theorem 265
9.6 Cauchy’s integral formula 269
9.7 Taylor series 271
9.8 Laurent series 273
9.9 Classification of isolated singularities 276
9.10 Branch points and Riemann surfaces 278
9.11 Residue theorem 285
9.12 Evaluation of definite integrals 288
9.13 Cauchy principal value 297
9.14 Focus: Kramers–Kronig relations 299
9.15 Focus: optical vortices 302
9.16 Additional reading 308
9.17 Exercises 308
10 Advanced complex analysis 312
10.1 Introduction 312
10.2 Analytic continuation 312
10.3 Stereographic projection 316
10.4 Conformal mapping 325
10.5 Significant theorems in complex analysis 332
10.6 Focus: analytic properties of wavefields 340
10.7 Focus: optical cloaking and transformation optics 345
10.8 Exercises 348
11 Fourier transforms 350
11.1 Introduction: Fraunhofer diffraction 350
x Contents
11.2 The Fourier transform and its inverse 352
11.3 Examples of Fourier transforms 354

11.4 Mathematical properties of the Fourier transform 358
11.5 Physical properties of the Fourier transform 365
11.6 Eigenfunctions of the Fourier operator 372
11.7 Higher-dimensional transforms 373
11.8 Focus: spatial filtering 375
11.9 Focus: angular spectrum representation 377
11.10 Additional reading 382
11.11 Exercises 383
12 Other integral transforms 386
12.1 Introduction: the Fresnel transform 386
12.2 Linear canonical transforms 391
12.3 The Laplace transform 395
12.4 Fractional Fourier transform 400
12.5 Mixed domain transforms 402
12.6 The wavelet transform 406
12.7 The Wigner transform 409
12.8 Focus: the Radon transform and computed axial tomography (CAT) 410
12.9 Additional reading 416
12.10 Exercises 416
13 Discrete transforms 419
13.1 Introduction: the sampling theorem 419
13.2 Sampling and the Poisson sum formula 423
13.3 The discrete Fourier transform 427
13.4 Properties of the DFT 430
13.5 Convolution 432
13.6 Fast Fourier transform 433
13.7 The z-transform 437
13.8 Focus: z-transforms in the numerical solution of Maxwell’s equations 445
13.9 Focus: the Talbot effect 449
13.10 Exercises 456

14 Ordinary differential equations 458
14.1 Introduction: the classic ODEs 458
14.2 Classification of ODEs 459
14.3 Ordinary differential equations and phase space 460
14.4 First-order ODEs 469
14.5 Second-order ODEs with constant coefficients 474
14.6 The Wronskian and associated strategies 476
14.7 Variation of parameters 478
14.8 Series solutions 480
14.9 Singularities, complex analysis, and general Frobenius solutions 481
Contents xi
14.10 Integral transform solutions 485
14.11 Systems of differential equations 486
14.12 Numerical analysis of differential equations 488
14.13 Additional reading 501
14.14 Exercises 501
15 Partial differential equations 505
15.1 Introduction: propagation in a rectangular waveguide 505
15.2 Classification of second-order linear PDEs 508
15.3 Separation of variables 517
15.4 Hyperbolic equations 519
15.5 Elliptic equations 525
15.6 Parabolic equations 530
15.7 Solutions by integral transforms 534
15.8 Inhomogeneous problems and eigenfunction solutions 538
15.9 Infinite domains; the d’Alembert solution 539
15.10 Method of images 544
15.11 Additional reading 545
15.12 Exercises 545
16 Bessel functions 550

16.1 Introduction: propagation in a circular waveguide 550
16.2 Bessel’s equation and series solutions 552
16.3 The generating function 555
16.4 Recurrence relations 557
16.5 Integral representations 560
16.6 Hankel functions 564
16.7 Modified Bessel functions 565
16.8 Asymptotic behavior of Bessel functions 566
16.9 Zeros of Bessel functions 567
16.10 Orthogonality relations 569
16.11 Bessel functions of fractional order 572
16.12 Addition theorems, sum theorems, and product relations 576
16.13 Focus: nondiffracting beams 579
16.14 Additional reading 582
16.15 Exercises 582
17 Legendre functions and spherical harmonics 585
17.1 Introduction: Laplace’s equation in spherical coordinates 585
17.2 Series solution of the Legendre equation 587
17.3 Generating function 589
17.4 Recurrence relations 590
17.5 Integral formulas 592
17.6 Orthogonality 594
17.7 Associated Legendre functions 597
xii Contents
17.8 Spherical harmonics 602
17.9 Spherical harmonic addition theorem 605
17.10 Solution of PDEs in spherical coordinates 608
17.11 Gegenbauer polynomials 610
17.12 Focus: multipole expansion for static electric fields 611
17.13 Focus: vector spherical harmonics and radiation fields 614

