Hassani S. Mathematical Methods: For Students of Physics and Related Fields

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CONTENTS xxiii

31.3UniversalityofChaos........................773

31.3.1 Feigenbaum Numbers . . . . . . . . . . . . . . . . . . . 773

31.3.2 Fractal Dimension . . . . . . . . . . . . . . . . . . . . . 775

31.4Problems ..............................778

32 Probability Theory 781

32.1BasicConcepts ...........................781

32.1.1 ASetTheoryPrimer....................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

32.3PoissonDistribution ........................797

32.4 Continuous Random Variable . . . . . . . . . . . . . . . . . . . 801

32.4.1 TransformationofVariables................804

32.4.2 NormalDistribution ....................806

32.5Problems ..............................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

chapteristoreviewtheordinary 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 vectors

are position r, displacement Δr,

velocity v,momentump, 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

Vectors will be denoted by Roman letters printed in boldface type.

4 Coordinate Systems and Vectors

+ a

+ b

Δr

ΔR

Δr

(a) (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

lyingalongthesamelineas 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, Δr

from

A to B is added to the displacement Δr

from B to C to give ΔR their

resultant, or their sum, i.e., the displacement from A to C:Δr

+Δr

=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.

If the components of

the points P and Q in Figure 1.2 are, respectively, (p

)and(q

), 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,namelyq − p.Thus,

for some real number t,wemusthavevector form of the

parametric

equation of a line

x − p = t(q − p)orx = 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 =(q

− p

)t + p

y =(q

− p

)t + p

z =(q

− p

)t + p

which is the usual parametric equation for a line.



We shall discuss components and coordinates in greater detail later in this chapter. For

now, the knowledge gained in calculus is suﬃcient for our discussion.

1.1 Vectors in a Plane and in Space 5

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

e with a subscript which

designates its direction. Thus, if we divided the vector a by its length |a| we

get the unit vector

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

, we can construct the vector: a = |a|

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

,and

are the unit vectors pointing in the positive

directions of the x-, y-, and z-axes, respectively.

We shall introduce unit

vectors in other coordinate systems whenwediscussthosecoordinatesystems

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,andθ is the angle between

the two vectors. This deﬁnition is motivated by many physical situations.

These unit vectors are usually denoted by i, j,andk, 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.

6 Coordinate Systems and Vectors

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,itdis-

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 demonstrates

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 adot(or

inner) product on a vector space.

The deﬁnition of the dot product leads directly to a · a = |a|

|a| =

√

a ·a, (1.2)

whichisusefulincalculatingthelength of sums or diﬀerences of vectors.

Figure 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.

1.1 Vectors in a Plane and in Space 7

a+b

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 (a

)and(b

) represent the components of a and b along the

axes x, y,andz,then

dot product in

terms of

components

a ·b = a

+ a

. (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|,weget

a ·b

|a|

= |b|cos θ or



|a|



· b = |b|cos θ ⇒

· b = |b|cos θ.

Noting that |b|cos θ is simply the projection of b along a, we conclude

ausefulrelationto

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

, 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:Ifa 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,

8 Coordinate Systems and Vectors

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.

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,andz 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 =(α

+ α

) ×(β

+ β

)

= α

+ α

But, by the last property of Equation (1.4), we have

=0.

Also, if we assume that

,and

form a so-called right-handed set,

i.e., if

right-handed set

of unit vectors

= −

, (1.5)

= −

then we obtain

a ×b =(α

− α

)

+(α

− α

)

+(α

− α

)

This 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.

1.1 Vectors in a Plane and in Space 9

det

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 form

cross product in

terms of the

determinant of

components

a ×b =det

⎛

⎝

⎞

⎠

. (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

⎛

⎝

11−2

20 3

⎞

⎠

− 7

− 2

and then computing its length

|a × b| =



+(−7)

+(−2)

√

62.



|a| cos θ

|a| sin 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.

No 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.

10 Coordinate Systems and Vectors

Example 1.1.3. The volume of a parallelepiped deﬁned by three non-coplanar

vectors, a, b,andc,isgivenby|a ·(b × c)|. This can be seen from Figure 1.6(b),

where it is clear thatvolume 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)

+(b × c)

= |(β

− β

)α

+(β

− β

)α

+(β

− β

)α

which can be written in determinant form asvolume of a

parallelepiped as

the determinant of

the components of

its side vectors

volume = |a · (b × c)| =



det

⎛

⎝

⎞

⎠



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 pair

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.

See Chapter 18.