Mathematical methods for physicists, 6th edition, arfken weber

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V e ~ d 3 r = / B d a , (Gauss),

L ( ~ x A ) . d a = A-dl,

(Stokes)

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(@V@- y?V@) da,

(Green)


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Cumed Orthogonal Coordinates
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(Euler-Mascheroni number)
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MATHEMATICAL
METHODS FOR
PHYSICISTS
SIXTH EDITION
George B. Arfken
Miami University
Oxford, OH

Hans J. Weber
University of Virginia
Charlottesville, VA

Amsterdam Boston Heidelberg London New York Oxford
Paris San Diego San Francisco Singapore Sydney Tokyo


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MATHEMATICAL
METHODS FOR
PHYSICISTS
SIXTH EDITION


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Printed in the United States of America
05 06 07 08 09 10 9 8 7 6

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1


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CONTENTS

Preface
1

2

xi

Vector Analysis

1.1
Definitions, Elementary Approach . . . . . .
1.2
Rotation of the Coordinate Axes . . . . . . .
1.3
Scalar or Dot Product . . . . . . . . . . . .
1.4
Vector or Cross Product . . . . . . . . . . .
1.5
Triple Scalar Product, Triple Vector Product
1.6
Gradient, ∇ . . . . . . . . . . . . . . . . . .
1.7
Divergence, ∇ . . . . . . . . . . . . . . . . .
1.8
Curl, ∇× . . . . . . . . . . . . . . . . . . .
1.9
Successive Applications of ∇ . . . . . . . .
1.10 Vector Integration . . . . . . . . . . . . . . .
1.11 Gauss’ Theorem . . . . . . . . . . . . . . . .
1.12 Stokes’ Theorem . . . . . . . . . . . . . . .
1.13 Potential Theory . . . . . . . . . . . . . . .
1.14 Gauss’ Law, Poisson’s Equation . . . . . . .
1.15 Dirac Delta Function . . . . . . . . . . . . .
1.16 Helmholtz’s Theorem . . . . . . . . . . . . .
Additional Readings . . . . . . . . . . . . .

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101

Vector Analysis in Curved Coordinates and Tensors
2.1

Orthogonal Coordinates in R3 . . . . . . . . . .
2.2
Differential Vector Operators . . . . . . . . . .
2.3
Special Coordinate Systems: Introduction . . .
2.4
Circular Cylinder Coordinates . . . . . . . . . .
2.5
Spherical Polar Coordinates . . . . . . . . . . .

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103

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115
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vi

Contents

2.6
2.7
2.8
2.9
2.10
2.11

3

4

5

6

Tensor Analysis . . . . . . . .
Contraction, Direct Product .
Quotient Rule . . . . . . . . .
Pseudotensors, Dual Tensors
General Tensors . . . . . . . .
Tensor Derivative Operators .
Additional Readings . . . . .

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133
139
141

142
151
160
163

Determinants and Matrices
3.1
Determinants . . . . . . . . . . . . . .
3.2
Matrices . . . . . . . . . . . . . . . . .
3.3
Orthogonal Matrices . . . . . . . . . .
3.4
Hermitian Matrices, Unitary Matrices
3.5
Diagonalization of Matrices . . . . . .
3.6
Normal Matrices . . . . . . . . . . . .
Additional Readings . . . . . . . . . .

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165
165
176
195
208
215
231
239

Group Theory
4.1
Introduction to Group Theory . . . . . . . .
4.2
Generators of Continuous Groups . . . . . .
4.3
Orbital Angular Momentum . . . . . . . . .

4.4
Angular Momentum Coupling . . . . . . . .
4.5
Homogeneous Lorentz Group . . . . . . . .
4.6
Lorentz Covariance of Maxwell’s Equations
4.7
Discrete Groups . . . . . . . . . . . . . . . .
4.8
Differential Forms . . . . . . . . . . . . . .
Additional Readings . . . . . . . . . . . . .

