Table Of ContentEssentials of Math Methods
for Physicists
Hans Weber
and
George B. Arfken
ACADEMIC PRESS
^n imprint of Elsevier Science
Amsterdam Boston London New York Oxford Paris
San Diego San Francisco Singapore Sydney Tokyo
PREFACE
This text is designed to accompany the usual introductory physics courses and
prepare undergraduate students for the level of mathematics expected in more
advanced undergraduate physics and engineering courses. One of its goals is
to guide the student in learning the mathematical language physicists use by
leading them through worked examples and then practicing problems. The
pedagogy is that of introducing concepts, designing and refining methods and
practice them repeatedly in physics examples and problems. Geometric and
algebraic approaches and methods are included and are more or less emphasized
in a variety of settings to accommodate different learning styles of students.
Sometimes examples are solved in more than one way. Theorems are usually
derived sketching the underlying ideas and describing the relevant mathematical
relations, so that one can recognize the assumptions they are based on and
their limitations. These proofs are not rigorous in the sense of the professional
mathematician, and no attempt was made to formulate theorems in their most
general form or under the least restrictive assumptions.
An important objective of this text is to train the student to formulate physi
cal phenomena in mathematical language, starting from intuitive and qualitative
ideas. The examples in the text have been worked out so as to develop the math
ematical treatment along with the physical intuition. A precise mathematical
formulation of physical phenomena and problems is always the ultimate goal.
Text Overview
In Chapter 1 the basic concepts of vector algebra and vector analysis are in
troduced and applied to classical mechanics and electrodynamics. The second
chapter deals with the extension of vector algebra and analysis to curved or
thogonal coordinates, again with applications from classical mechanics and elec
trodynamics. These chapters lay the foundations for differential equations in
Chapters 8, 9 and 16, variational calculus in Chapter 18 and nonlinear analy
sis in Chapter 19. Chapter 3 extends high school algebra of one or two linear
equations to determinants and matrix solutions of general systems of linear equa
tions, eigenvalues and eigenvectors, linear transformations in real and complex
vector spaces. These chapters are extended to function spaces of solutions of
differential equations in Chapter 9, thereby laying the mathematical foundations
7
8
for and formulation of quantum mechanics. Chapter 5 gives a fairly extensive
treatment of series which forms the basis for the special functions Chapters 10-
13, and also complex functions in Chapters 6-7. Chapter 17 on probability and
statistics is basic for the experimentally oriented physicist. Some of its content
can be studied right after Chapters 1,2, but later sections are based on Chapters
8 and 10. Chapter 19 on non-linear methods can be studied right after Chapter
8 complementing and extending Chapter 8 in many directions. Chapters 10-13
on special functions contain many examples of physics problems requiring solu
tions of differential equations that can also be incorporated in Chapters 8 and
16. Chapters 14-15 on Fourier analysis are indispensible for a more advanced
treatment of partial differential equations in Chapter 16.
Historical remarks are included that describe some physicists and mathe
maticians who introduced the ideas and methods which later generations per
fected to the tools we now use routinely. We hope they provide motivation for
students and generate some appreciation of the efforts, devotion, and courage
of past and present scientists.
Pathways through the Material
As the text contains more than enough material for a two-semester undergrad
uate course, an opportunity is provided for the instructor to select topics to
suit the particular level of the class. Chapters 1-3 and 5-8 provide a basis for
a one-semester course in mathematical physics. By omitting some topics, such
as symmetries and group theory and tensors, it is possible, in a one-semester
course, to include also parts of Chapters 10-13 on special functions, or Chap
ter 14-15 on Fourier analysis, or Chapter 17 on probability and statistics, or
Chapter 18 on variational calculus or Chapter 19 on non-linear methods.
A two-semester course can treat tensors and symmetries in Chapters 2 and 4,
and special functions in Chapters 10-13 more extensively, as well as variational
calculus of Chapter 18 in support of classical and quantum mechanics.
Problem-Solving Skills
Students should study the text until they are sure they understand the physical
interpretation, can derive equations with the book closed, make predictions in
special cases and recognize the limits of applicability of the theories and equa
tions of physics. However, physics and engineering courses routinely demand
an even higher level of understanding involving active learning where students
can apply the material to solve problems, because it is common knowledge that
we only learn the mathematical language physicists use by practising it solving
problems repeatedly.
The problem sets at the end of sections and chapters are arranged in the
order in which the material is covered in the text. A sufficient variety and
level of difficulty of problems has been aimed for to assure that anyone who
9
conscientiously solves them has mastered the material in the text beyond mere
understanding of step-by-step derivations. More difficult problems that require
some modification of routine methods are also included in various sets to en
gage the creative powers of the student, a skill that later is expected of the
professional physicist.