17.14 Exercises 618
18 Orthogonal functions 622
18.1 Introduction: Sturm–Liouville equations 622
18.2 Hermite polynomials 627
18.3 Laguerre functions 641
18.4 Chebyshev polynomials 650
18.5 Jacobi polynomials 654
18.6 Focus: Zernike polynomials 655
18.7 Additional reading 662
18.8 Exercises 662
19 Green’s functions 665
19.1 Introduction: the Huygens–Fresnel integral 665
19.2 Inhomogeneous Sturm–Liouville equations 669
19.3 Properties of Green’s functions 674
19.4 Green’s functions of second-order PDEs 676
19.5 Method of images 685
19.6 Modal expansion of Green’s functions 689
19.7 Integral equations 693
19.8 Focus: Rayleigh–Sommerfeld diffraction 701
19.9 Focus: dyadic Green’s function for Maxwell’s equations 704
19.10 Focus: scattering theory and the Born series 709
19.11 Exercises 712
20 The calculus of variations 715
20.1 Introduction: principle of Fermat 715
20.2 Extrema of functions and functionals 718
20.3 Euler’s equation 721
20.4 Second form of Euler’s equation 727
20.5 Calculus of variations with several dependent variables 730
20.6 Calculus of variations with several independent variables 732
20.7 Euler’s equation with auxiliary conditions: Lagrange multipliers 734

20.8 Hamiltonian dynamics 739
20.9 Focus: aperture apodization 742
20.10 Additional reading 745
20.11 Exercises 745
21 Asymptotic techniques 748
21.1 Introduction: foundations of geometrical optics 748
Contents xiii
21.2 Definition of an asymptotic series 753
21.3 Asymptotic behavior of integrals 756
21.4 Method of stationary phase 763
21.5 Method of steepest descents 766
21.6 Method of stationary phase for double integrals 771
21.7 Additional reading 772
21.8 Exercises 773
Appendix A The gamma function 775
A.1 Definition 775
A.2 Basic properties 776
A.3 Stirling’s formula 778
A.4 Beta function 779
A.5 Useful integrals 780
Appendix B Hypergeometric functions 783
B.1 Hypergeometric function 784
B.2 Confluent hypergeometric function 785
B.3 Integral representations 785
References 787
Index 793

Preface
Why another textbook on Mathematical Methods for Scientists? Certainly there are quite a
few good, indeed classic texts on the subject. What can another text add that these others

have not already done?
I began to ponder these questions, and my answers to them, over the past several years
while teaching a graduate course on Mathematical Methods for Physics and Optical Science
at the University of North Carolina at Charlotte. Although every student has his or her
own difficulties in learning mathematical techniques, a few problems amongst the students
have remained common and constant. The foremost among these is the “wall” between the
mathematics the students learn in math class and the applications they study in other classes.
The Fourier transform learned in math class is internally treated differently than the Fourier
transform used in, say, Fraunhofer diffraction. The end result is that the student effectively
learns the same topic twice, and is unable to use the intuition learned in a physics class to
help aid in mathematical understanding, or to use the techniques learned in math class to
formulate and solve physical problems.
To try and correct for this, I began to devote special lectures to the consequences of the
math the students were studying. Lectures on complex analysis would be followed by dis-
cussions of the analytic properties of wavefields and the Kramers–Kronig relations. Lectures
on infinite series could be highlighted by the discussion of the Fabry–Perot interferometer.
Students in my classes were uniformly dissatisfied with the standard textbooks. Part of
this dissatisfaction arises from the broad topics from which examples are drawn: quantum
physics, field theory, general relativity, optics, mechanics, and thermodynamics, to name
a few. Even the most dedicated theoretical physics students do not have a great physical
intuition about all these subfields, and consequently many of the examples are no more
useful in their minds than problems in abstract mathematics.
Given that students in my class are studying optics, I have focused most of my attention
on methods directly related to optical science. Here again the standard texts became a
problem, as there is not a perfect overlap between important methods for general physics
and important methods for optics. For example, group theory is not commonly used among
most optics researchers, and Fourier transforms, essential to the optics researcher, are not
used as much by the rest of the general physics community. Teaching to an optics crowd
would require that the emphasis on material be refocused. It was in view of these various
xv