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241
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291
304
319

Infinite Series
5.1
Fundamental Concepts . . . . . . . . . . . . .
5.2
Convergence Tests . . . . . . . . . . . . . . .
5.3
Alternating Series . . . . . . . . . . . . . . . .

5.4
Algebra of Series . . . . . . . . . . . . . . . .
5.5
Series of Functions . . . . . . . . . . . . . . .
5.6
Taylor’s Expansion . . . . . . . . . . . . . . .
5.7
Power Series . . . . . . . . . . . . . . . . . .
5.8
Elliptic Integrals . . . . . . . . . . . . . . . .
5.9
Bernoulli Numbers, Euler–Maclaurin Formula
5.10 Asymptotic Series . . . . . . . . . . . . . . . .
5.11 Infinite Products . . . . . . . . . . . . . . . .
Additional Readings . . . . . . . . . . . . . .

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401

Functions of a Complex Variable I Analytic Properties, Mapping
6.1
Complex Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2
Cauchy–Riemann Conditions . . . . . . . . . . . . . . . . . . . . . . .
6.3
Cauchy’s Integral Theorem . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents
6.4
6.5
6.6
6.7
6.8

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8

9

Cauchy’s Integral Formula
Laurent Expansion . . . .
Singularities . . . . . . . .
Mapping . . . . . . . . . .
Conformal Mapping . . .
Additional Readings . . .

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425
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451
453

Functions of a Complex Variable II
7.1
Calculus of Residues . . . . .
7.2
Dispersion Relations . . . . .
7.3
Method of Steepest Descents .
Additional Readings . . . . .

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455
455
482
489
497

The Gamma Function (Factorial Function)
8.1
Definitions, Simple Properties . . . . .
8.2
Digamma and Polygamma Functions .
8.3
Stirling’s Series . . . . . . . . . . . . .
8.4
The Beta Function . . . . . . . . . . .

8.5
Incomplete Gamma Function . . . . .
Additional Readings . . . . . . . . . .

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520
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533

Differential Equations
9.1
Partial Differential Equations . . . . . . . . . .
9.2
First-Order Differential Equations . . . . . . .
9.3
Separation of Variables . . . . . . . . . . . . . .
9.4
Singular Points . . . . . . . . . . . . . . . . . .
9.5
Series Solutions—Frobenius’ Method . . . . . .
9.6

A Second Solution . . . . . . . . . . . . . . . . .
9.7
Nonhomogeneous Equation—Green’s Function
9.8
Heat Flow, or Diffusion, PDE . . . . . . . . . .
Additional Readings . . . . . . . . . . . . . . .

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535
535
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554
562
565
578
592
611
618

10 Sturm–Liouville Theory—Orthogonal Functions
10.1 Self-Adjoint ODEs . . . . . . . . . . . . . .

10.2 Hermitian Operators . . . . . . . . . . . . .
10.3 Gram–Schmidt Orthogonalization . . . . . .
10.4 Completeness of Eigenfunctions . . . . . . .
10.5 Green’s Function—Eigenfunction Expansion
Additional Readings . . . . . . . . . . . . .

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621
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642
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662
674

11 Bessel Functions

11.1 Bessel Functions of the First Kind, Jν (x) . .
11.2 Orthogonality . . . . . . . . . . . . . . . . .
11.3 Neumann Functions . . . . . . . . . . . . .
11.4 Hankel Functions . . . . . . . . . . . . . . .
11.5 Modified Bessel Functions, Iν (x) and Kν (x)

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675
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699
707
713


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viii

Contents
11.6
11.7

Asymptotic Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . .
Spherical Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . .
Additional Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . .

