guide to geophysical equations

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A Students Guide to Geophysical Equations

The advent of accessible student computing packages has meant that geophysics students

can now easily manipulate datasets and gain rst-hand modeling experience essential

in developing an intuitive understanding of the physics of the Earth. Yet to gain a more

in-depth understanding of the physical theory, and to be able to develop new models and

solutions, it is necessary to be able to derive the relevant equations from rst principles.

This compact, handy book lls a gap left by most modern geophysics textbooks,

which generally do not have space to derive all of the important formulae, showing the

intermediate steps. This guide presents full derivations for the classical equations of

gravitation, gravity, tides, Earth rotation, heat, geomagnetism, and foundational seismol-

ogy, illustrated with simple schematic diagrams. It supports students through the suc-

cessive steps and explains the logical sequence of a derivation facilitating self-study

and helping students to tackle homework exercises and prepare for exams.

w i l l i a m l o w r i e was born in Hawick, Scotland, and attended the University of

Edinburgh, where he graduated in 1960 with rst-class honors in physics. He achieved a

masters degree in geophysics at the University of Toronto and, in 1967, a doctorate at the

University of Pittsburgh. After two years in the research laboratory of Gulf Oil Company

he became a researcher at the Lamont-Doherty Geological Observatory of Columbia

University. In 1974 he was elected professor of geophysics at the ETH Zrich (Swiss

Federal Institute of Technology in Zurich), Switzerland, where he taught and researched

until retirement in 2004. His research in rock magnetism and paleomagnetism consisted

of deducing the Earths magnetic eld in the geological past from the magnetizations of

dated rocks. The results were applied to the solution of geologic-tectonic problems, and

to analysis of the polarity history of the geomagnetic eld. Professor Lowrie has authored

135 scientic articles and a second edition of his acclaimed 1997 textbookFundamentals

of Geophysicswas published in 2007. He has been President of the European Union of

Geosciences (19879) and Section President and Council member of the American

Geophysical Union (20002). He is a Fellow of the American Geophysical Union and

a Member of the Academia Europaea.


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A Students Guide

to Geophysical Equations

W I L L I A M L O W R I E

Institute of Geophysics

Swiss Federal Institute of Technology

Zurich, Switzerland


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c a m b r i d g e u n i v e r s i t y p r e s s

Cambridge, New York, Melbourne, Madrid, Cape Town,

Singapore, So Paulo, Delhi, 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/9781107005846

William Lowrie 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 catalogue record for this publication is available from the British Library

Library of Congress Cataloguing in Publication data

Lowrie, William, 1939

A students guide to geophysical equations / William Lowrie.

p. cm.

Includes bibliographical references and index.

ISBN 978-1-107-00584-6 (hardback)1. Geophysics Mathematics Handbooks, manuals, etc.

2. Physics Formulae Handbooks, manuals, etc. 3. Earth Handbooks, manuals, etc.

I. Title.

QC809.M37L69 2011

550.1051525dc22

2011007352

ISBN 978 1 107 00584 6 Hardback

ISBN 978 0 521 18377 2 Paperback

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.
http://www.cambridge.org/http://www.cambridge.org/9781107005846http://www.cambridge.org/9781107005846http://www.cambridge.org/


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This book is dedicated to Marcia


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Contents

Preface pagexi

Acknowledgments xiii

1 Mathematical background 1

1.1 Cartesian and spherical coordinates 1

1.2 Complex numbers 1

1.3 Vector relationships 4

1.4 Matrices and tensors 8

1.5 Conservative force,eld, and potential 17

1.6 The divergence theorem (Gausss theorem) 18

1.7 The curl theorem (Stokestheorem) 20

1.8 Poissons equation 23

1.9 Laplaces equation 26

1.10 Power series 28

1.11 Leibnizs rule 32

1.12 Legendre polynomials 32

1.13 The Legendre differential equation 34

1.14 Rodriguesformula 41

1.15 Associated Legendre polynomials 43

1.16 Spherical harmonic functions 491.17 Fourier series, Fourier integrals, and Fourier transforms 52

Further reading 58

2 Gravitation 59

2.1 Gravitational acceleration and potential 59

2.2 Keplers laws of planetary motion 60

2.3 Gravitational acceleration and the potential of a solid

sphere 66

2.4 Laplaces equation in spherical polar coordinates 69

2.5 MacCullaghs formula for the gravitational potential 74

Further reading 85

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3 Gravity 86

3.1 The ellipticity of the Earths gure 86

3.2 The geopotential 88

3.3 The equipotential surface of gravity 91

3.4 Gravity on the reference spheroid 96

3.5 Geocentric and geographic latitude 102

3.6 The geoid 106

Further reading 115

4 The tides 116

4.1 Origin of the lunar tide-raising forces 116

4.2 Tidal potential of the Moon 119

4.3 Loves numbers and the tidal deformation 1244.4 Tidal friction and deceleration of terrestrial and lunar

rotations 130

Further reading 136

5 Earths rotation 137

5.1 Motion in a rotating coordinate system 138

5.2 The Coriolis and Etvs effects 140

5.3 Precession and forced nutation of Earths rotation axis 142

5.4 The free, Eulerian nutation of a rigid Earth 1555.5 The Chandler wobble 157

Further reading 169

6 Earths heat 170

6.1 Energy and entropy 171

6.2 Thermodynamic potentials and Maxwells relations 172

6.3 The melting-temperature gradient in the core 176

6.4 The adiabatic temperature gradient in the core 178

6.5 The Grneisen parameter 179

6.6 Heatow 182

Further reading 197

7 Geomagnetism 198

7.1 The dipole magnetic eld and potential 198

7.2 Potential of the geomagnetic eld 200

7.3 The Earths dipole magnetic eld 205

7.4 Secular variation 213

7.5 Power spectrum of the internal eld 214

7.6 The origin of the internal eld 217

Further reading 225

8 Foundations of seismology 227

8.1 Elastic deformation 227

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8.2 Stress 228

8.3 Strain 233

8.4 Perfectly elastic stressstrain relationships 239

8.5 The seismic wave equation 244

8.6 Solutions of the wave equation 252

8.7 Three-dimensional propagation of plane P- and S-waves 254

Further reading 258

Appendix A Magnetic poles, the dipoleeld, and current loops 259

Appendix B Maxwells equations of electromagnetism 265

References 276

Index 278

Contents ix


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Preface

This work was written as a supplementary text to help students understand the

mathematical steps in deriving important equations in classical geophysics. It is

not intended to be a primary textbook, nor is it intended to be an introduction to

modern research in any of the topics it covers. It originated in a set of handouts, a

kindof do-it-yourself manual, that accompanied a course I taught on theoretical

geophysics. The lecture aids were necessary for two reasons. First, my lectures

were given in German and there were no comprehensive up-to-date texts in the

language; the recommended texts were in English, so the students frequentlyneeded clarication. Secondly, it was often necessary to explain classical theory

in more detail than one nds in a multi-topic advanced textbook. To keep such a

book as succinct as possible, the intermediate steps in the mathematical derivation

of a formula must often be omitted. Sometimes the unassisted student cannotll

in the missing steps without individual tutorial assistance, which is usually in

short supply at most universities, especially at large institutions. To help my

students in these situations, the do-it-yourself text that accompanied my lec-

tures explained missing details in the derivations. This is the background against

which I prepared the present guide to geophysical equations, in the hope that it

might be helpful to other students at this level of study.

The classes that I taught to senior grades were largely related to potential

theory and primarily covered topics other than seismology, since this was the

domain of my colleagues and better taught by a true seismologist than by a

paleomagnetist! Theoretical seismology is a large topic that merits its own

treatment at an advanced level, and there are several textbooks of classical

and modern vintage that deal with this. However, a short chapter on the

relationship of stress, strain, and the propagation of seismic waves is included

here as an introduction to the topic.

Computer technology is an essential ingredient of progress in modern geo-

physics, but a well-trained aspiring geophysicist must be able to do more than

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apply advanced software packages. A fundamental mathematical understanding

is needed in order to formulate a geophysical problem, and numerical computa-

tional skills are needed to solve it. The techniques that enabled scientists to

understand much about the Earth in the pre-computer era also underlie much of

modern methodology. For this reason, a university training in geophysics still

requires the student to work through basic theory. This guide is intended as a

companion in that process.

Historically, most geophysicists came from the eld of physics, for which

geophysics was an applied science. They generally had a sound training in

mathematics. The modern geophysics student is more likely to have begun

studies in an Earth science discipline, the mathematical background might be

heavily oriented to the use of tailor-made packaged software, and some studentsmay be less able to handle advanced mathematical topics without help or

tutoring. To ll these needs, the opening chapter of this book provides a

summary of the mathematical background for topics handled in subsequent

chapters.

