Table Of ContentLaplace's equation is one of the simplest partial dilTcrelllial
equations arising in ekctricity and magnetism, /luid
mechanics, heat conduction and other branches of
mathematical physics. A study of the equation and its
solutions is essential to the understanding of these suh
jects. This book not only provides this study hut docs so
by methods which the reader can later apply to more
complex physical situations.
Solutions of Laplace's Equations. $1' 00
SOLUTIONS OF
LAPLACE'S EQUATION
LIBRARY OF MATHEMATICS
edited by
WALTER LEDERMANN
D.Se., Ph.D., F.R.S.Ed., Professor of
Mathematics, University of Sus~ex
Linear Equations P. M. Cohn
Sequences and Series J. A. Green
Differential Calculus P. J. Hilton
Elementary Differential
Equations and Operators G. E. H. Reuter
Partial Derivatives P. J. Hilton
Complex Numbers W. Ledermann
Principles of Dynamics M. B. Glauert
Electrical and Mechanical
Oscillations D. S. Jones
Vibrating Strings D. R. Bland
Vibrating Systems R. F. Chisnell
Fourier Series I. N. Sneddon
Solutions of Laplace's Equation D. R. Bland
Solid Geometry P. M. Cohn
Numerical Approximation B. R. Morton
Integral Calculus W. Ledermann
Sets and Groups J. A. Green
Differential Geometry K. L. Wardle
Probability Theory A. M. Arthurs
SOLUTIONS OF
LAPLACE'S EQUATION
BY
D. R. BLAND
ROUTLEDGE AND KEGAN PAUL
LONDON
First published 1961
by Routledge & Kegan Paul Ltd
Broadway House, 68-74 Carter Lane
London, E.C.4
by Latimer, Trend i-J Co Ltd, Plymouth
ISBN-13: 978-0-7100-4353-5 e-ISBN-13: 978-94-011-7694-1
DOl: 10.1007/978-94-011-7694-1
C D. R. Bland 1961
No part of this book may be reproduced
in any form without permission from
the publisher, except for the quotation
of brief passages in criticism
Second impression 1965
Preface
THIS book is an introduction both to Laplace's equation
and its solutions and to a general method of treating partial
differential equations. Chapter 1 discusses vector fields and
shows how Laplace's equation arises for steady fields which
are irrotational and solenoidal. In the second chapter the
method of separation of variables is introduced and used to
reduce each partial differential equation, Laplace's equa
tion in different co-ordinate systems, to three ordinary
differential equations. Chapters 3 and 5 are concerned with
the solutions of two of these ordinary differential equations,
which lead to treatments of Bessel functions and Legendre
polynomials. Chapters 4 and 6 show how such solutions
are combined to solve particular problems. This general
method of approach has been adopted because it can be
applied to other scalar and vector fields arising in the physi
cal sciences; special techniques applicable only to the solu
tions of Laplace's equation have been omitted. In particular
generating functions have been relegated to exercises. After
mastering the content of this book, the reader will have
methods at his disposal to enable him to look for solutions
of other partial differential equations.
The author would like to thank Dr. W. Ledermann for
his criticism of the first draft of this book.
D. R.BLAND
The University,
Sussex.
v
Contents
Preface page v
1. Occurrence and Derivation of Laplace's Equation
1. Situations in which Laplace's equation arises 1
2. Laplace's equation in orthogonal curvilinear
co-ordinates 8
3. Laplace's equation in particular co-ordinate
systems 12
2. The Method of Separation of Variables
1. Rectangular Cartesian co-ordinates 15
2. Temperature distribution in a rectangular
metal block 18
3. The analogous electrostatic problem 23
4. Cylindrical polar co-ordinates 25
5. Spherical polar co-ordinates 26
3. Bessel Functions
1. An infinite series solution of Bessel's equation 29
2. Bessel functions of the second kind 35
3. Derivatives of Bessel functions and recu"ence
formulae 37
4. Modified Bessel functions 39
vii
CONTENTS
5. Behaviour of Bessel functions at zero and
infinity 42
6. Series of zero order Bessel functions 43
4. Solutions using Cylindrical Polar Co-ordinates
1. Form of solutions of Laplace's equation 48
2. An infinite cylinder in a uniform field 49
3. A particular solid of revolution in a uniform
field 52
4. Axi-symmetric temperature distributions in a
cylinder 55
5. Legendre Polynomials
1. Solution in series of Legendre's equation 61
2. Associated Legendre functions 69
3. Derivatives and recurrence formulae for
Legendre polynomials 71
4. Series of Legendre polynomials 73
6. Solutions using Spherical Polar Co-ordinates
1. Form of solutions of Laplace's equation 78
2. Sphere moving in a liquid at rest at infinity 78
3. A charged conducting sphere in a uniform electric
field 79
4. Dielectric sphere in a uniform electric field 81
5. Axi-symmetric temperature distributions tn a
hollow sphere 83
6. Flow past a nearly spherical body 85
7. Sources, sinks and doublets 90
8. Doublet in a fluid bounded by a sphere 93
9. Doublet in a cavity in a dielectric medium 94
viii
CHAPTER ONE
Occurrence and Derivation of
Laplace's Equation
1. SITUATIONS IN WHICH LAPLACE'S EQUATION ARISES
If a vector v can be associated with each point in a given
space, then v is said to be a vector field. Two everyday ex
amples are an electric field and a velocity field in a moving
fluid. In an electric field, the electric intensity E at any
point is a vector whose magnitude and direction are equal
to the magnitude and direction of the force which would
be exerted on a unit charge if it were placed at that point;
generally the vector E varies from point to point. In a fluid
the velocity v at any point is the velocity of the particle
instantaneously situated at that point. Although in general
vector fields are functions of time as well as of space, in this
book we shall be concerned only with space dependence
and we shall assume that all vector fields with which we deal
are steady, i.e. they do not change with time.
We shall now restrict ourselves to special types of steady
vector fields, namely those that are both irrotational and
solenoidal. First let us explain the meaning of these terms.
Consider a curve C in space and let P and Q be two neigh-
-+
bouring points on the curve. Let the vector PQ be ds. Let
the vector field vat P be equal to v(P). Then we can form
the scalar product v(P). ds. We can form this product for
all adjoining pairs of neighbouring points on the curve.
Suppose the curve starts at A and ends at B. Then, adding
together the scalar product from each pair of neighbouring
points, we obtain ~v . ds where, in each term of the sum,
v is the field vector at the first of the two points and ds is
1
OCCURRENCE AND DERIVATION
;!9C B
A
FIG. I
the vector joining them. As ds-+O, the number of terms in
the sum becomes infinite and in the limit the sum becomes
an integral. It is called the 'line integral' of v from A to B
along the curve C and is denoted by
f:v.
ds.
C
In general the value of a line integral of a vector field be
tween two points will depend on the path between them.
The special property that distinguishes irrotational vector
fields from other vector fields is simply that the value of the
line integral of the vector field between any two points is
independent of the path chosen between them. In the dia
gram above, this means that, if the curve C' had been chosen
instead of C, then
f:v. ds=f:v .
ds. (1)
C' c
The line integral of an irrotational vector field around any
closed path is zero. This statement will now be proved.
O
R
P Q
FIG. a
2
