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cannot learn a language by just studying a textbook. In order to truly learn how to use a

language one has to go abroad and start using a language. By the same token one cannot

learn how to use mathematics in physics by just studying textbooks or attending lectures,

the only way to achieve this is to venture into the unknown and apply mathematics to

physical problems.

It is the goal of this course to do exactly that a number of problems is presented

in order to apply mathematical techniques and knowledge to physical concepts. These

examples are not presented as well-developed theory. Instead, these examples are presented

as a number of problems that elucidate the issues that are at stake. In this sense this

book o ers a guided tour material for learning is presented but true learning will only

take place by active exploration.

Since this book is written as a set of problems you may frequently want to consult

other material to refresh or deepen your understanding of material. In many places we

will refer to the book of Boas 11]. In addition, the books of Butkov 14] and Arfken 3] are

excellent. When you are a physics of geophysics student you should seriously consider

buying a comprehensive textbook on mathematical physics, it will be of great bene t to

you.

In addition to books, colleagues either in the same eld or other elds can be a great

source of knowledge and understanding. Therefore, don't hesitate to work together with

others on these problems if you are in the fortunate positions to do so. This may not

only make the work more enjoyable, it may also help you in getting \unstuck" at di cult

moments and the di erent viewpoints of others may help to deepen yours.

This book is set up with the goal of obtaining a good working knowledge of mathe-

matical geophysics that is needed for students in physics or geophysics. A certain basic

knowledge of calculus and linear algebra is needed for digesting the material presented

here. For this reason, this book is meant for upper-level undergraduate students or lower-

level graduate students, depending on the background and skill of the student.

5

CHAPTER 1. INTRODUCTION

6

At this point the book is still under construction. New sections are reg-

ularly added, and both corrections and improvements will be made. If you

are interested in this material therefore regularly check the latest version at

Samizdat Press. The feedback of both teachers and students who use this

material is vital in improving this manuscript, please send you remarks to:

Roel Snieder

Dept. of Geophysics

Utrecht University

P.O. Box 80.021

3508 TA Utrecht

The Netherlands

telephone: +31-30-253.50.87

fax: +31-30-253.34.86

email: snieder@geo.uu.nl

Acknowledgements: This manuscript has been prepared with the help of a large number

of people. The feedback of John Scales and his proposal to make this manuscript available

via internet is very much appreciated. Barbara McLenon skillfully drafted the gures.

The patience of Joop Hoofd, John Stockwell and Everhard Muyzert in dealing with my

computer illiteracy is very much appreciated. Numerous students have made valuable

comments for improvements of the text. The input of Huub Douma in correcting errors

and improving the style of presentation is very much appreciated.

Chapter 2

Summation of series

2.1 The Taylor series

In many applications in mathematical physics it is extremely useful to write the quantity

of interest as a sum of a large number of terms. To x our mind, let us consider the

motion of a particle that moves along a line as time progresses. The motion is completely

described by giving the position x(t) of the particle as a function of time. Consider the

four di erent types of motion that are shown in gure 2.1.

constant constant constant variable

position velocity acceleration acceleration

x(t) x(t) x(t) x(t)

(a) (b) (c) (d)

t t t t

Figure 2.1: Four di erent kinds of motion of a particle along a line as a function of time.

The simplest motion is a particle that does not move, this is shown in panel (a). In

this case the position of the particle is constant:

x(t) = x0 : (2.1)

The value of the parameter x0 follows by setting t = 0, this immediately gives that

x0 = x (0) : (2.2)

In panel (b) the situation is shown of a particle that moves with a constant velocity, in

that case the position is a linear function of time:

x(t) = x0 + v0 t : (2.3)

Again, setting t = 0 gives the parameter x0 , which is given again by (2.2). The value of

the parameter v0 follows by di erentiating (2.3) with respect to time and by setting t = 0.

7

CHAPTER 2. SUMMATION OF SERIES

8

Problem a: Do this and show that

v0 = dx (t = 0) : (2.4)

dt

This expression re ects that the velocity v0 is given by the time-derivative of the position.

Next, consider a particle moving with a constant acceleration a0 as shown in panel (c).

As you probably know from classical mechanics the motion is in that case a quadratic

function of time:

1

x(t) = x0 + v0 t + 2 a0t2 : (2.5)

Problem b: Evaluate this expression at t = 0 to show that x0 is given by (2.2). Di er-

entiate (2.5) once with respect to time and evaluate the result at t = 0 to show that

v0 is again given by (2.4). Di erentiate (2.5) twice with respect to time, set t = 0

to show that a0 is given by:

d2 x (t = 0) :

a0 = dt2 (2.6)

This result re ects the fact that the acelleration is the second derivative of the

position with respect to time.

Let us now consider the motion shown in panel (d) where the acceleration changes

with time. In that case the displacement as a function of time is not a linear function of

time (as in (2.3) for the case of a constant velocity) nor is it a quadratic function of time

(as in (2.5) for the case of a constant acceleration). Instead, the displacement is in general

a function of all possible powers in t:

1

X

x(t) = c0 + c1t + c2 t2 + cn tn :

= (2.7)

n=0

This series, where a function is expressed as a sum of terms with increasing powers of the

independent variable, is called a Taylor series. At this point we do not know what the

constants cn are. These coe cients can be found in exactly the same way as in problem

b where you determined the coe cients a0 and v0 in the expansion (2.5).

Problem c: Determine the coe cient cm by di erentiating expression (2.7) m-times with

respect to t and by evaluating the result at t = 0 to show that

1 dm x (x = 0) :

c= (2.8)

m m! dtm

Of course there is no reason why the Taylor series can only be used to describe the

displacement x(t) as a function of time t. In the literature, one frequently uses the Taylor

series do describe a function f(x) that depends on x. Of course it is immaterial how we

call a function. By making the replacements x f and t x the expressions (2.7) and

! !

(2.8) can also be written as:

1

Xn

f(x) = cnx (2.9)

n=0

2.1. THE TAYLOR SERIES 9

with

1 dn f (x = 0) :

cn = n! dxn (2.10)

You may nd this result in the literature also be written as

1

X xn dnf df (x = 0) + 1 d2 f (x = 0) +

f(x) = n! dxn (x = 0) = f(0) + x dx (2.11)

2 dx2

n=0

Problem d: Show by evaluating the derivatives of f (x) at x = 0 that the Taylor series

of the following functions are given by:

1 1

sin (x) = x 3! x3 + 5! x5 (2.12)

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