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Acknowledgments

In writing this book I have beneted from the help and support of various

people. At an early stage, anonymous proposal reviewers gave me useful

suggestions, not all of which have been acted on, but all of which were

appreciated. Each chapter was read and checked by an obliging colleague.

I wish to thank Dave Chapman, Rob Coe, Ramon Egli, Chris Finlay, Valentin

Gischig, Klaus Holliger, Edi Kissling, Emile Klingel, Alexei Kuvshinov,

Germn Rubino, Rolf Sidler, and Doug Smylie for their corrections and sug-

gestions for improvement. The responsibility for any errors that escaped scru-tiny is, of course, mine. I am very grateful to Derrick Hasterok and Dave

Chapman for providing me with an unpublished gure from Derricks Ph.D.

thesis. Dr. Susan Francis, Senior Commissioning Editor at Cambridge

University Press, gave me constant support and friendly encouragement

throughout the many months of writing, for which I am sincerely grateful.

Above all, I thank my wife Marcia for her generous tolerance of the intrusion

of this project into our retirement activities.

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1

Mathematical background

1.1 Cartesian and spherical coordinates

Two systems of orthogonal coordinates are used in this book, sometimes

interchangeably. Cartesian coordinates (x, y, z) are used for a system with

rectangular geometry, and spherical polar coordinates (r, , ) are used for

spherical geometry. The relationship between these reference systems is shown

inFig. 1.1(a). The convention used here for spherical geometry is dened as

follows: the radial distance from the origin of the coordinates is denoted r, the

polar angle (geographic equivalent: the co-latitude) lies between the radius

and thez-axis (geographic equivalent: Earths rotation axis), and the azimuthal

angle in the xy plane is measured from the x-axis (geographic equivalent:

longitude). Position on the surface of a sphere (constantr) is described by the

two anglesand. The Cartesian and spherical polar coordinates are linked as

illustrated inFig. 1.1(b)by the relationships

xr sin cos yr sin sin zr cos

(1:1)

1.2 Complex numbers

The numbers we most commonly use in daily life are realnumbers. Some of

them are also rationalnumbers. This means that they can be expressed as the

quotient of two integers, with the condition that the denominator of the quotient

must not equal zero. When the denominator is 1, the real number is an integer.

Thus 4, 4/5, 123/456 are all rational numbers. A real number can also be

irrational, which means it cannot be expressed as the quotient of two integers.

1


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Familiar examples are , e (the base of natural logarithms), and some square

roots, such as2,3,5, etc. The irrational numbers are real numbers that do

not terminate or repeat when expressed as decimals.

In certain analyses, such as determining the roots of an equation, it is

necessary to nd the square root of a negative real number, e.g. (y2), where

yis real. The result is an imaginarynumber. The negative real number can be

written as (1)y2, and its square root is then(1)y. The quantity(1) is written

iand is known as the imaginary unit, so that(y2) becomes iy.

A complex number comprises a real part and an imaginary part. For example,

z = x+ iy, in whichx and y are both real numbers, is a complex number with a

real part xand animaginary part y. The composition of a complex number can

be illustrated graphically with the aid of thecomplex plane(Fig. 1.2). The real

part is plotted on the horizontal axis, and the imaginary part on the vertical axis.

The two independent parts are orthogonal on the plot and the complex numberz

r

z

xy

(a)

r

y= rsinsin

x= rsincos

z= rcos

rsin

(b)

Fig. 1.1. (a) Cartesian and spherical polar reference systems. (b) Relationshipsbetween the Cartesian and spherical polar coordinates.

+x

+y z= x+ iy

Realaxis

Imaginaryaxis

r

rcos

rsin

Fig. 1.2. Representation of a complex number on an Argand diagram.

2 Mathematical background


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is represented by their vector sum, dening a point on the plane. The distancer

of the point from the origin is given by

r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 y2p (1:2)The line joining the point to the origin makes an angle with the real (x-)axis,

and so rhas real and imaginary components rcos and rsin , respectively. The

complex numberzcan be written in polar form as

zr cos isin (1:3)It is often useful to write a complex number in the exponential form introduced

by Leonhard Euler in the late eighteenth century. To illustrate this we make useof innite power series; this topic is described inSection 1.10. The exponential

function, exp(x), of a variablexcan be expressed as a power series as in (1.135).

On substitutingx = i, the power series becomes

exp i 1 i i 2

2! i

3

3! i

4

4! i

5

5! i

6

6!

1

i 2

2! i 4

4! i 6

6! i

i 3

3! i 5

5! 1

2

2!

4

4!

6

6!

i

3

3!

5

5!

(1:4)

Comparison with (1.135) shows that the rst bracketed expression on the

right is the power series for cos ; the second is the power series for sin .

Therefore

exp i cos isin (1:5)On inserting (1.5) into (1.3), the complex numberzcan be written in exponential

form as

zr exp i (1:6)The quantityris themodulusof the complex number and is itsphase.

Conversely, using (1.5) the cosine and sine functions can be dened as the

sum or difference of the complex exponentials exp(i) and exp(i):

cos exp i expi 2

sin exp i expi 2i

(1:7)

1.2 Complex numbers 3


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1.3 Vector relationships

A scalar quantity is characterized only by its magnitude; a vector has bothmagnitude and direction; a unit vector has unit magnitude and the direction of

the quantity it represents. In this overview the unit vectors for Cartesian

coordinates (x, y, z) are written (ex, ey, ez); unit vectors in spherical polar

coordinates (r,,) are denoted (er,e,e). The unit vector normal to a surface

is simply denotedn.

1.3.1 Scalar and vector products

The scalar product of two vectors a and b is dened as the product of their

magnitudes and the cosine of the angle between the vectors:

a bab cos (1:8)If the vectors are orthogonal, the cosine of the angle is zero and

a b0 (1:9)The vector product of two vectors is another vector, whose direction is perpen-

dicular to both vectors, such that a right-handed rule is observed. The magnitudeof the vector product is the product of the individual vector magnitudes and the

sine of the angle between the vectors:

a bj j ab sin (1:10)Ifa and b are parallel, the sine of the angle between them is zero and

a b0 (1:11)

Applying these rules to the unit vectors (ex, ey, ez), which are normal to each

other and have unit magnitude, it follows that their scalar products are

ex ey ey ez ez ex0ex ex ey ey ez ez1

(1:12)

The vector products of the unit vectors are

ex

ey

ez

ey ez exez ex eyex ex ey ey ez ez0

(1:13)



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A vectorawith components (ax,ay,az) is expressed in terms of the unit vectors

(ex,ey,ez) as

aaxex ayey azez (1:14)The scalar product of the vectorsaand bis found by applying the relationships

in (1.12):

a b axex ayey azez

bxex byey bzez

axbx ayby azbz (1:15)The vector product of the vectorsa and b is found by using (1.13):

a b axex ayey azez bxex byey bzez aybz azby

ex azbx axbz ey axby aybx

ez (1:16)

This result leads to a convenient way of evaluating the vector product of two

vectors, by writing their components as the elements of a determinant, as

follows:

a

b

ex ey ez

ax ay az

bx by bz (1:17)

The following relationships may be established, in a similar manner to the

above, for combinations of scalar and vector products of the vectors a,b, andc:

a b c b c a c a b (1:18)

a b c b c a c a b (1:19)

a

b

c

b c a

a b c

(1:20)

1.3.2 Vector differential operations

The vector differential operator is dened relative to Cartesian axes (x,y,z)

as

r ex x

ey y

ez z

(1:21)

The vector operator determines the gradient of a scalar function, which may

be understood as the rate of change of the function in the direction of each of the

reference axes. For example, the gradient of the scalar function with respect to

Cartesian axes is the vector



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r ex x

ey y

ez z

(1:22)

The vector operator can operate on either a scalar quantity or a vector. The

scalar product of with a vector is called thedivergenceof the vector. Applied

to the vectora it is equal to

r a ex x

ey y

ez z

axex ayey azez

axx

ayy

azz

(1:23)

If the vectora is dened as the gradient of a scalar potential , as in(1.22), we

can substitute potential gradients for the vector components (ax, ay, az). This

gives

r r x

x

y

y

z

z

(1:24)

By convention the scalar product ( ) on the left is written 2. The resulting

identity is very important in potential theory and is encountered frequently. In

Cartesian coordinates it is

r2 2

x2

2

y2

2

z2 (1:25)

The vector product of with a vector is called the curl of the vector. The

curl of the vector a may be obtained using a determinant similar to (1.17):

r aex ey ez

=x =y =zax ay az

(1:26)In expanded format, this becomes

r a azy

ayz

ex ax

z az

x

ey ay

x ax

y

ez (1:27)

The curl is sometimes called the rotation of a vector, because of its physical

interpretation (Box 1.1). Some commonly encountered divergence and curl

operations on combinations of the scalar quantity and the vectors a and b

are listed below:

r a r a r a (1:28)



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r a b b r a a r b (1:29)

r a r a r a (1:30)

r a b ar b br a a r b b r a (1:31)

r r 0 (1:32)

Box 1.1. The curl of a vector

The curl of a vector at a given point is related to the circulation of the vector

about that point. This interpretation is best illustrated by an example, in

which a uid is rotating about a point with constant angular velocity .