Computer Software
Those problems in the text that can be solved analytically can also be solved
by modern symbolic computer software, such as Macsyma, Mathcad, Maples,
Mathematica, Reduce, and others, because these programs include the routine
methods of mathematical physics texts. Once the student has developed an
analytical result these powerful programs are useful for checking and plotting
the results. Finding an analytical solution by computer without understanding
how it is derived is pointless. When computers are used too early in solving a
problem, many instructors have found that students can be led astray by the
computer in many directions. The available computer software is so diverse as
to preclude any detailed discussion of it. Each instructor willing to make use
of computers in the course will have to make a choice of a particular software
and provide an introduction for the students. Many problems and examples in
the text may then be adapted to it. However, their real utility and power lies
in the graphics software they include and the ability to solve those problems
approximately and numerically that do not allow for an analytical solution.
Special training is needed and can be provided using the text to train students
in approximation methods, such as series and asymptotic expansions, or integral
representations that are suitable for further symbolic computer manipulations.
Acknowledgements
Many reviewers deserve special praise, Frank Harris from Florida in particular,
although not all of their advice was always followed. We are particularly grateful
to our untiring editor, Barbara Holland, for much help with them. Despite great
care we expect some misprints and errors to have survived the collective scrutiny
of reviewers, proof readers, editors and authors. While we are grateful for their
efforts we assume responsibility and welcome readers to point them out to us.
Chapter 1
VECTOR ANALYSIS
1.1 ELEMENTARY APPROACH
In science and engineering we frequently encounter quantities that have only
magnitude: mass, time, and temperature. This magnitude remains the same,
no matter how we orient the coordinate axes that we may use. These quanti
ties we label scalar quantities. In contrast, many interesting physical quanti
ties have magnitude or length and, in addition, an associated direction. Quan
tities with magnitude and direction are labeled vectors. Their length and the
angle between any vectors remain unaffected by the orientation of coordinates
we may choose. To distinguish vectors from scalars, we identify vector quan
tities with boldface type, that is, V. Vectors are useful in solving systems of
linear equations (Chapter 3). They are not only helpful in Euclidean geom
etry but indispensable in classical mechanics and engineering because force,
velocity, acceleration and angular momentum are vectors. Electrodynamics is
unthinkable without vector fields like the electric and magnetic fields.
Practical problems of mechanics and geometry like looking for the shortest
distance between straight lines or parameterizing the orbit of a particle will
lead us to the differentiation of vectors and to vector analysis. Vector analysis
is a powerful tool to formulate equations of motions of particles and then solve
them in mechanics and engineering, or field equations of electrodynamics.
In this Section 1.1 we learn to add and subtract vectors geometrically and
algebraically in terms of their rectangular components.
A vector may be geometrically represented by an arrow with length pro
portional 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 (head to
tail rule). Vector C is then represented by an arrow drawn from the rear of A
1
2 CHAPTER 1. VECTOR ANALYSIS
Figure 1.1: Triangle law of vector addition.
to the point of B. This procedure, the triangle law of addition, assigns mean
ing to Eq. (1.1) and is illustrated in Fig. 1.1. By completing the parallelogram
(sketch it!), we see that
C = A + B = B + A. (1.2)
In words, vector addition is commutative.
For the sum of three vectors
D = A + B + C,
illustrated in Fig. 1.2, we first add A and B,
A + B = E.
Then this sum is added to C,
D = E + C.
Alternatively, 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),
so that these alternative ways of summing three vectors lead to the same
vector, or vector addition is associative.
A direct physical example of the parallelogram addition law is provided
by a weight suspended by two cords in Fig. 1.3. If the junction point is in
equilibrium, the vector sum of the two forces Fi and F must cancel the down
2
ward force of gravity, F . Here the parallelogram addition law is subject to
3
immediate experimental verification.1 Such a balance of forces is of immense
importance for the stability of buildings, bridges, airplanes in flight, etc.
1 Strictly speaking the parallelogram addition was introduced as a definition. Experi
ments show that forces are vector quantities that are combined by parallelogram addition,
as required by the equilibrium condition of zero resultant force.
1.1. ELEMENTARY APPROACH
Figure 1.2: Vector addition is associative.
4^-
/.V rfß/e
Figure 1.3: Equilibrium of forces. Fi 4- F = -F
2 3
Subtraction is handled by defining the negative of a vector as a vector of
the same magnitude but with reversed direction. Then
A -B = A + (-B).