719
725
739

12 Legendre Functions

12.1 Generating Function . . . . . . . . . . . . .
12.2 Recurrence Relations . . . . . . . . . . . . .
12.3 Orthogonality . . . . . . . . . . . . . . . . .
12.4 Alternate Definitions . . . . . . . . . . . . .
12.5 Associated Legendre Functions . . . . . . .
12.6 Spherical Harmonics . . . . . . . . . . . . .
12.7 Orbital Angular Momentum Operators . . .
12.8 Addition Theorem for Spherical Harmonics
12.9 Integrals of Three Y’s . . . . . . . . . . . . .
12.10 Legendre Functions of the Second Kind . . .
12.11 Vector Spherical Harmonics . . . . . . . . .
Additional Readings . . . . . . . . . . . . .

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741
741

749
756
767
771
786
793
797
803
806
813
816

13 More Special Functions
13.1 Hermite Functions . . . . . . . . . .
13.2 Laguerre Functions . . . . . . . . . .
13.3 Chebyshev Polynomials . . . . . . .
13.4 Hypergeometric Functions . . . . . .
13.5 Confluent Hypergeometric Functions
13.6 Mathieu Functions . . . . . . . . . .
Additional Readings . . . . . . . . .

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817
817
837
848
859
863
869
879

14 Fourier Series
14.1 General Properties . . . . . . . . . . . . .
14.2 Advantages, Uses of Fourier Series . . . .
14.3 Applications of Fourier Series . . . . . . .
14.4 Properties of Fourier Series . . . . . . . .
14.5 Gibbs Phenomenon . . . . . . . . . . . . .
14.6 Discrete Fourier Transform . . . . . . . .
14.7 Fourier Expansions of Mathieu Functions
Additional Readings . . . . . . . . . . . .

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881
881
888
892
903
910
914
919
929

15 Integral Transforms
15.1 Integral Transforms . . . . . . . . . . . .
15.2 Development of the Fourier Integral . . .
15.3 Fourier Transforms—Inversion Theorem
15.4 Fourier Transform of Derivatives . . . .
15.5 Convolution Theorem . . . . . . . . . . .
15.6 Momentum Representation . . . . . . . .
15.7 Transfer Functions . . . . . . . . . . . .
15.8 Laplace Transforms . . . . . . . . . . . .

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931
931
936
938
946
951
955
961
965

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Contents
15.9
15.10
15.11
15.12

Laplace Transform of Derivatives
Other Properties . . . . . . . . .
Convolution (Faltungs) Theorem
Inverse Laplace Transform . . . .
Additional Readings . . . . . . .

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ix

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. 971
. 979
. 990
. 994
. 1003

16 Integral Equations
16.1 Introduction . . . . . . . . . . . . . . . . . . . . .
16.2 Integral Transforms, Generating Functions . . . .
16.3 Neumann Series, Separable (Degenerate) Kernels
16.4 Hilbert–Schmidt Theory . . . . . . . . . . . . . .
Additional Readings . . . . . . . . . . . . . . . .

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1005
1005
1012
1018
1029
1036

17 Calculus of Variations
17.1 A Dependent and an Independent Variable . .
17.2 Applications of the Euler Equation . . . . . .

17.3 Several Dependent Variables . . . . . . . . . .
17.4 Several Independent Variables . . . . . . . . .
17.5 Several Dependent and Independent Variables
17.6 Lagrangian Multipliers . . . . . . . . . . . . .
17.7 Variation with Constraints . . . . . . . . . . .
17.8 Rayleigh–Ritz Variational Technique . . . . .
Additional Readings . . . . . . . . . . . . . .

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1037
1038

1044
1052
1056
1058
1060
1065
1072
1076

18 Nonlinear Methods and Chaos
18.1 Introduction . . . . . . . . . . . . . . . . . . . .
18.2 The Logistic Map . . . . . . . . . . . . . . . . .
18.3 Sensitivity to Initial Conditions and Parameters
18.4 Nonlinear Differential Equations . . . . . . . .
Additional Readings . . . . . . . . . . . . . . .