At distancerfrom the point the linear velocity of the uidv is equal to r.

Taking the curl ofv, and applying the identity (1.31) with constant,

r v r w r wr r w r r (1)To evaluate the rst term on the right, we use rectangular coordinates (x,y,z):

wr r w ex

x ey

y ez

z

xexyey zez w ex ex ey ey ez ez

3w (2)The second term is

w r r x x

y y

z z

xexyey zez

xex

yey

zez

w (3)

Combining the results gives

r v2w (4)

w12r v (5)

Because of this relationship between the angular velocity and the linear

velocity of a

uid, the curl operation is often interpreted as the rotationofthe uid. When v = 0 everywhere, there is no rotation. A vector that

satises this condition is said to beirrotational.



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r r a 0 (1:33)

r r a r r a r2a (1:34)

It is a worthwhile exercise to establish these identities from basic principles,especially (1.19) and (1.31)(1.34), which will be used in later chapters.

1.4 Matrices and tensors

1.4.1 The rotation matrix

Consider two sets of orthogonal Cartesian coordinate axes (x,y,z) and (x0,y0,

z0) that are inclined to each other as in Fig. 1.3. Thex0-axis makes angles (1,1,

1) with each of the (x,y,z) axes in turn. Similar sets of angles (2,2,2) and

(3,3,3) are dened by the orientations of they0- andz0-axes, respectively, to

the (x,y,z) axes. Let the unit vectors along the (x,y,z) a n d (x0,y0,z0)axesbe(n1,

n2,n3) and (m1,m2,m3), respectively. The vectorr can be expressed in either

system, i.e.,r = r(x,y,z) =r(x0,y0,z0), or, in terms of the unit vectors,

rxn1yn2 zn3x0m1y0m2 z0m3 (1:35)We can write the scalar product (r m1) as

r m1xn1 m1yn2 m1 zn3 m1x0 (1:36)The scalar product (n1 m1) = cos 1=11denes11as thedirection cosineof

thex0-axis with respect to thex-axis (Box 1.2). Similarly, (n2 m1) = c o s1 = 12

z0z

x0x

y0

y

1

1

1

2

3

2

2

3

3

m

1 n1

m2

n2

m3n3

Fig. 1.3. Two sets of Cartesian coordinate axes, (x, y, z) and (x0, y0, z0), with

corresponding unit vectors (n1, n2, n3)and(m1, m2, m3), rotated relative to each other.



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and (n3 m1) = cos 1= 13dene 12and 13as the direction cosines of the

x0-axis with respect to they- andz-axes, respectively. Thus, (1.36) is equivalent to

x011x 12y 13z (1:37)On treating they0- andz0-axes in the same way, we get their relationships to the

(x,y,z) axes:

y021x 22y 23zz031x 32y 33z

(1:38)

The three equations can be written as a single matrix equation

x0y0z0

24 35 11 12 1321 22 2331 32 33

24 35 xyz

24 35M xyz

24 35 (1:39)The coefcients nm(n = 1, 2, 3; m = 1, 2, 3) are the cosines of the interaxial

angles. By denition,12=21,23=32, and31=13, so the square matrixM

is symmetric. It transforms the components of the vector in the (x, y, z)

coordinate system to corresponding values in the (x0,y0,z0) coordinate system.

It is thus equivalent to a rotation of the reference axes.

Because of the orthogonality of the reference axes, useful relationships existbetween the direction cosines, as shown inBox 1.2. For example,

11 2 12 2 13 2 cos21 cos21 cos21 1

r2 x2 y2 z2 1

(1:40)

and

1121 1222 1323cos 1cos 2 cos1cos2 cos 1cos 20

(1:41)The last summation is zero because it is the cosine of the right angle between the

x0-axis and they0-axis.

These two results can be summarized as

X3k1

mknk 1; mn0; m6n

(1:42)

1.4.2 Eigenvalues and eigenvectors

The transpose of a matrix Xwith elements nm is a matrix with elements mn(i.e., the elements in the rows are interchanged with corresponding elements in



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Box 1.2. Direction cosines

The vectorr is inclined at angles,, and, respectively, to orthogonal

reference axes (x,y,z) with corresponding unit vectors (ex,ey,ez), as inFig.

B1.2. The vectorr can be written

rxexyey zez (1)

where (x,y,z) are the components ofrwith respect to these axes. The scalar

products ofr withex,ey, andezare

z

x

r

y

ex

ey

ez

Fig. B1.2. Angles,, and dene the tilt of a vectorr relative to orthogonal

reference axes (x, y, z), respectively. The unit vectors (ex, ey, ez) dene thecoordinate system.

r exxr cos r eyyr cosr ezzr cos

(2)

Therefore, the vectorr in (1) is equivalent to

r r cos ex r cos ey r cos ez (3)

The unit vectoru in the direction ofr has the same direction as r but its

magnitude is unity:

u rr cos ex cos ey cos ezlex mey nez (4)

where (l,m,n) are the cosines of the angles that the vectorr makes with

the reference axes, and are called the direction cosines ofr. They are

useful for describing the orientations of lines and vectors.



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the columns). The transpose of a (3 1) column matrix is a (1 3) row matrix.

For example, ifXis a column matrix given by

Xx

yz

24 35 (1:43)

then its transpose is the row matrix XT, where

XT x y z (1:44)The matrix equation XTMX = K, where K is a constant, denes a quadric

surface:

XTMX x y z 11 12 1321 22 2331 32 33

24 35 xyz

24 35K (1:45)The symmetry of the matrix leads to the equation of this surface:

The scalar product of two unit vectors is the cosine of the angle they form.

Letu1and u2be unit vectors representing straight lines with direction

cosines (l1,m1,n1) and (l2,m2,n2), respectively, and letbe the anglebetween the vectors. The scalar product of the vectors is

u1 u2cos l1ex m1ey n1ez

l2ex m2ey n2ez

(5)

Therefore,

cos l1l2 m1m2 n1n2 (6)

The square of a unit vector is the scalar product of the vector with itself and isequal to 1:

u u r rr2

1 (7)

On writing the unit vectoru as in (4), and applying the orthogonality

conditions from (2), we nd that the sum of the squares of the direction

cosines of a line is unity:

lex mey nez lex mey nez l2 m2 n2 1 (8)



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fx;y; z 11x2 22y2 33z2 212xy 223yz 231zxK (1:46)When the coefcientsnmare all positive real numbers, the geometric expres-

sion of the quadratic equation is an ellipsoid. The normal direction n to the

surface of the ellipsoid at the point P(x,y,z) is the gradient of the surface. Using

the relationships between (x,y,z) and (x0,y0,z0) in(1.39) and the symmetry of

the rotation matrix,nm= mnforn m, the normal direction has components

f

x2 11x 12y 13z 2x0

f

y2 21x 22y 23z 2y0

f

z2 31x 32y 33z 2z0 (1:47)

and we write

nx;y; z rf ex fx

ey fx

ez fx

(1:48)

nx;y; z 2x0exy0ey z0ez 2rx0;y0; z0 (1:49)The normalnto the surface at P(x,y,z) in the original coordinates is parallel to

the vectorr at the point (x0,y0,z0) in the rotated coordinates (Fig. 1.4).

The transformation matrixMhas the effect of rotating the reference axes from

one orientation to another. A particular matrix exists that will cause the direc-

tions of the (x0, y0, z0) axes to coincide with the (x, y, z) axes. In this case the

normal to the surface of the ellipsoid is one of the three principal axes of the

x y

z P

tangentplane

(x0,y0,z0)

(x,y,z)r

n

er

Fig. 1.4. Location of a point (x, y, z) on an ellipsoid, where the normal n to the

surface is parallel to the radius vector at the point (x0,y0,z0).



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1.4.3 Tensor notation

Equations describing vector relationships can become cumbersome when writ-

ten in full or symbolic form. Tensor notation provides a succinct alternative wayof writing the equations. Instead of the alphabetic indices used in theprevious

section, tensor notation uses numerical indices that allow summations to be

expressed in a compact form.

