A Course of Higher Mathematics: Advanced Calculus

637 Pages·1964·22.632 MB·English

by  V. I. Smirnov

A COURSE OF
HIGHER
MATHEMATICS
V. I. Smirnov
Volume II
ADVANCED
CALCULUS
A COURSE OF
Higher Mathematics
VOLUME II
V. I. SMIRNOV
Translated by
D. E. BROWN
Translation edited by
I. N. SNEDDON
Simson Professor in Mathematics
University of Glasgow
PERGAMON PRESS
OXFORD • LONDON • EDINBURGH • NEW YORK
PARIS • FRANKFURT
ADDISON-WESLEY PUBLISHING COMPANY, INC.
READING, MASSACHUSETTS PALO ALTO LONDON
1964
Copyright © 1964
PERGAMON PRESS LTD.
U. S. A. Edition distributed by
ADDISON-WESLEY PUBLISHING COMPANY, INC.
Beading, Massachusetts • Palo Alto • London
PERGAMON PRESS
International Series of Monographs in
PURE AND APPLIED MATHEMATICS
Volume 58
Library of Congress Catalog Card No. 63-10134
This translation has been made from
the Sixteenth (revised) Russian Edition of
V. I. Smirnov's book
Kypc ebiciueii MameMamuKU (Kurs vysshei matematiki),
published in 1958 by Fizmatgiz, Moscow
MADE IN GREAT BRITAIN
INTRODUCTION
SOME account of the history of this five-volume course of higher
mathematics has been given in the Introduction to Vol. I of the
present English edition.
The first Russian edition of the present volume appeared (in 1926)
under the joint authorship of Professor Smirnov and the late Professor
J. D. Tamarkin but later editions, prepared after Professor Tamarkin
had settled in the U.S.A. and consisting of a drastic revision of (and
many additions to) the original material, contained only Professor
Smirnov's name. This volume is made up of a course of advanced
calculus which is of great use to students of mathematics and which
provides the physicists and engineers with a complete set of those
tools, based upon the theory of functions of real variables, which
are indispensable for the study of the classical branches of mathe­
matical physics.
It consists essentially of five distinct parts, although there are
strong links connecting all of them. There is a full discussion of the
solution of ordinary differential equations with many applications to
the treatment of physical problems. This is followed by an account
of the properties of multiple integrals and of line integrals, with a
valuable section on the theory of measurable sets and of multiple
integrals.
The mathematics necessary to the discussion of problems in classical
field theories is discussed in a section on vector algebra and vector
analysis; the methods developed are illustrated not only by applica­
tions to physics but also by an account of the elements of differential
geometry in three-dimensional space.
After this there comes an elementary but full account of Fourier
series.
The principles and techniques developed in these sections are then
applied to the discussion of the solution of the partial differential
equations of classical mathematical physics.
The clarity of Prof. Smirnov's exposition and the width of his
knowledge of the mathematical techniques effective in the study of
the physical sciences makes the whole course a most valuable one for
xi
xii INTRODUCTION
the student anxious not only to learn the methods of advanced cal­
culus but also to understand the influences which have motivated
their development.
I. N. SNEDDON
PREFACE TO THE SIXTH EDITION
THIS edition of the second volume differs considerably from the pre­
vious one. The first chapter of the previous edition, containing the
theory of complex numbers, the principles of higher algebra, and
integration of functions, was transferred to the first volume. At the
same time, all material referring to the principles of vector algebra
was taken from Volume I to Volume II. This material was incorporated
in Chapter IV, together with vector analysis
The presentation of the remaining chapters underwent substantial
changes. This refers particularly to Chapters HI, VT and VII. A special
paragraph containing the theory of dimensions and the rigorous theory
of multiple integrals was added to Chapter HI. A certain re-distribution
of material was carried out in Chapter VI, and a proof was added of
the closure equation on the basis of Weierstrass' theorem on polyno­
mial approximation to continuous functions. Chapter VII now contains
additional material on the propagation of spherical and cylindrical
waves and Kirchhoff's formula for the solution of the wave equation.
The explanation of linear differential equations with constant coef­
ficients is introduced at first without using the symbolic method.
First paragraphs of each chapter have retained their explanatory
character. The book is arranged in such a way that the basic material
in larger type can be studied without reference to the examples or
complementary theoretical material printed in small type.
I should like to express my deep gratitude to Prof. Fikhtengol'ts,
who has read the manuscript of this edition, for his valuable sugges­
tions concerning the style and arrangement of the book.
V. SMIRNOV
xiii
PREFACE TO THE FOURTEENTH EDITION
THE GENERAL arrangement of the present edition is the same as that
of the previous edition. However, small alterations were introduced
in many places with the aim of clarifying the style and achieving
greater readability.
Most substantial changes were carried out in Paragraph 9 (Chapter
HI), "Supplementary remarks on the theory of multiple integrals".
In Chapter VII, devoted to simple problems of mathematical physics,
the formulation of conditions for the solution of a series of basic