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1079
1079
1080
1085
1088
1107

19 Probability
19.1 Definitions, Simple Properties
19.2 Random Variables . . . . . .
19.3 Binomial Distribution . . . .
19.4 Poisson Distribution . . . . .
19.5 Gauss’ Normal Distribution .
19.6 Statistics . . . . . . . . . . . .
Additional Readings . . . . .
General References . . . . . .

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1109
1109
1116
1128
1130
1134
1138
1150
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1153


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www.elsolucionario.net

PREFACE

Through six editions now, Mathematical Methods for Physicists has provided all the mathematical methods that aspirings scientists and engineers are likely to encounter as students
and beginning researchers. More than enough material is included for a two-semester undergraduate or graduate course.
The book is advanced in the sense that mathematical relations are almost always proven,
in addition to being illustrated in terms of examples. These proofs are not what a mathematician would regard as rigorous, but sketch the ideas and emphasize the relations that
are essential to the study of physics and related fields. This approach incorporates theorems that are usually not cited under the most general assumptions, but are tailored to the
more restricted applications required by physics. For example, Stokes’ theorem is usually
applied by a physicist to a surface with the tacit understanding that it be simply connected.
Such assumptions have been made more explicit.

PROBLEM-SOLVING SKILLS
The book also incorporates a deliberate focus on problem-solving skills. This more advanced level of understanding and active learning is routine in physics courses and requires
practice by the reader. Accordingly, extensive problem sets appearing in each chapter form
an integral part of the book. They have been carefully reviewed, revised and enlarged for
this Sixth Edition.

PATHWAYS THROUGH THE MATERIAL
Undergraduates may be best served if they start by reviewing Chapter 1 according to the
level of training of the class. Section 1.2 on the transformation properties of vectors, the

cross product, and the invariance of the scalar product under rotations may be postponed
until tensor analysis is started, for which these sections form the introduction and serve as
xi


www.elsolucionario.net
xii

Preface
examples. They may continue their studies with linear algebra in Chapter 3, then perhaps
tensors and symmetries (Chapters 2 and 4), and next real and complex analysis (Chapters 5–7), differential equations (Chapters 9, 10), and special functions (Chapters 11–13).
In general, the core of a graduate one-semester course comprises Chapters 5–10 and
11–13, which deal with real and complex analysis, differential equations, and special functions. Depending on the level of the students in a course, some linear algebra in Chapter 3
(eigenvalues, for example), along with symmetries (group theory in Chapter 4), and tensors (Chapter 2) may be covered as needed or according to taste. Group theory may also be
included with differential equations (Chapters 9 and 10). Appropriate relations have been
included and are discussed in Chapters 4 and 9.
A two-semester course can treat tensors, group theory, and special functions (Chapters 11–13) more extensively, and add Fourier series (Chapter 14), integral transforms
(Chapter 15), integral equations (Chapter 16), and the calculus of variations (Chapter 17).

CHANGES TO THE SIXTH EDITION
Improvements to the Sixth Edition have been made in nearly all chapters adding examples
and problems and more derivations of results. Numerous left-over typos caused by scanning into LaTeX, an error-prone process at the rate of many errors per page, have been
corrected along with mistakes, such as in the Dirac γ -matrices in Chapter 3. A few chapters have been relocated. The Gamma function is now in Chapter 8 following Chapters 6
and 7 on complex functions in one variable, as it is an application of these methods. Differential equations are now in Chapters 9 and 10. A new chapter on probability has been
added, as well as new subsections on differential forms and Mathieu functions in response
to persistent demands by readers and students over the years. The new subsections are
more advanced and are written in the concise style of the book, thereby raising its level to
the graduate level. Many examples have been added, for example in Chapters 1 and 2, that
are often used in physics or are standard lore of physics courses. A number of additions
have been made in Chapter 3, such as on linear dependence of vectors, dual vector spaces

and spectral decomposition of symmetric or Hermitian matrices. A subsection on the diffusion equation emphasizes methods to adapt solutions of partial differential equations to
boundary conditions. New formulas have been developed for Hermite polynomials and are
included in Chapter 13 that are useful for treating molecular vibrations; they are of interest
to the chemical physicists.