Let the Cartesian coordinates (x,y,z) be replaced by coordinates (x1,x2, x3)

and let the corresponding unit vectors be (e1, e2, e3). The vectora in (1.14)

becomes

aa1e1 a2e2 a3e3 Xi1;2;3

aiei (1:56)

A convention introduced by Einstein drops the summation sign and tacitly

assumes that repetition of an index implies summation over all values of the

index, in this case from 1 to 3. The vectora is then written explicitly

aaiei (1:57)Alternatively, the unit vectors can be implied and the expression ai is under-

stood to represent the vectora. Using the summation convention, (1.15) for the

scalar product of two vectorsa and b is

a ba1b1 a2b2 a3b3aibi (1:58)Suppose that two vectorsa and b are related, so that each component ofa is a

linear combination of the components ofb. The relationship can be expressed in

tensor notation as

aiTijbj (1:59)

The indicesiandjidentify components of the vectorsaandb; each index takeseach of the values 1, 2, and 3 in turn. The quantity Tij is a second-order (or

second-rank) tensor, representing the array of nine coefcients (i.e., 32). A

vector has three components (i.e., 31) a n d i s a rst-order tensor; a scalar property

has a single (i.e., 30) value, its magnitude, and is a zeroth-order tensor.

To write the cross product of two vectors we need to dene a new quantity,

the Levi-Civitapermutation tensorijk. It has the value +1 when a permutation

of the indices is even (i.e., 123 = 231 = 312= 1) and the value 1 when a

permutation of the indices is odd (i.e., 132 = 213 = 321 = 1). If any pair of

indices is equal,ijk= 0. This enables us to write the cross product of two vectors

in tensor notation. Letu be the cross product of vectors a and b:

u a b a2b3 a3b2 e1 a3b1 a1b3 e2 a1b2 a2b1 e3 (1:60)



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In tensor notation this is written

uiijkajbk (1:61)This can be veried readily for each component ofu. For example,

u1123a2b3 132a3b2a2b3 a3b2 (1:62)The tensor equivalent to the unit matrix dened in (1.55) is known as

Kroneckers symbol,ij, or alternatively theKronecker delta.It has the values

ij 1; if ij0; if i6j

(1:63)

Kroneckers symbol is convenient for selecting a particular component of a

tensor equation. For example, (1.54) can be written in tensor form using the

Kronecker symbol:

ijij

xj0 (1:64)This represents the set of simultaneous equations in (1.50). Likewise, the

relationship between direction cosines in (1.42) simplies to

mknkmn (1:65)in which a summation over the repeated index is implied.

1.4.4 Rotation of coordinate axes

Letvkbe a vector related to the coordinates xlby the tensorTkl

vkTklxl (1:66)

A second set of coordinates xn is rotated relative to the axes xl so that the

direction cosines of the angles between corresponding axes are the elements of

the tensornl:

x0nnlxl (1:67)Let the same vector be related to the rotated coordinate axes xn by the tensor

Tkn:

v0kT0knx0n (1:68)vk and vk are the same vector, expressed relative to different sets of axes.

Therefore,

v0kknvnknTnlxl (1:69)



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r e1 x1

e2 x2

e3 x3

ei xi

(1:78)

Several shorthand forms of this equation are in common use; for example,

xi r i;ii (1:79)

Thedivergenceof the vectora is written in tensor notation as

r a a1x1

a2x2

a3x3

aixi

iai (1:80)

Thecurl(orrotation) of the vectora becomes

r a e1 a3x2

a2x3

e2 a1

x3 a3

x1

e3 a2

x1 a1

x2

(1:81)

r a iijkak

xjijkjak (1:82)

1.5 Conservative force,eld, and potential

If the work done in moving an object from one point to another against a force is

independent of the path between the points, the force is said to be conservative.

No work is done if the end-point of the motion is the same as the starting point;

this condition is called the closed-path test of whether a force is conservative. In

a real situation, energy may be lost, for example to heat or friction, but in an

ideal case the total energyEis constant. The workdWdone against the force F is

converted into a gain dEP in the potential energy of the displaced object. Thechange in the total energydEis zero:

dEdEP dW0 (1:83)The change in potential energy when a force with components (Fx, Fy, Fz)

parallel to the respective Cartesian coordinate axes (x,y,z) experiences elemen-

tary displacements (dx,dy,dz) is

dEP dW Fx dx Fy dy Fzdz (1:84)The value of a physical force may vary in the space around its source. For

example, gravitational and electrical forces decrease with distance from a

source mass or electrical charge, respectively. The region in which a physical

1.5 Conservative force,eld, and potential 17


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is distributed in the volume with mean density , a volume integral can replace

the enclosed mass:

ZZZV

r aG dV 4GZZZ

V

dV (1:114)

ZZZV

r aG 4G dV0 (1:115)

For this to be generally true the integrand must be zero. Consequently,

r aG

4G (1:116)

The gravitational acceleration is the gradient of the gravitational potential UGas

in (1.88):

r rUG 4G (1:117)

r2UG4G (1:118)Equation (1.118) is known as Poissons equation, after Simon-Denis Poisson

(17811840), a French mathematician and physicist. It describes the gravita-

tional potential of a mass distribution at a point that is within the mass distri-

bution. For example, it may be used to compute the gravitational potential at a

point inside the Earth.

1.9 Laplaces equation

Another interesting case is the potential at a point outside the mass distribution.

Let S be a closed surface outside the point mass m. The radius vectorrfrom the

point mass mnow intersects the surface S at two points A and B, where it forms

angles1and2with the respective unit vectorsn1andn2normal to the surface

(Fig. 1.10). Leterbe a unit vector in the radial direction. Note that the outward

normaln1forms an obtuse angle with the radius vector at A. The gravitational

acceleration at A is a1and its ux through the surface area dS1is

dN1

a1 n1dS1

Gm

r2

1 r n1

dS1 (1:119)

dN1 Gmr21

cos 1 dS1Gm cos 1dS1

r21(1:120)



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dN1Gm d (1:121)The gravitational acceleration at B is a2 and its ux through the surface area dS2 is

dN2 a2 n2dS2 Gm cos 2dS2r22

(1:122)

dN2 Gm d (1:123)The total contribution of both surfaces to the gravitational ux is

dNdN1 dN20 (1:124)Thus, the total ux of the gravitational acceleration aGthrough a surface S that

does not include the point massm is zero. By invoking the divergence theorem

we have for this situationZV

r aG dVZS

aG n dS0 (1:125)

For this result to be valid for any volume, the integrand must be zero:

r aG r rUG 0 (1:126)

r2UG 0 (1:127)Equation (1.127) is Laplaces equation, named after Pierre Simon, Marquis de

Laplace (17491827), a French mathematician and physicist. It describes the

gravitational potential at a point outside a mass distribution. For example, it is

applicable to the computation of the gravitational potential of the Earth at an

external point or on its surface.

In Cartesian coordinates, which are rectilinear, Laplaces equation has the

simple form

S

m a1

n2

d

dS1

1

2

dS2n1

a2A B

er

Fig. 1.10. Representation of the gravitational ux through a closed surface S that

does not enclose the source of the ux (the point massm).

1.9 Laplaces equation 27


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f0 a0; dfdx

x0

a1

d2

fdx2

x0

2a2; d3

fdx3

x0

3 2 a3dnf

dxn

x0

nn 1n 2 . . . 3 2 1 ann!an

(1:133)

On inserting these values for the coefcients into (1.130) we get the power

series for(x):

f

x

f

0

x

df

dx

x0

x2

2!

d2f

dx

2 x0

x3

3!

d3f

dx

3 x0

xn

n!

dnf

dxn

x0

(1:134)

This is the MacLaurin series for(x) about the origin,x = 0. It was derived in the

eighteenth century by the Scottish mathematician Colin MacLaurin (1698

1746) as a special case of a Taylor series.

The MacLaurin series is a convenient way to derive series expressions for

several important functions. In particular,

sin xx x3

3! x

5

5! x

7

7! 1 n1 x

2n1

2n 1 !

cos x1 x2

2! x

4

4! x

6

6! 1 n1 x

2n2

2n 2 !

expx ex 1 x x2

2! x

3

3! x

n1

n 1 !

ln 1

x

loge 1

x

x

x2

2

x3

3

x4

4 1

n1 xn

n

(1:135)

1.10.2 Taylor series

We can write the power series in (1.134) for(x) centered on any new origin, for

examplex = x0. To do this we substitute (x x0) forx in the above derivation.

The power series becomes

fx fx0 x x0 dfdx xx0

x x0 2

2!

d2f

dx2 xx0 x x0

3

3!

d3f

dx3

xx0

x x0 n

n!

dnf

dxn

xx0

(1:136)

1.10 Power series 29


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This is called a Taylor series, after an English mathematician, Brooks Taylor

(16851731), who described its properties in 1712.

The MacLaurin and Taylor series are both approximations to the function

(x). The remainder between the true function and its power series is a measure

of how well the function is expressed by the series.

