Table Of ContentThis is a sequel to the author's Integral Calculus. The
notion of integration is extended to functions of several
variables based on the intuitive idea of summation, but
explicit treatment is limited to curvilinear, double, triple
and surface integrals. Techniques are presented for
evaluating those integrals, particularly those which occur
in Physics.
Numerous examples are worked in the text and exercises,
with solutions, are appended to each chapter.
MULTIPLE INTEGRALS
LIBRARY OF MATHEMATICS
edited by
WALTER LEDERMANN
D.Se., Ph.D., F.R.S.Ed., Professor of
Mathematics, University of Sussex
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 Systems R. F. Chisnell
Vibrating Strings D. R. Bland
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
Multiple Integrals W. Ledermann
MULTIPLE
INTEGRALS
BY
WALTER LEDERMANN
LONDON: Routledge & Kegan Paul Ltd
NEW YORK: Dover Publications Inc
First Published 1966
in Great Britain by
Routledge & Kegan Paul Ltd
Broadway House, 68-74 Carter Lane
London, E.C.4
and in the U.S.A. by
Dover Publications Inc.
180 Varick Street
New York, 10014
Second impression 1968
© Walter Ledermann 1966
No part of this book may be reproduced
in any form without permission from
the publisher, except for the quotation
of briefp assages in criticism
ISBN-13: 978-0-7100-4358-0 e-ISBN-13: 978-94-011-6091-9
DOl: 10.1007/978-94-011-6091-9
Library of Congress Catalog
Card Number 66-21244
Preface
The aim of this book is to give an elementary treatment of
multiple integrals. The notions of integrals extended over a
curve, a plane region, a surface and a solid are introduced in
tum, and methods for evaluating these integrals are presented
in detail. Especial reference is made to the results required in
Physics and other mathematical sciences, in which multiple
integrals are an indispensable tool.
A full theoretical discussion of this topic would involve deep
problems of analysis and topology, which are outside the scope
of this volume, and concessions had to be made in respect of
completeness without, it is hoped, impairing precision and a
reasonable standard of rigour. As in the author's Integral
Calculus (in this series), the main existence theorems are first
explained informally and then stated exactly, but not proved.
Topological difficulties are circumvented by imposing some
what stringent, though no unrealistic, restrictions on the
regions of integration.
Numerous examples are worked out in the text, and each
chapter is followed by a set of exercises.
My thanks are due to my colleague Dr. S. Swierczkowski,
who read the manuscript and made valuable suggestions.
w.
LEDERMANN
The University of Sussex,
Brighton.
v
Contents
Preface page v
1. Line Integrals
1. Preliminary remarks about curves 1
2. Definition of a curvilinear integral 4
3. Area 12
Exercises 20
2. Double Integrals
1. Definition of a double integral 22
2. Double integrals over a rectangle 26
3. Double integrals over normal regions 31
4. Green's Theorem in two dimensions 36
5. Change of variables 41
6. Transformation of double integrals 49
7. Improper integrals 56
8. Volumes 62
Exercises 66
3. Surface Integrals
1. Preliminary remarks about surfaces 68
2. The area of a surface 72
3. Surface integrals 75
Exercises 78
4. Volume Integrals
1. Definition of a volume integral 80
2. Reduction to repeated integral 81
3. Gauss's Theorem (divergence theorem) 84
4. Change of variables 88
5. Stokes's Theorem 93
Exercises 96
Appendix 99
Solutions to Exercises 103
Index 106
vi
CHAPTER ONE
Line Integrals
1. PRELIMINARY REMARKS ABOUT CURVES
If 4>(t) and 1jl(t) are continuous functions of t, defined in an
interval oc <:; t <:; (J, the equations
x = 4>(t), y = 1jJ(t) (oc <:; t < (J) (1)
determine a path in the (x,y)-plane. We may think of 1 as the
time and interpret (1) as the motion of a point whose coordin
ates at time tare (4)(t), 1jJ(t». For brevity, we often refer to this
point as the point t of the path. The initial point and the end
point of the path are A = (4)(oc), 'IJ'(a» and B = (4)({J), 1jJ({J)
respectively. For a closed path, or loop, we have that
4>( oc) = 4>({J), 1jJ( oc) = 1jJ({J) •
•I n most cases the functions 4> and 'IJ' are not only continuous
but possess continuous derivatives, except at a finite number
of points. It may, however, be necessary to employ separate
formulae to describe different sections of the path. For ex
ample, the path presented in Fig. 1 is defined as follows: .
< <
t (0 t 1)
<
{1 (1 1 <2)
x = 1 + sin ~(t - 2) (2 < t < 3)
{t
Y = (0 <:; ~ < 2)
< <
3 - cos "2(1 - 2) (2 t 3).
1
LINE INTEGRALS
Y
(2,3)
__ ________
~ ~x
o
Figure 1.
The curve which constitutes the geometrical picture of th~ path
is sometimes called the track of the path, but we observe that
the equations (1) not only specify the curve but also indicate
the sense and 'speed' with which the point traverses the curve.
Thus, if 11 < ta, the point 11 precedes the point ta. If we are
interested in the curve rather than in the path, we say that (1)
gives a parametric representation of the curve, the variable I
being called the parameter.
It should be noted that a given curve possesses many para
metric equations. For example, each of the two pairs of
equations
(i) x = cos nl, y = sin nl (0 < t <: 1)
1 - u2 2u
(ii)x=--,y=-- (0 <u <(0)
+ +
1 u2 1 u2
represents the semi-circle in the upper half-plane starting at (1,0)
and ending at ( -1,0) (in (ii) the end-point corresponds to u = (0).
Thus (i) and (ii) are distinct paths which correspond to the
2