ACKNOWLEDGMENTS
We have benefited from the advice and help of many people. Some of the revisions are in response to comments by readers and former students, such as Dr. K. Bodoor and J. Hughes.
We are grateful to them and to our Editors Barbara Holland and Tom Singer who organized
accuracy checks. We would like to thank in particular Dr. Michael Bozoian and Prof. Frank
Harris for their invaluable help with the accuracy checking and Simon Crump, Production
Editor, for his expert management of the Sixth Edition.


www.elsolucionario.net

CHAPTER 1

VECTOR ANALYSIS

1.1

DEFINITIONS, ELEMENTARY APPROACH
In science and engineering we frequently encounter quantities that have magnitude and
magnitude only: mass, time, and temperature. These we label scalar quantities, which remain the same no matter what coordinates we use. In contrast, many interesting physical
quantities have magnitude and, in addition, an associated direction. This second group
includes displacement, velocity, acceleration, force, momentum, and angular momentum.
Quantities with magnitude and direction are labeled vector quantities. Usually, in elementary treatments, a vector is defined as a quantity having magnitude and direction. To distinguish vectors from scalars, we identify vector quantities with boldface type, that is, V.
Our vector may be conveniently represented by an arrow, with length proportional to the
magnitude. The direction of the arrow gives the direction of the vector, the positive sense
of direction being indicated by the point. In this representation, vector addition

C=A+B

(1.1)

consists in placing the rear end of vector B at the point of vector A. Vector C is then
represented by an arrow drawn from the rear of A to the point of B. This procedure, the
triangle law of addition, assigns meaning to Eq. (1.1) and is illustrated in Fig. 1.1. By
completing the parallelogram, we see that
C = A + B = B + A,
as shown in Fig. 1.2. In words, vector addition is commutative.
For the sum of three vectors
D = A + B + C,
Fig. 1.3, we may first add A and B:
A + B = E.
1

(1.2)


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2

Chapter 1 Vector Analysis

FIGURE 1.1

Triangle law of vector
addition.

FIGURE 1.2 Parallelogram law of

vector addition.

FIGURE 1.3 Vector addition is
associative.
Then this sum is added to C:
D = E + C.
Similarly, we may first add B and C:
B + C = F.
Then
D = A + F.
In terms of the original expression,
(A + B) + C = A + (B + C).
Vector addition is associative.
A direct physical example of the parallelogram addition law is provided by a weight
suspended by two cords. If the junction point (O in Fig. 1.4) is in equilibrium, the vector


www.elsolucionario.net
1.1 Definitions, Elementary Approach

FIGURE 1.4

3

Equilibrium of forces: F1 + F2 = −F3 .

sum of the two forces F1 and F2 must just cancel the downward force of gravity, F3 . Here
the parallelogram addition law is subject to immediate experimental verification.1
Subtraction may be handled by defining the negative of a vector as a vector of the same
magnitude but with reversed direction. Then

A − B = A + (−B).
In Fig. 1.3,
A = E − B.
Note that the vectors are treated as geometrical objects that are independent of any coordinate system. This concept of independence of a preferred coordinate system is developed
in detail in the next section.
The representation of vector A by an arrow suggests a second possibility. Arrow A
(Fig. 1.5), starting from the origin,2 terminates at the point (Ax , Ay , Az ). Thus, if we agree
that the vector is to start at the origin, the positive end may be specified by giving the
Cartesian coordinates (Ax , Ay , Az ) of the arrowhead.
Although A could have represented any vector quantity (momentum, electric field, etc.),
one particularly important vector quantity, the displacement from the origin to the point
1 Strictly speaking, the parallelogram addition was introduced as a definition. Experiments show that if we assume that the

forces are vector quantities and we combine them by parallelogram addition, the equilibrium condition of zero resultant force is
satisfied.
2 We could start from any point in our Cartesian reference frame; we choose the origin for simplicity. This freedom of shifting
the origin of the coordinate system without affecting the geometry is called translation invariance.