1.10.3 Binomial series

Finite series

An important series is the expansion of the function (x) = (a + x)n. Ifn is a

positive integer, the expansion of(x) is a nite series, terminating after (n + 1)

terms. Evaluating the series for some low values ofn gives the following:

n0 : a x 0 1n1 : a x 1 a xn2 : a x 2 a2 2ax x2n3 : a x 3 a3 3a2x 3ax2 x3n4 : a x 4 a4 4a3x 6a2x2 4ax3 x4

(1:137)

The general expansion of(x) is therefore

a x n an nan1x n n 1 1 2 a

n2x2

n n 1 . . . n k 1 k!

ankxk xn(1:138)

The coefcient of the general kth term is equivalent to

n n 1 . . . n k 1 k!

n!

k! n k ! (1:139)

This is called thebinomial coefcient.

When the constanta is equal to 1 and n is a positive integer, we have the

useful series expansion

1 x nXnk0

n!

k! n k !xk (1:140)

Innite series

If the exponent in (1.140) is not a positive integer, the series does not terminate,

but is an innite series. The series for(x) = (1 +x)p, in which the exponentpis

not a positive integer, may be derived as a MacLaurin series:



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1.11 Leibnizs rule

Assume thatu(x) and v(x) are differentiable functions ofx. The derivative oftheir product is

d

dx u x v x u x dv x

dx v x du x

dx (1:144)

If we dene the operatorD = d/dx, we obtain a shorthand form of this equation:

D uv u Dv v Du (1:145)We can differentiate the product (uv) a second time by parts,

D2 uv D D uv u D2v Du Dv Dv Du v D2uu D2v 2 Du Dv v D2u (1:146)

and, continuing in this way,

D3 uv u D3v 3 Du D2v 3 D2u Dv v D3uD4 uv u D4v 4 Du D3v

6 D2u

D2v

4 D2u

Dv v D4u

(1:147)

The coefcients in these equations are the binomial coefcients, as dened in

(1.139). Thus aftern differentiations we have

Dn uv Xnk0

n!

k! n k ! Dku

Dnkv

(1:148)

This relationship is known as Leibnizs rule, after Gottfried Wilhelm Leibniz

(16461716), who invented innitesimal calculus contemporaneously with

Isaac Newton (16421727); each evidently did so independently of the other.

1.12 Legendre polynomials

LetrandRbe the sides of a triangle that enclose an angle and letube the side

opposite this angle (Fig. 1.11). The angle and sides are related by the cosine rule

u2 r2 R2 2rR cos (1:149)Inverting this expression and taking the square root gives

1

u 1

R

"1 2

r

R

!cos

r

R

!2#1=2(1:150)



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Equation (1.156) is known as the generating function for the polynomials Pn(x).

Using this result, and substituting h = r/R and x = cos , we nd that (1.151)

becomes

1

u 1

R

X1n0

r

R

!nPn cos (1:157)

The polynomialsPn(x) orPn(cos ) are called Legendre polynomials, after the

French mathematician Adrien-Marie Legendre (17521833). The dening

equation (1.157) is called the reciprocal-distance formula. An alternative for-

mulation is given inBox 1.4.

1.13 The Legendre differential equation

The Legendre polynomials satisfy an important second-order partial differential

equation, which is called the Legendre differential equation. To derive this

equation we will carry out a sequence of differentiations, starting with the

generating function in the form

1 2xh h2 1=2 (1:158)Differentiating this function once with respect toh gives

h x h 1 2xh h2 3=2 x h 3 (1:159)

Differentiating twice with respect to x gives

xh 1

2xh

h2 3=2 h

3

3 1

h

x

(1:160)

2

x2 3h2

x3h25

5 1

3h2

2

x2

(1:161)

Next we perform successive differentiations of the product (h) with respect to

h. The rst gives

h h h

h h x h 3 (1:162)



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On repeating the differentiation, and taking (1.159) into account, we get

2

h2 h

h h x h 32

h x 2h 3

x h 3 3h x h 25 x 2h 3 2x 3h 3 3h x h 25 (1:163)

Now substitute for3, from (1.160), and 5, from (1.161), giving

Box 1.4. Alternative form of the reciprocal-distance formula

The sides and enclosed angle of the triangle in Fig. 1.11are related by the

cosine rule

u2 r2 R2 2rR cos (1)

Instead of takingRoutside the brackets as in (1.150), we can moveroutside

and write the expression foru as

1

u

1

r

1

2

R

r cos

R

r

2

" #1=2

(2)

Following the same treatment as inSection 1.12, but now withhR=randx = cos , we get

1

u1

r 1 2xh h2 1=2 1

r 1 t 1=2 (3)

wheret= 2xh h2. The function (1 t)1/2 is expanded as a binomial series,

which again gives an innite series inh, in which the coefcient ofhn

isPn(x). The dening equation is as before:

x; h 1 2xh h2 1=2 X1n0

hnPn x (4)

On substitutinghR=rand x = cos , we nd an alternative form for thegenerating equation for the Legendre polynomials:

1u1

r

X1n0

Rr

nPn cos (5)



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2

h2 h 2x 3h 1

h

x

3h x h 2 1

3h2

2

x2

(1:164)

Multiply throughout byh:

h

2

h2 h 2x 3h

x

x h 2

2

x2

2x x

3h

x

x h 2

2

x2

(1:165)

The second term on the right can be replaced as follows, again using (1.160) and

(1.161):

3h

x3h23 1

2

2

x2

1 2xh h2 2x2

(1:166)

On substituting into (1.165) and gathering terms, we get

h 2

h2

h

x

h

2

1

2xh

h2 h i

2

x2 2x

x (1:167)

h

2

h2 h x2 1 2

x2

2x

x

(1:168)

The Legendre polynomialsPn(x) are dened in (1.156) as the coefcients ofhn

in the expansion of as a power series. On multiplying both sides of (1.156) by

h, we get

h X

1

n0 hn

1

Pnx (1:169)We differentiate this expression twice and multiply by h to get a result that can

be inserted on the left-hand side of (1.168):

h h

X1n0

n 1 hnPnx (1:170)

h

2

h2 h X

1

n0n n

1

hnP

nx

(1:171)

Using (1.156), we can now eliminate and convert (1.168) into a second-order

differential equation involving the Legendre polynomials Pn(x),



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Thus

d

dx 1

x2 PlP0

nP0

l

Pn n n 1 l l 1 PlPn0 (1:186)Now integrate each term in this equation with respect to x over the range

1x 1. We get

1 x2 PlP0n P0lPn 1x1 n n 1 l l 1 Z1

x1PlPn dx0

(1:187)

The rst term is zero on evaluation of (1 x2) atx = 1; thus the second termmust also be zero. Forn l the condition for orthogonality of the Legendre

polynomials is Z 1x1

PnxPlxdx0 (1:188)

1.13.2 Normalization of the Legendre polynomials

A function is said to be normalized if the integral of the square of the function overits range is equal to 1. Thus we must evaluate the integral

R1x1 Pn x 2 dx. We

begin by recalling the generating function for the Legendre polynomials given in

(1.156), which we rewrite forPn(x) andPl(x) individually:X1n0

hnPn x 1 2xh h2 1=2

(1:189)

X1l0 h

l

Pl x 1 2xh h2 1=2 (1:190)

Multiplying these equations together gives

X1l0

X1n0

hnlPn x Pl x 1 2xh h2 1

(1:191)

Now let l= n and integrate both sides with respect to x, taking into account

(1.188):

X1n0

h2nZ1

x1Pn x 2 dx

Z1x1

dx

1 h2 2xh (1:192)



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The right-hand side of this equation is a standard integration that results in a

natural logarithm:

Z dxa bx

1b

ln a bx (1:193)

The right-hand side of (1.192) therefore leads to

Z1x1

dx

1 h2 2xh 1

2h ln 1 h2 2xh

1

x1

1

2h ln 1 h2

2h ln 1 h2 2h (1:194)and

Z1x1

dx

1 h2 2xh 1

h 1

2ln 1 h 2 1

2ln 1 h 2

1h

ln 1 h ln 1 h (1:195)

Using the MacLaurin series for the natural logarithms as in (1.135), we get

ln 1 h h h2

2 h

3

3 h

4

4 1 n1h

n

n (1:196)

ln 1 h h h2

2 h

3

3 h

4

4 1 n1h

n

n (1:197)

Subtracting the second equation from the rst gives

Z1x1

dx

1 h2 2xh2

h h h

3

3 h

5

5

2

h

X1n0

h2n1

2n 1 (1:198)

Inserting this result into (1.192) gives

X1n0

h2nZ1

x1Pn x 2 dx 2

h

X1n0

h2n1

2n 1 (1:199)

X1n0

h2nZ1

x1Pn x 2 dx 2

2n 1

0@

1A0 (1:200)



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x2 1 Dn2f 2x n 1 Dn1f 2 n 1n12

Dnf2nx Dn1f

2n n

1

Dnf

(1:207)

On gathering terms and bringing all to the left-hand side, we have

x2 1 Dn2f 2x Dn1f nn 1Dnf0 (1:208)Now we deney(x) such that

y x

Dnf

dn

dxn

x2

1 n (1:209)

and we have

x2 1 d2ydx2

2x dydx

nn 1y0 (1:210)

On comparing with (1.174), we see that this is the Legendre equation. The

Legendre polynomials must therefore be proportional to y(x), so we can write

Pnx cndn

dxn x

2

1 n (1:211)The quantitycnis a calibration constant. To determine cnwe rst write

dn

dxn x2 1 n dn

dxn x 1 n x 1 n (1:212)

then we apply Leibnizs rule to the product on the right-hand side of the

equation:

dndxn

x2 1 n Xnm0

n!m! n m !

dmdxm

x 1 ndnmdxnm

x 1 n (1:213)

The successive differentiations of (x 1)n give

d

dx x 1 n n x 1 n1

d2

dx2 x 1 n n n 1 x 1 n2

dn1dxn1

x 1 n n n 1 n 2 . . . 321 x 1 n! x 1 dn

dxn x 1 n n!