problems was clarified. References to matters explained in detail in
Volume IV have been added in several places in Chapter VH.
V. SMIRNOV
xiv
CHAPTER I
ORDINARY DIFFERENTIAL EQUATIONS
§ 1. Equations of the first order
1. General principles. A differential equation is defined as an
equation which contains, in addition to independent variables and
unknown functions, derivatives or differentials of the unknown
functions [1, 51]. If the functions appearing in a differential equation
depend on a single independent variable, the equation is called an
ordinary differential equation. On the other hand, if partial derivatives
of the functions with respect to certain of the independent variables
appear in the equation, it is called a partial differential equation.
We confine ourselves to ordinary differential equations in the present
chapter, the greater part of which is devoted to the case of a single
equation containing one unknown function.
Let x be the independent variable, and y the required function of
this variable. The general form of the differential equation becomes:
&(x,v,y'y",...,yin)) =o.
The order of the differential equation is defined as the order n of
the highest order derivative of the function that appears in the equation.
We shall consider ordinary differential equations of the first order
in the present article. The general form of this equation is:
<%,</, */') = <> (i)
or, on solving with respect to y'\
y'= /(*,»). (2)
If a function
V = <P{x) (3)
satisfies the differential equation, i.e. if the equation reduces to an
identity on replacing y and y' by <p(x) and g/(a?), the function tp(x)
is said to be a solution of the differential equation.
1
2 OKDINAKY DIFFERENTIAL EQUATIONS [2
The problem of finding a solution of a differential equation is
alternatively referred to as the task of integrating the equation.
If x and y are considered as the coordinates of points on a plane,
differential equation (1) [or (2)] expresses a relationship between
coordinates of points on a certain curve and the slopes of the tangents
to the curve at these points. A curve corresponds to the solution (3)
of the equation, the points and tangential slopes of which satisfy
the equation. This curve is referred to as an integral curve of the given
differential equation.
In the simplest case, when the right-hand side of equation (2) does
not contain y, a differential equation is obtained of the form:
y = t(x).
Finding the solutions of this equation is the primary task of the
integral calculus [I, 86], and the total set of solutions is given by the
formula:
y = $f{x)dz+C,
where C is an arbitrary constant. We thus obtain in this elementary
case a solution of the differential equation containing an arbitrary
constant. We shall see that a solution containing an arbitrary constant
is also obtained in the general case of a first order differential equa­
tion; such a solution is referred to as the general solution of the equa­
tion. On assigning different numerical values to the arbitrary constant,
we obtain the various so-called particular solutions of the equation.
We give in the following sections some particular types of first
order equation, integration of which leads to evaluation of indefinite
integrals — or, as it may sometimes be expressed, their integration
reduces to quadrature.^
2. Equations with separable variables. On replacing y' in equation (2)
by the quotient dy/dx, multiplying both sides by dx, and carrying all
terms to the left-hand side, we can write (2) in the form:
M(x y) dx + N (x, y) dy = 0, (4)
9
which will be more convenient in some cases. Both variables x and y
play an identical role here in the equation, so that (4) does not bind
us to the choice of unknown function: we can take either x or y for
this, as we wish.
t Evaluation of an integral has a direct connection with evaluation of an
area, whence the term "quadrature".
2] EQUATIONS WITH SEPARABLE VARIABLES 3
We assume that each of the functions M(x, y) and N(x, y) consists
of the product of two factors, one of which depends only on x, and
the other only on y:
M^x) M2(y) dx + N^x) N2(y) dy = 0. (*)
On dividing both sides of the equation by M (y) N^x), we reduce it
2
to the form:
M(x)
t dx + dy = 0, (6)
N(x) M (y)
t 2
so that the coefficient of dx now depends only on x, and the coefficient
of dy only on y. Equation (5) is called an equation with separable
variables [1, 93], whilst the method itself of reduction to the form (6)
is called separation of the variables.
The left-hand side of equation (6) is the differential of the follow­
ing expression:
f Ml{x) dx 1 f N*{v) dv
J N x) aX + ) M (y) ay>
l{ 2
and the equating to zero of the differential of this expression means
that the expression itself is equal to an arbitrary constant:
(7)
J N(x) J M (y) "
t 2
where C is the arbitrary constant. This formula gives an infinite se
of solutions and, from the geometrical point of view, is the implici
equation of a family of integral curves. If
the quadratures are carried out in (7) and
we solve the equation with respect to y,
we obtain the explicit equation of the
family of integral curves (the solution of
the differential equation):
= cp{x, C).
y
Example. The area OAMN, bounded by the
FIG. 1
coordinate axes, the segment AM of a curve
and its ordinate MN (Fig. 1), is equal in mag­
nitude to a rectangular area OBCN with the same base ON = x and with
height rj:
x x
\ydx = xrj; rj = \y dx. (8)