The graphical representation of vector A by an arrow suggests using coor
dinates as a second possibility. Arrow A (Fig. 1.4), starting from the origin,2
terminates at the point (A A,A). Thus, if we agree that the vector is to
Xi y z
start at the origin, the positive end may be specified by giving the rectangu
lar or Cartesian coordinates (A A, A) of the arrow head, its components.
Xi y z
Although A could have represented any vector quantity (momentum, elec
tric field, etc.), one particularly important vector quantity, the distance from
the origin to the point (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 end point:
r o (x,y,z). (1.3)
Defining the magnitude r of vector r as its geometrical length, we find that
Fig. 1.4 shows that the end point coordinates and the magnitude are related
2We could start from any point; we choose the origin for simplicity. This freedom
of shifting the origin of the coordinate system without affecting the geometry is called
translation invariance.
CHAPTER 1. VECTOR ANALYSIS
(A ,A ,A )
x x y τ'
(A,A,0)
x y
Figure 1.4: Components and direction cosines of A.
by
x = rcosa, y = rcosß, z = rcosj. (1.4)
cosa, cos/3, and cos7 are called the direction cosines, a being the angle
between the given vector and the positive x-axis, and so on. The (Cartesian)
components A,A, and A can also be viewed as the projections of A on
x y z
the respective axes.
Thus, any vector A may be resolved into its components (or projected
onto the coordinate axes) to yield A = A cos a, etc., as in Eq. (1.4). We refer
x
to the vector as a single quantity A or to its components (A ,A,A). Note
x y z
that the subscript x in A denotes the x component and not a dependence on
x
the variable x. The choice between using A or its components (A , A A ) is
x yi 2
essentially a choice between a geometric or an algebraic representation.
The geometric "arrow in space" often aids in visualization. The algebraic
set of components is usually more suitable for precise numerical or algebraic
calculations. (This is illustrated in Examples 1.1.1-3 and applies also to the
Exercises 1.1.1, 1.1.3, 1.1.5 and 1.1.6 below.)
Vectors enter physics in two distinct forms.
• 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.
• Vector A may be defined over some extended region; that is, A and its
components may be functions of position: A = A(xyz), and so on.
x x ì ì
Imagine a vector A attached to each point (x, y, 2), whose length and direction
change with position. Examples include the velocity of air around the wing of
a plane in flight varying from point to point and electric and magnetic fields
(made visible by iron filings). Thus, vectors defined at each point of a region
are usually characterized as a vector field. The concept of the vector defined
over a region and being a function of position will be extremely important in
Section 1.2 and in later sections where we differentiate and integrate vectors.
A unit vector has length 1 and may point in any direction. Coordinate
unit vectors are implicit in the projection of A onto the cordinate axes to
--'*.
Û
/■3
(ΊΓΙΑΛΛ^
1.1. ELEMENTARY APPROACH 5
define its Cartesian components. Now we define x explicitly as a vector of unit
magnitude pointing in the positive x-direction, y, a vector of unit magnitude
in the positive ^-direction, and z a vector of unit magnitude in the positive
z-direction. Then xA is a vector with magnitude equal to A and in the
x x
positive x-direction, that is, the projection of A onto the x-direction, etc. By
vector addition
A = xA + y A + zA (1.5)
x y Zi
which states that a vector equals the vector sum of its components or projec
tions. Note that, if A vanishes, all of its components must vanish individually;
that is, if
A = 0, then A - A = A = 0.
x y z
Finally, by the Pythagorean theorem, the length of vector A is
A=(Al + Al + Al)1/2 (1.6)
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 x =
(l,0,0),y = (0,l,0)andz = (0,0,l).
Equation (1.5) means that the three unit vectors x, y, and z span the
real three-dimensional space: Any constant vector may be written as a linear
combination of x, y, and z. Since x, y, and z are linearly independent (no
one is a linear combination of the other two), they form a basis for the real
three-dimensional space.
Complementary to the geometrical technique, algebraic addition and sub
traction of vectors may now be carried out in terms of their components.
For
A = StA + y Ay + zA and B = xB + yB + zB,
x z x y z
A ± B = x(A ± B) + y(A ± B) + z(A ± B),, (1.7)
x x y y z z :
Biographical Data
Descartes, René. French mathematician and philosopher, born in La
Haye on March 31, 1596, died in Stockholm, Sweden on November 2, 1650.
Cartesius is the latinized version of his name at a time when Latin was the
language of sciences, although he mainly wrote in French. He discovered his
love of mathematics in the army with plenty of time for research. He intro
duced the concept of rectangular coordinates thereby converting geometry to
algebraic equations of the coordinates of points, now called analytic geome
try. Thus he paved the way for Newton's and Leibniz's calculus. He coined:
"Cogito, ergo sum, " which translates to "I think, therefore I am. "