(1:214)



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1 x2 ddx

P00n 22x d

dxP0n n n 1 12

d

dxPn0 (1:222)

Next, we differentiate this expression again, observing the same rules andgathering terms,

1x2 d2dx2

P00n 2x d

dxP00n

4 d

dxP0n 4x

d2

dx2P0n

n n 1 2 d

2

dx2Pn 0

(1:223)

1 x2

d2

dx2P00n 2x

d2

dx2P0n 4x

d2

dx2P0n n n 1 2 4

d2

dx2Pn0

(1:224)

1 x2 d2dx2

P00n 6xd2

dx2P0n n n 1 6

d2

dx2Pn0 (1:225)

which, as we did with (1.222), we can write in the form

1 x2 d2dx2

P00n 23xd2

dx2P0n n n 1 23

d2

dx2Pn0 (1:226)

On following step-by-step in the same manner, we get after the third

differentiation

1 x2 d3dx3

P00n 2 4 xd3

dx3P0n n n 1 3 4

d3

dx3Pn0 (1:227)

Equations (1.222), (1.226), and (1.227) all have the same form. The higher-

order differentiation is accompanied by systematically different constants. By

extension, differentiating (1.219)m times (wherem n) yields the differential

equation

1 x2 dmdxm

P00n 2 m 1 x dm

dxmP0n n n 1 m m 1

dm

dxmPn0(1:228)

Now let themth-order differentiation ofPnbe written as

dm

dxmPnx Qx

1

x2

m=2

(1:229)

Substitution of this expression into (1.228) gives a new differential equation

involving Q(x). We need to determine bothdm=dxmP0n anddm=dxmP00n , sorst we differentiate (1.229) with respect tox:



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The functionsQ(x) involve two parameters, the degreenand orderm, and are

writtenPn,m(x). Thus

1 x2 d2dx2

Pn;m x 2x ddx

Pn;m x n n 1 m2

1 x2

Pn;m x 0

(1:237)

This is the associated Legendre equation. The solutions Pn,m(x) orPn,m(cos ),

wherex = cos , are calledassociated Legendre polynomials, and are obtained

from the ordinary Legendre polynomials using the denition ofQ in (1.229):

Pn;mx 1 x2 m=2 dm

dxmPnx (1:238)

Substituting Rodriguesformula (1.218) forPn(x) into this equation gives

Pn;mx 1 x2 m=2

2nn!

dnm

dxnm x2 1 n (1:239)

The highest power ofx in the function (x2 1)n isx2n. After 2ndifferentiations

the result will be a constant, and a further differentiation will give zero.Therefore n + m 2n, and possible values ofm are limited to the range 0 m n.

1.15.1 Orthogonality of associated Legendre polynomials

For succinctness we again writePn,m(x) as simplyPn,m. The dening equations

for the associated Legendre polynomialsPn,mand Pl,mare

1 x2 Pn;m 00 2x Pn;m 0 n n 1 m21 x2 Pn;m0 (1:240)1 x2 Pl;m 00 2x Pl;m 0 l l 1 m2

1 x2

Pl;m0 (1:241)

As for the ordinary Legendre polynomials, we multiply (1.240) by Pl,m and

(1.241) byPn,m:

1 x2

Pn;m 00Pl;m 2x Pn;m 0Pl;m n n 1 m

2

1 x2 Pn;mPl;m0(1:242)



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1.16 Spherical harmonic functions

Several geophysical potential

elds

for example, gravitation and geomagnet-ism satisfy the Laplace equation. Spherical polar coordinates are best suited

for describing a global geophysical potential. The potential can vary with

distance r from the Earths center and with polar angular distance and

azimuth (equivalent to co-latitude and longitude in geographic terms)

on any concentric spherical surface. The solution of Laplaces equation in

spherical polar coordinates for a potential U may be written (see Section

2.4.5,(2.104))

UX1n0

Xnm0

Anrn Bn

rn1

amn cosm bmn sinm

Pmn cos (1:252)

HereAn,Bn,amn, and b

mn are constants that apply to a particular situation. On the

surface of the Earth, or an arbitrary sphere, the radial part of the potential of a

point source at the center of the sphere has a constant value and the variation

over the surface of the sphere is described by the functions in and . We are

primarily interested in solutions outside the Earth, for which An is zero. Also we

can set the constantBnequal toRn+1

, whereRis the Earths mean radius. Thepotential is then given by

UX1n0

Xnm0

R

r

n1amn cosm bmn sinm

Pmn cos (1:253)

Let thespherical harmonic functionsCmn ; and Smn ; be dened as

C

m

n ; cosm Pm

n cos Smn ; sinm Pmn cos

(1:254)

The variation of the potential over the surface of a sphere may be described by

these functions, or a more general spherical harmonic function Ymn ; thatcombines the sine and cosine variations:

Ymn ; Pmn cos cosm

sin

m

(1:255)

Like their constituent parts the sine, cosine, and associated Legendre

functions spherical harmonic functions are orthogonal and can be

normalized.

1.16 Spherical harmonic functions 49


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1.16.1 Normalization of spherical harmonic functions

Normalization of the functions Cmn ; and Smn ; requires integrating thesquared value of each function over the surface of a unit sphere. The element ofsurface area on a unit sphere is d = sin dd (Box 1.3) and the limits of

integration are 0 and 0 2. The integral is

ZZS

Cmn ; 2

d Z

0

Z20

Cmn ; 2

sin dd

Z

0

Z20

cosmPmn cos 2 sin dd (1:256)

Letx = cos in the associated Legendre polynomial, so thatdx = sin dand

the limits of integration are 1 x 1. The integration becomes

Z1

x1 Z2

0

cos2md

8>:

9>=>; P

mn x

2dx Z

1

x1

Pmn x 2

dx (1:257)

Normalization of the associated Legendre polynomials gives the result in

(1.248), thus

ZZS

Cmn ; 2

d 22n 1

n m!n m ! (1:258)

The normalization of the function Smn ; by this method delivers the sameresult.

Spherical harmonic functions make it possible to express the variation of a

physical property (e.g., gravity anomalies,g(, )) on the surface of the Earth as

an innite series, such as

g ; X1n0

Xnm0

amnCmn ; bmnSmn ;

(1:259)

The coefcients amn and bmn may be obtained by multiplying the functiong ;

by Cmn ; or Smn ; , respectively, and integrating the product over thesurface of the unit sphere. The normalization properties give



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nodal lines of latitude and the longitudinal lines separate sectors in which the

potential is greater or less than the uniform value. An example of a sectorial

spherical harmonic is Y5

5

;

, shown inFig. 1.12(b).

In the general case (m 0, n m) the potential varies with both latitude and

longitude. There aren m nodal lines of latitude andm nodal great circles (2m

meridians) of longitude. The appearance of the spherical harmonic resembles a

patchwork of alternating regions in which the potential is greater or less than the

uniform value. An example of atesseralspherical harmonic isY45 ; , whichis shown inFig. 1.12(c).