xvi Preface
observations that I decided that a new mathematical methods book, with an emphasis on
optics, would be useful.
Optics as both an industry and a field of study in its own right has grown dramatically
over the past two decades. Optics programs at universities have grown in size and number in
recent years. The University of Rochester and the University of Arizona are schools which
have had degrees for some time, while the University of Central Florida and the University
of North Carolina at Charlotte have started programs within recent years. With countless
electrical engineering programs emphasizing studies in optics, it seems likely that more
optics degrees will follow in the years to come. A textbook which serves such programs
and optical researchers in general seems to have the potential to be a popular resource.
My goal, then, was to write a textbook on mathematical methods for physics and optical
science, with an emphasis on those techniques relevant to optical scientists and engineers.
The level of the book is intended for an advanced undergraduate or beginning graduate
level class on math methods. One of my main objectives was to write a “leaner” book than
many of the 1000+ page math books currently available, and do so by pushing much of the
abstract mathematical subtlety into references. Instead, the emphasis is placed on making the
connection between the mathematical techniques and the optics problems they are intimately
related to. To make this connection, most chapters begin with a short introduction which
illustrates the relevance of the mathematical technique being considered, and ends with one
or more applications, in which the technique is used to solve a problem. Physical examples
within the chapters are drawn predominantly from optics, though examplesfrom other fields
will be used when appropriate.
Abook of this type will address a number of mathematical techniques which are normally
not compiled into a single volume. It is hoped that this book will therefore serve not only
as a textbook but also potentially as a reference book for researchers in optics.
Another “wall” in students’ understanding is making the connection between the topics
learned in class and research results in the literature. A number of exercises in each chapter
are essay-style questions, in which a journal article must be read and its relevance to the
mathematical method discussed. I have also endeavored to provide an appreciable number of

exercises in each chapter, with some similar problems to facilitate teaching a class multiple
times. Some more advanced chapters have fewer exercises, mainly because it is difficult to
find exercises that are simultaneously solvable and enlightening.
Early chapters cover the basics that are essential for any student of the physical sci-
ences, including vectors, curvilinear coordinate systems, differential equations, sequences
and series, matrices, and this part of the book might be used for any math methods for
physics course. Later chapters concentrate on techniques significant to optics, including
Fourier analysis, asymptotic methods, Green’s functions, and more general types of integral
transform.
A book of this sort requires a lot of help, and I have sought plenty of insight from col-
leagues. I would like to thank Professor John Schotland, Professor Daniel James, Professor
Tom Suleski, Dr Choon How Gan,Mike Fairchild and Casey Rhodes for helpful suggestions
during the course of writing. I give special thanks to Professor Taco Visser and Dr Damon
Preface xvii
Diehl, each of whom read significant sections of the manuscript and provided corrections,
and to Professor Emil Wolf, who gave me encouragement and inspiration during the writing
process. I am grateful to Professor John Foley who some time ago gave me access to his
collection of math methods exercises, which were useful in developing my own problems.
Professor Daniel S. Jones generously provided a photograph of X-ray diffraction, and Pro-
fessor Visser provided a figure on the Poincaré sphere. I would also like to express my
appreciation to the very helpful people at Cambridge University Press, including Simon
Capelin, John Fowler, Megan Waddington, and Lindsay Barnes. Special thanks goes to
Frances Nex for her careful editing of the text.
I also have to thank a number of people for their help in keeping me sane during the
writing process! Among them, let me thank my guitar instructor Toby Watson, my skating
coach Tappie Dellinger, and my friends at Skydive Carolina, particularly my regular jump
buddies Nancy Newman, Mickey Turner, John Solomon, Robyn Porter, Mike Reinhart, and
Heiko Lotz! I would also like to give a “shout out” to Eric Smith and Mahy El-Kouedi for
their friendship and support.
Finally, let me thank my wife Beth Szabo for her support, understanding and patience

during this rather strenuous writing process.