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4

Chapter 1 Vector Analysis

FIGURE 1.5

Cartesian components and direction cosines of A.

(x, y, z), is denoted by the special symbol r. We then have a choice of referring to the displacement as either the vector r or the collection (x, y, z), the coordinates of its endpoint:
r ↔ (x, y, z).

(1.3)

Using r for the magnitude of vector r, we find that Fig. 1.5 shows that the endpoint coordinates and the magnitude are related by
x = r cos α,

y = r cos β,

z = r cos γ .

(1.4)

Here cos α, cos β, and cos γ are called the direction cosines, α being the angle between the
given vector and the positive x-axis, and so on. One further bit of vocabulary: The quantities Ax , Ay , and Az are known as the (Cartesian) components of A or the projections
of A, with cos2 α + cos2 β + cos2 γ = 1.
Thus, any vector A may be resolved into its components (or projected onto the coordinate axes) to yield Ax = A cos α, etc., as in Eq. (1.4). We may choose to refer to the vector
as a single quantity A or to its components (Ax , Ay , Az ). Note that the subscript x in Ax
denotes the x component and not a dependence on the variable x. The choice between
using A or its components (Ax , Ay , Az ) is essentially a choice between a geometric and
an algebraic representation. Use either representation at your convenience. The geometric
“arrow in space” may aid in visualization. The algebraic set of components is usually more
suitable for precise numerical or algebraic calculations.
Vectors enter physics in two distinct forms. (1) Vector A may represent a single force
acting at a single point. The force of gravity acting at the center of gravity illustrates this
form. (2) Vector A may be defined over some extended region; that is, A and its components may be functions of position: Ax = Ax (x, y, z), and so on. Examples of this sort
include the velocity of a fluid varying from point to point over a given volume and electric
and magnetic fields. These two cases may be distinguished by referring to the vector defined over a region as a vector field. The concept of the vector defined over a region and


www.elsolucionario.net

1.1 Definitions, Elementary Approach

5

being a function of position will become extremely important when we differentiate and
integrate vectors.
At this stage it is convenient to introduce unit vectors along each of the coordinate axes.
Let xˆ be a vector of unit magnitude pointing in the positive x-direction, yˆ , a vector of unit
magnitude in the positive y-direction, and zˆ a vector of unit magnitude in the positive zdirection. Then xˆ Ax is a vector with magnitude equal to |Ax | and in the x-direction. By
vector addition,
A = xˆ Ax + yˆ Ay + zˆ Az .

(1.5)

Note that if A vanishes, all of its components must vanish individually; that is, if
A = 0,

then Ax = Ay = Az = 0.

This means that these unit vectors serve as a basis, or complete set of vectors, in the threedimensional Euclidean space in terms of which any vector can be expanded. Thus, Eq. (1.5)
is an assertion that the three unit vectors xˆ , yˆ , and zˆ span our real three-dimensional space:
Any vector may be written as a linear combination of xˆ , yˆ , and zˆ . Since xˆ , yˆ , and zˆ are
linearly independent (no one is a linear combination of the other two), they form a basis
for the real three-dimensional Euclidean space. Finally, by the Pythagorean theorem, the
magnitude of vector A is
|A| = A2x + A2y + A2z

1/2

.