1.17 Fourier series, Fourier integrals, andFourier transforms

1.17.1 Fourier series

Analogously to the representation of a continuous function by a power series

(Section 1.10), it is possible to represent a periodic function by an innite sum

of terms consisting of the sines and cosines of harmonics of a fundamental

frequency. Consider a periodic function(t) with periodthat is dened in the

interval 0 t , so that (a)(t) is

nite within the interval; (b) (t) is periodicoutside the interval, i.e., (t + ) = (t); and (c) (t) is single-valued in the

interval except at a nite number of points, and is continuous between these

points. Conditions (a)(c) are known as the Dirichlet conditions. If they are

satised,(t) can be represented as

ft a02

X1n1

ancos nt bnsin nt (1:262)

where = 2/ and the factor 1

2 in the rst term is included for reasons ofsymmetry. This representation of (t) is known as a Fourier series. The

orthogonal properties of sine and cosine functions allow us to nd the coef-

cientsan andbn of thenth term in the series by multiplying (1.262) by sin(nt)

or cos(nt) and integrating over a full period:

an2

Z=2t

=2

ftcos nt dt

bn2

Z=2t=2

ftsin nt dt(1:263)



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Instead of using trigonometric functions, we can replace the sine and

cosine terms with complex exponentials using the denitions in (1.7), i.e., we

write

cos nt exp int expint 2

sin nt exp int expint 2i

(1:264)

Using these relationships in (1.262) yields

ft X1n0

an2

exp int expint bn2i

exp int expint

X1n0

an ibn2

exp int

X1n0

an ibn2

expint

(1:265)

The summation indices are dummy variables, so in the second sum we can

replacen by n, and extend the limits of the sum to n = ; thus

ft X1n0

an ibn2

exp int

X0

n1

an ibn2

exp int

X1

n1

an an i bn bn 2

exp int (1:266)

If we denecnas the complex number

cn an

a

n

i bn

b

n

2 (1:267)the Fourier series (1.262) can be written in complex exponential form as

ft X1

n1cnexp int (1:268)

In this case the harmonic coefcientscnare given by

cn1

Z=2t=2

ftexpint dt (1:269)

1.17 Fourier series, integrals, and transforms 53


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Box 1.5. Transition from Fourier series to Fourier integral

The complex exponential Fourier series for a function(t) is

ft X1

n1cnexp int (1)

where the complex coefcientscnare given by

cn1

Z=2

t=2

ftexpint dt (2)

In these expressions = 2/is the fundamental frequency and is the

fundamental period. From one value ofn to the next, the harmonic frequency

changes by = 2/, so the factor preceding the second equation can be

replaced by 1/= /(2). To avoid confusion when we insert (2) into (1), we

change the dummy variable of the integration tou, giving

cn

2Z=2

u=2fuexpinu du (3)

After insertion, (1) becomes

ft X1

n1

2

Z=2u=2

fuexpinu du

0B@

1CA

exp int

X1

n1

2

Z=2u=2

fuexp in t u du0B@

1CA (4)

We now dene the function within the integral as

F Z=2

u=2

fuexp in t u du (5)

The initial Fourier series becomes



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f t 12 X

1

1

F (6)

We now let the incremental frequency become very small, tending in the

limit to zero; this is equivalent to letting the periodbecome innite. The

indexnis dropped becauseis now a continuous variable; the discrete sum

becomes an integral and the functionf(t) is

f t 12

Z1

1

F d (7)

while the functionF() from (5) becomes

F Z1

u1fuexp i t u du (8)

On insertingF() into (7) we get

f t 12

Z11

Z1u1

fuexp i t u du24 35d

12

Z11

Z1u1

fuexpiu du24

35 exp it d (9)

The quantity in square brackets, on changing the variable from uback tot, is

g 12

Z1t1

ftexpit dt (10)

and the original expression can now be written

f t Z1

1g exp it d (11)

The equivalence of these two equations is known as the Fourier integral

theorem.



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Fourier series that represent odd or even functions consist of sums of sines or

cosines, respectively. In the same way, there are sine and cosine Fourier integrals

that represent odd and even functions, respectively. Suppose that the function (t)

iseven, and let us replace the complex exponential in (1.272) using(1.5):

g 12

Z1t1

ftcos t isin t dt (1:273)

The sine function is odd, so, if(t) is even, the product(t)sin(t) is odd, and the

integral of the second term is zero. The product(t)cos(t) is even, and we can

convert the limits of integration to the positive interval:

g 12

Z1t1

ftcos t dt

1

Z1t0

ftcos t dt (1:274)

Thus, if(t) is even, then g() is also even. Similarly, one nds that, if(t) is

odd,g() is also odd.

Now we expand the exponential in (1.271) and apply the same conditions of

evenness and oddness to the products:

ft Z1

1g cos t isin t d

2Z1

0

g cos t d (1:275)

If we were to substitute (1.275) back into (1.274), the integration would be

preceded by a constant 2/, the product of the two constants in these equations.

Equations (1.275) and (1.274) form a Fourier-transform pair, and it does not

matter how the factor 2/ is divided between them. We will associate it here

entirely with the second equation, so that we have the pair of equations

ft Z1

0

g cos t d

g 2

Z1t0

ftcos t dt(1:276)



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The even functions(t) andg() areFourier cosine transformsof each other.

A similar treatment for a function (t) that is oddleads to a similar pair of

equations in which the Fourier transform g() is alsooddand

ft Z1

0g sin t d

g 2

Z1t0

ftsin t dt(1:277)

The odd functions(t) andg() areFourier sine transformsof each other.

f u r t h e r r e a d i n g

Boas, M. L. (2006).Mathematical Methods in the Physical Sciences, 3rd edn. Hoboken,

NJ: Wiley, 839 pp.

James, J. F. (2004). A Students Guide to Fourier transforms, 2nd edn. Cambridge:

Cambridge University Press, 135 pp.



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2

Gravitation

2.1 Gravitational acceleration and potential

The Universal Law of Gravitation deduced by Isaac Newton in 1687 describes

the force of gravitational attraction between two point masses m and Msepa-

rated by a distancer. Let a spherical coordinate system (r,,) be centered on

the point mass M. The force of attraction F exerted on the point mass m acts

radially inwards towards M, and can be written

F G mMr2

er (2:1)

In this expression, Gis the gravitational constant (6.674 21 1011 m3 kg1 s2),

eris the unit radial vector in the direction of increasing r, and the negative sign

indicates that the force acts inwardly, towards the attracting mass. The gravita-

tional accelerationaGat distanceris the force on a unit mass at that point:

aG GM

r2 er (2:2)

The acceleration aGmay also be written as the negative gradient of a gravita-

tional potentialUG

aG rUG (2:3)

The gravitational acceleration for a point mass is radial, thus the potential

gradient is given by

UGr

G Mr2

(2:4)

UG GM

r (2:5)

59


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In Newtons time the gravitational constant could not be veried in a laboratory

experiment. The attraction between heavy masses of suitable dimensions is weak

and the effects of friction and air resistance relatively large, so the rst successful

measurement of the gravitational constant by Lord Cavendish was not made until

more than a century later, in 1798. However, Newton was able to conrm the

validity of the inverse-square law of gravitation in 1687 by using existing

astronomic observations of the motions of the planets in the solar system.

These had been summarized in three important laws by Johannes Kepler in

1609 and 1619. The small sizes of the planets and the Sun, compared with the

immense distances between them, enabled Newton to consider these as point

masses and this allowed him to verify the inverse-square law of gravitation.

2.2 Keplers laws of planetary motion

Johannes Kepler (15711630), a German mathematician and scientist, formu-

lated his laws on the basis of detailed observations of planetary positions by

Tycho Brahe (15461601), a Danish astronomer. The observations were made

in the late sixteenth century, without the aid of a telescope. Kepler found that the

observations were consistent with the following three laws (Fig. 2.1).

1. The orbit of each planet is an ellipse with the Sun at one focus.

2. The radius from the Sun to a planet sweeps over equal areas in equal intervals

of time.

S

Q

Q*

p

a

b

r P(r,)

P*

aphelion perihelion

Fig. 2.1. Illustration of Keplers laws of planetary motion. The orbit of each planetis an ellipse with the Sun at its focus (S); a,b, andpare the semi-major axis, semi-

minor axis, and semi-latus rectum, respectively. The area swept by the radius to a

planet in a given time is constant (i.e., area SPQ equals area SP*Q*); the square of

the period is proportional to the cube of the semi-major axis. After Lowrie (2007).

60 Gravitation


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3. The square of the period is proportional to the cube of the semi-major axis of

the orbit.

The fundamental assumption is that the planets move under the inuence of acentral, i.e., radially directed force. For a planet of mass mat distancerfrom the

Sun the forceF can be written

F md2r

dt2 frer (2:6)

The angular momentumh of the planet about the Sun is

h

r

m

dr

dt (2:7)

Differentiating with respect to time, the rate of change of angular momentum is

dh

dt m

d

dt r

dr

dt

m

dr

dt

dr

dt

m r

d2r

dt2

(2:8)

The rst term on the right-hand side is zero, because the vector product of a

vector with itself (or with a vector parallel to itself) is zero. Thus

dh

dt r m

d2r

dt2 (2:9)

On substituting from (2.6) and applying the same condition, we have

dh

dt r frer fr r er 0 (2:10)

This equation means thath is a constant vector; the angular momentum of the

system is conserved. On taking the scalar product ofh and r, we obtain

r h r r mdr

dt

(2:11)

Rotating the sequence of the vectors in the triple product gives

r h mdr

dt r r 0 (2:12)

This result establishes that the vectorr describing the position of a planet isalways perpendicular to its constant angular momentum vectorh and therefore

denes a plane. Every planetary orbit is therefore a plane that passes through the

Sun. The orbit of the Earth denes theeclipticplane.