1
Vector algebra
1.1 Preliminaries
In introductory physics, we often deal with physical quantities that can be described by a
single number. The temperature of a heated body, the mass of an object, and the electric
potential of an insulated metal sphere are all examples of such scalar quantities.
Descriptions of physical phenomena are not always (indeed, rarely)that simple, however,
and often we must use multiple, but related, numbers to offer a complete description of an
effect. The next level of complexity is the introduction of vector quantities.
A vector may be described as a conceptual object having both magnitude and direction.
Graphically, vectors can be represented by an arrow:
The length of the arrow is the magnitude of the vector, and the direction of the arrow
indicates the direction of the vector.
Examples of vectors in elementary physics include displacement, velocity, force, momen-
tum, and angular momentum, though the concept can be extended to more complicated and
abstract systems. Algebraically, we will usually represent vectors by boldface characters,
i.e. F for force, v for velocity, and so on.
It is worth noting at this point that the word “vector” is used in mathematics with some-
what broader meaning. In mathematics, a vector space is defined quite generally as a set
of elements (called vectors) together with rules relating to their addition and scalar mul-
tiplication of vectors. In this sense, the set of real numbers form a vector space, as does
any ordered set of numbers, including matrices, to be discussed in Chapter 4, and complex
numbers, to be discussed in Chapter 9. For most of this chapter we reserve the term “vector”
for quantities which possess magnitude and direction in three-dimensional space, and are
independent of the specific choice of coordinate system in a manner to be discussed in
1
2 Vector algebra
A

A
B
B
C
Figure 1.1 The parallelogram law of vector addition. Adding B to A (the addition above the C-line)
is equivalent to adding A to B (the addition below the C-line).
Section 1.2. We briefly describe vector spaces at the end of this chapter, in Section 1.5.The
interested reader can also consult Ref. [Kre78, Sec. 2.1].
Vector addition is commutative and associative; commutativity refers to the observation
that the addition of vectors is order independent, i.e.
A +B =B +A = C. (1.1)
This can be depicted graphically by the parallelogram law of vector addition, illustrated
in Fig. 1.1. A pair of vectors are added “tip-to-tail”; that is, the second vector is added to
the first by putting its tail at the end of the tip of the first vector. The resultant vector is
found by drawing an arrow from the origin of the first vector to the tip of the second vector.
Associativity refers to the observation that the addition of multiple vectors is independent
of the way the vectors are grouped for addition, i.e.
(A +B) +C =A +(B +C). (1.2)
This may also be demonstrated graphically if we first define the following vector additions:
E ≡A +B, (1.3)
D ≡E +C, (1.4)
F ≡B +C. (1.5)
The vectors and their additions are illustrated in Fig. 1.2. It can be immediately seen that
E +C = A +F. (1.6)
So far, we have introduced vectors as purely geometrical objects which are independent
of any specific coordinate system. Intuitively, this is an obvious requirement: where I am
standing in a room (my “position vector”) is independent of whether I choose to describe
it by measuring it from the rear left corner of the room or the front right corner. In other
words, the vector has a physical significance which does not change when I change my
method of describing it.

1.1 Preliminaries 3
A
B
C
D
E
F
Figure 1.2 The trapezoid rule of vector addition. It makes no difference if we first add A and B, and
then C, or first add B and C, and then A.
By choosing a coordinate system, however, we may create a representation of the vector
in terms of these coordinates. We start by considering a Cartesian coordinate system with
coordinates x, y, z which are all mutually perpendicular and form a right-handed coordinate
system.
1
For a given Cartesian coordinate system, the vector A, which starts atthe origin and
ends at the point with coordinates (A
x
,A
y
,A
z
), is completely described by the coordinates
of the end point.
It is highly convenient to express a vector in terms of these components by use of unit
vectors
ˆ
x,
ˆ
y,
ˆ

z, vectors of unit magnitude pointing in the directions of the positive coordinate
axes,
A = A
x
ˆ
x +A
y
ˆ
y +A
z
ˆ
z. (1.7)
This equation indicates that a vector equals the vector sum of its components. In three
dimensions, the position vector r which measures the distance from a chosen origin is
written as
r =x
ˆ
x +y
ˆ
y +z
ˆ
z, (1.8)
where x, y, and z are the lengths along the different coordinate axes.
The sum of two vectors can be found by taking the sum of their individual components.
This means that the sum of two vectors A and B can be written as
A +B =(A
x
+B
x
)