(1.6)

Note that the coordinate unit vectors are not the only complete set, or basis. This resolution
of a vector into its components can be carried out in a variety of coordinate systems, as
shown in Chapter 2. Here we restrict ourselves to Cartesian coordinates, where the unit
vectors have the coordinates xˆ = (1, 0, 0), yˆ = (0, 1, 0) and zˆ = (0, 0, 1) and are all constant
in length and direction, properties characteristic of Cartesian coordinates.
As a replacement of the graphical technique, addition and subtraction of vectors may
now be carried out in terms of their components. For A = xˆ Ax + yˆ Ay + zˆ Az and B =
xˆ Bx + yˆ By + zˆ Bz ,
A ± B = xˆ (Ax ± Bx ) + yˆ (Ay ± By ) + zˆ (Az ± Bz ).

(1.7)

It should be emphasized here that the unit vectors xˆ , yˆ , and zˆ are used for convenience.
They are not essential; we can describe vectors and use them entirely in terms of their
components: A ↔ (Ax , Ay , Az ). This is the approach of the two more powerful, more
sophisticated definitions of vector to be discussed in the next section. However, xˆ , yˆ , and
zˆ emphasize the direction.
So far we have defined the operations of addition and subtraction of vectors. In the next
sections, three varieties of multiplication will be defined on the basis of their applicability:
a scalar, or inner, product, a vector product peculiar to three-dimensional space, and a
direct, or outer, product yielding a second-rank tensor. Division by a vector is not defined.


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6

Chapter 1 Vector Analysis

Exercises
1.1.1

Show how to find A and B, given A + B and A − B.

1.1.2

The vector A whose magnitude is 1.732 units makes equal angles with the coordinate
axes. Find Ax , Ay , and Az .

1.1.3

Calculate the components of a unit vector that lies in the xy-plane and makes equal
angles with the positive directions of the x- and y-axes.

1.1.4

The velocity of sailboat A relative to sailboat B, vrel , is defined by the equation vrel =
vA − vB , where vA is the velocity of A and vB is the velocity of B. Determine the
velocity of A relative to B if
vA = 30 km/hr east
vB = 40 km/hr north.
ANS. vrel = 50 km/hr, 53.1◦ south of east.

1.1.5

A sailboat sails for 1 hr at 4 km/hr (relative to the water) on a steady compass heading
of 40◦ east of north. The sailboat is simultaneously carried along by a current. At the
end of the hour the boat is 6.12 km from its starting point. The line from its starting point

to its location lies 60◦ east of north. Find the x (easterly) and y (northerly) components
of the water’s velocity.
ANS. veast = 2.73 km/hr, vnorth ≈ 0 km/hr.

1.1.6

A vector equation can be reduced to the form A = B. From this show that the one vector
equation is equivalent to three scalar equations. Assuming the validity of Newton’s
second law, F = ma, as a vector equation, this means that ax depends only on Fx and
is independent of Fy and Fz .

1.1.7

The vertices A, B, and C of a triangle are given by the points (−1, 0, 2), (0, 1, 0), and
(1, −1, 0), respectively. Find point D so that the figure ABCD forms a plane parallelogram.
ANS. (0, −2, 2) or (2, 0, −2).

1.1.8

A triangle is defined by the vertices of three vectors A, B and C that extend from the
origin. In terms of A, B, and C show that the vector sum of the successive sides of the
triangle (AB + BC + CA) is zero, where the side AB is from A to B, etc.

1.1.9

A sphere of radius a is centered at a point r1 .
(a) Write out the algebraic equation for the sphere.
(b) Write out a vector equation for the sphere.
ANS.

(a) (x − x1 )2 + (y − y1 )2 + (z − z1 )2 = a 2 .
(b) r = r1 + a, with r1 = center.
(a takes on all directions but has a fixed magnitude a.)


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1.2 Rotation of the Coordinate Axes

7

1.1.10

A corner reflector is formed by three mutually perpendicular reflecting surfaces. Show
that a ray of light incident upon the corner reflector (striking all three surfaces) is reflected back along a line parallel to the line of incidence.
Hint. Consider the effect of a reflection on the components of a vector describing the
direction of the light ray.