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2.2.1 Keplers Second Law

Let the position of a planet in its orbit be described by polar coordinates ( r,)

with respect to the Sun. The coordinates are dened so that the angleis zero atthe closest approach of the planet to the Sun (perihelion). The angular momen-

tum at an arbitrary point of the orbit has magnitude

h mr2d

dt (2:13)

In a short interval of time t the radius vector from the Sun to the planet

moves through a small angle and denes a small triangle. The area A of

the triangle is

DA 1

2r2 D (2:14)

The rate of change of the area swept over by the radius vector is

dA

dt lim

Dt!0

DA

Dt

limDt!0

1

2r2D

Dt

(2:15)

dA

dt

1

2 r

2d

dt (2:16)

On inserting from (2.13) we get

dA

dt

h

2m (2:17)

Thus the area swept over by the radius vector in a given time is constant. This is

Keplers Second Law of planetary motion.

2.2.2 Keplers First Law

If just the gravitational attraction of the Sun acts on the planet (i.e., we ignore the

interactions between the planets), the total energy Eof the planet is constant.

The total energy E is composed of the planets orbital kinetic energy and its

potential energy in the Suns gravitational eld:

1

2 m

dr

dt 2

1

2 mr2 d

dt 2

Gm

S

r E (2:18)

The rst term here is the planets linear (radial) kinetic energy, the second term

is its rotational kinetic energy (with mr2 being the planets moment of inertia

62 Gravitation


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about the Sun), and the third term is the gravitational potential energy. On

writing

drdt

drd

ddt

(2:19)

and rearranging terms we get

dr

d

2d

dt

2 r2

d

dt

2 2G

S

r 2

E

m (2:20)

Now, to simplify later steps, we make a change of variables, writing

u 1r

(2:21)

Then

dr

d

d

d

1

u

1

u2du

d

r2

du

d

(2:22)

Substituting from (2.22) into (2.20) gives

r2ddt

2dud

2 r2 ddt

2 2G Sr 2Em (2

:23)

With the result of (2.13) we have

r2d

dt

h

m

r d

dt

1

r

h

m

u

h

m

(2:24)

On replacing these expressions, ( 2.23) becomes

h

m

2du

d

2 u2

h

m

2 2uGS 2

E

m (2:25)

du

d

2 u2 2uGS

m2

h2 2

Em

h2 (2:26)

The rest of the evaluation is straightforward, if painstaking. First we add a

constant to each side,

du

d

2 u2 2uGS

m2

h2 GS

m2

h2

2 2

Em

h2 GS

m2

h2

2(2:27)



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du

d

2 u GS

m2

h2

2 2

Em

h2 GS

m2

h2

2(2:28)

Next, we move the second term to the right-hand side of the equation, giving

du

d

2 2

Em

h2 GS

m2

h2

2 u GS

m2

h2

2(2:29)

du

d

2 GS

m2

h2

21

2Eh2

G2S2m3

u GS

m2

h2

2(2:30)

Now, we dene some combinations of these terms, as follows:

u0 GSm2

h2 (2:31)

e2 1 2Eh2

G2S2m3 (2:32)

Using these dened terms, (2.30) simplies to a more manageable form:

du

d 2

u2

0e

2

u u0

2

(2:33)

du

d

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu20e

2 u u0 2

q (2:34)

The solution of this equation, which can be tested by substitution, is

u u0 1 e cos (2:35)

The angleis dened to be zero at perihelion. The negative square root in (2.34)

is chosen because, asincreases,rincreases andu must decrease. Let

p 1

u0

h2

GSm2 (2:36)

r p

1 e cos (2:37)

This is the polar equation of an ellipse referred to its focus, and is the proof of

Keplers First Law of planetary motion. The quantityeis the eccentricity of the

ellipse, whilep is thesemi-latus rectumof the ellipse, which is half the length of

a chord passing through the focus and parallel to the minor axis (Fig. 2.1).

These equations show that three types of trajectory around the Sun are

possible, depending on the value of the total energyEin (2.18). If the kinetic

64 Gravitation


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energy is greater than the potential energy, the value ofEin (2.32) is positive,

andeis greater than 1; the path of the object is a hyperbola. If the kinetic energy

and potential energy are equal, the total energy is zero and e is exactly 1; the path

is a parabola. In each of these two cases the object can escape to innity, and the

paths are called escape trajectories. If the kinetic energy is less than the potential

energy, the total energy Eis negative and the eccentricity is less than 1. In this

case (corresponding to a planet or asteroid) the object follows an elliptical orbit

around the Sun.

2.2.3 Keplers Third Law

It is convenient to describe the elliptical orbit in Cartesian coordinates (x, y),

centered on the mid-point of the ellipse, instead of on the Sun. Dene thex-axis

parallel to the semi-major axis a of the ellipse and they-axis parallel to the semi-

minor axisb. The equation of the ellipse in Fig. 2.1is

x2

a2

y2

b2 1 (2:38)

The semi-minor axis is related to the semi-major axis by the eccentricity e, so

that

b2 a2 1 e2

(2:39)

The distance of the focus of the ellipse from its center is by denition ae. The

lengthpof the semi-latus rectum is the value ofyfor a chord through the focus.

On settingy = p and x = ae in (2.38), we obtain

p2

b2 1

ae 2

a2 1 e2

(2:40)

p2 a2 1 e2 2

(2:41)

Now consider the application of Keplers Second Law to an entire circuit of the

elliptical orbit. The area of the ellipse isab, and the period of the orbit is T, so

dA

dt

ab

T (2:42)

Using (2.17),

h

m

2ab

T (2:43)



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From (2.36) and (2.43) we get the value of the semi-latus rectum,

p

1

GS

h

m 2

1

GS

2ab

T 2

(2:44)

Substituting from (2.41) gives

a 1 e2

42a2b2

GST2

42a4

GST2 1 e2

(2:45)

After simplifying, we nally get

T2

a3

42

GS (2:46)

The quantities on the right-hand side are constant, so the square of the period is

proportional to the cube of the semi-major axis, which is Keplers Third Law.

2.3 Gravitational acceleration and the potential

of a solid sphere

The gravitational potential and acceleration outside and inside a solid spheremay be calculated from the Poisson and Laplace equations, respectively.

2.3.1 Outside a solid sphere, using Laplaces equation

Outside a solid sphere the gravitational potentialUG satises Laplaces equation

(Section 1.9). If the density is uniform, the potential does not vary with the

polar angle or azimuth . Under these conditions, Laplaces equation in

spherical polar coordinates ( 2.67) reduces to

r r2

UG

r

0 (2:47)

This implies that the bracketed quantity that we are differentiating must be a

constant,C,

r2UG

r C (2:48)

UG

r

C

r2 (2:49)

The gravitational acceleration outside the sphere is therefore

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UG

r

4

3Gr (2:58)

aG r5R UG

r 4

3Gr

er (2:59)

This shows that the gravitational acceleration inside a homogeneous solid

sphere is proportional to the distance from its center. At the surface of the

sphere,r= R, and the gravitational acceleration is

aG R 4

3GR er

GM

R2 er (2:60)

where the massMof the sphere is

M4

3R3 (2:61)

To obtain the potentialinside the solid sphere, we must integrate ( 2.58). This

gives

UG 23Gr2 C2 (2:62)

The constant of integrationC2is obtained by noting that the potential must be

continuous at the surface of the sphere. Otherwise a discontinuity would exist

and the potential gradient (and force) would be innite. Equating (2.54) and (

2.62) atr= R gives

2

3GR2

C

2

GM

R

4

3GR2 (2:63)

C2 2GR2 (2:64)

The gravitational potential inside the uniform solid sphere is therefore given by

UG 2

3Gr2 2GR2 (2:65)

UG

2

3G r2 3R2 (2:66)

A schematic graph of the variation of the gravitational potential inside and

outside a solid sphere is shown in Fig. 2.2.

68 Gravitation


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2.4 Laplaces equation in spherical polar coordinates

In the above examples the sphere was assumed to have uniform density so that

only the radial term in Laplaces equation had to be solved. This is also the case

when density varies only with radius. In the Earth, however, lateral variations of

the density distribution occur, and the gravitational potential UG is then a

solution of the full Laplace equation

1

r2

rr2

UG

r

1

r2 sin

sin

UG

1

r2 sin2

2UG

2 0 (2:67)

This equation is solved using the method ofseparation of variables. This is a

valuable mathematical technique, which allows the variables in a partial differ-

ential equation to be separated so that only terms in one variable are on one side

of the equation and terms in other variables are on the opposite side. A trial

solution forUGis

UG r; ; < r (2:68)

Here , , and are all functions of a single variable only, namelyr,, and,

respectively. Multiplying (2.67) byr2 and inserting (2.68) forUGgives

rr2

guide to geophysical equations

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