ˆ
x +(A
y
+B
y
)
ˆ
y +(A
z
+B
z
)
ˆ
z. (1.9)
The magnitude (length) of a vector in terms of its components can be found by two successive
applications of the Pythagorean theorem. The magnitude A of the complete vector, also
written as |A|, is found to be
A =

A
2
x
+A
2
y
+A
2
z
. (1.10)
Another way to represent the vector in a particular coordinate system is by its magnitude

A and the angles α, β, γ that the vector makes with each of the positive coordinate axes.
1
If x is the outward-pointing index finger of the right hand, y is the folded-in ring finger and z is the thumb,
pointing straight up.
4 Vector algebra
x
y
z
A
αβ
γ
A
x
A
y
A
z
Figure 1.3 Illustration of the vector A, its components (A
x
,A
y
,A
z
), and the angles α, β, γ .
These angles and their relationship to the vector and its components are illustrated in Fig. 1.3.
The quantities cosα, cosβ, and cosγ are called direction cosines. It might seem that there
is an inconsistency with this representation, since we now evidently need four numbers
(A,α,β,γ ) to describe the vector, where we needed only three (A
x
,A

y
,A
z
) before. This
seeming contradiction is resolved by the observation that α, β, and γ are not independent
quantities; they are related by the equation,
cos
2
α +cos
2
β +cos
2
γ = 1. (1.11)
In the spherical coordinate system to be discussed in Chapter 3, we will see that we may
completely specify the position vector by its magnitude r and two angles θ and φ.
It is to be noted that we usually see vectors in physics in two distinct classes:
1. Vectors associated with the property of a single, localized object, such as the velocity of a car, or
the force of gravity acting on a moving projectile.
2. Vectors associated with the property of a nonlocalized “object” or system, such as the electric
field of a light wave, or the velocity of a fluid. In such a case, the vector quantity is a function of
position and possibly time and we may do calculus with respect to the position and time variables.
This vector quantity is usually referred to as a vector field.
Vector fields are extremely important quantities in physics and we will return to them often.
1.2 Coordinate system invariance
We have said that a vector is independent of any specific coordinate system – in other
words, that a vector is independent of how we choose to characterize it. This seems like an
obvious criterion, but there are physical quantities which have magnitude and direction but
1.2 Coordinate system invariance 5
are not vectors; an example of this in optics is the set of principle indices of refraction of an
anisotropic crystal. Thus, to define a vector properly, we need to formulate mathematically

this concept of coordinate system invariance. Furthermore, it is not uncommon to require,
in the solution of a physical problem, the transformation from one coordinate system to
another. We therefore take some time to study the mathematics relating to the behavior of
a vector under a change of coordinates.
The simplest coordinate transformation is a change of origin, leaving the orientation of
the axes unchanged. The only vector that depends explicitly upon the origin is the position
vector r, which is a measure of the vector distance from the origin. If the new origin of a
new coordinate system, described by position vector r

, is located at the position r
0
from
the old origin, the coordinates are related by the formula
r

=r −r
0
. (1.12)
Most other basic vectors depend uponthe displacement vector R =r
2
−r
1
, i.e. the change in
position, and therefore are unaffected by a change in origin. Examples include the velocity,
momentum, and force upon an object.
A less trivial example of a change of coordinate system is a change of the orientation of
coordinate axes, and its effect on a position vector r. For simplicity, we first consider the
two dimensional case. The vector r may be written in one coordinate system as r =x
ˆ
x +y

ˆ
y,
while in a second coordinate system this vector may be written as r

=x

ˆ
x

+y

ˆ
y

.The(x,y)
coordinate axes are rotated to a new location to become the (x

,y

) axes, while leaving the
vector r (in particular, the location of the tip of r) fixed. The question we ask: what are
the components of the vector r in the new coordinate system, which makes an angle φ with
the old system? The relation between the two systems is illustrated in Fig. 1.4.
x
y
x'
y'
r
φ
Figure 1.4 Illustration of the position vector r and its components in the two coordinate systems.