1.1.11

Hubble’s law. Hubble found that distant galaxies are receding with a velocity proportional to their distance from where we are on Earth. For the ith galaxy,
vi = H0 ri ,
with us at the origin. Show that this recession of the galaxies from us does not imply
that we are at the center of the universe. Specifically, take the galaxy at r1 as a new
origin and show that Hubble’s law is still obeyed.

1.1.12

1.2

Find the diagonal vectors of a unit cube with one corner at the origin and its three sides

lying
√ along Cartesian coordinates axes. Show that there are four diagonals with length
3. Representing these as vectors,
√ what are their components? Show that the diagonals
of the cube’s faces have length 2 and determine their components.

ROTATION OF THE COORDINATE AXES3
In the preceding section vectors were defined or represented in two equivalent ways:
(1) geometrically by specifying magnitude and direction, as with an arrow, and (2) algebraically by specifying the components relative to Cartesian coordinate axes. The second definition is adequate for the vector analysis of this chapter. In this section two more
refined, sophisticated, and powerful definitions are presented. First, the vector field is defined in terms of the behavior of its components under rotation of the coordinate axes. This
transformation theory approach leads into the tensor analysis of Chapter 2 and groups of
transformations in Chapter 4. Second, the component definition of Section 1.1 is refined
and generalized according to the mathematician’s concepts of vector and vector space. This
approach leads to function spaces, including the Hilbert space.
The definition of vector as a quantity with magnitude and direction is incomplete. On
the one hand, we encounter quantities, such as elastic constants and index of refraction
in anisotropic crystals, that have magnitude and direction but that are not vectors. On
the other hand, our naïve approach is awkward to generalize to extend to more complex
quantities. We seek a new definition of vector field using our coordinate vector r as a
prototype.
There is a physical basis for our development of a new definition. We describe our physical world by mathematics, but it and any physical predictions we may make must be
independent of our mathematical conventions.
In our specific case we assume that space is isotropic; that is, there is no preferred direction, or all directions are equivalent. Then the physical system being analyzed or the
physical law being enunciated cannot and must not depend on our choice or orientation
of the coordinate axes. Specifically, if a quantity S does not depend on the orientation of
the coordinate axes, it is called a scalar.

3 This section is optional here. It will be essential for Chapter 2.

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8

Chapter 1 Vector Analysis

FIGURE 1.6

Rotation of Cartesian coordinate axes about the z-axis.

Now we return to the concept of vector r as a geometric object independent of the
coordinate system. Let us look at r in two different systems, one rotated in relation to the
other.
For simplicity we consider first the two-dimensional case. If the x-, y-coordinates are
rotated counterclockwise through an angle ϕ, keeping r, fixed (Fig. 1.6), we get the following relations between the components resolved in the original system (unprimed) and
those resolved in the new rotated system (primed):
x ′ = x cos ϕ + y sin ϕ,
y ′ = −x sin ϕ + y cos ϕ.

(1.8)

We saw in Section 1.1 that a vector could be represented by the coordinates of a point;
that is, the coordinates were proportional to the vector components. Hence the components
of a vector must transform under rotation as coordinates of a point (such as r). Therefore
whenever any pair of quantities Ax and Ay in the xy-coordinate system is transformed into
(A′x , A′y ) by this rotation of the coordinate system with
A′x = Ax cos ϕ + Ay sin ϕ,
A′y = −Ax sin ϕ + Ay cos ϕ,

(1.9)

we define4 Ax and Ay as the components of a vector A. Our vector now is defined in terms
of the transformation of its components under rotation of the coordinate system. If Ax and
Ay transform in the same way as x and y, the components of the general two-dimensional
coordinate vector r, they are the components of a vector A. If Ax and Ay do not show this
4 A scalar quantity does not depend on the orientation of coordinates; S ′ = S expresses the fact that it is invariant under rotation

of the coordinates.