A COURSE OF HIGHER MATHEMATICS V. I. Smirnov Volume I ELEMENTARY CALCULUS ADIWES INTERNATIONAL SERIES IN MATHEMATICS A. J. LOHWATER Consulting Editor A COURSE OF Higher Mathematics VOLUME I V. I. SMIRNOV Translated by D. E. BROWN Translation edited and additions made 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. Reading, Massachusetts · Palo Alto · London PERGAMON PRESS International Series of Monographs in PURE AND APPLIED MATHEMATICS Volume 57 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 ebicmeu MameMamuKU (Kurs vysshei matematiki), published in 1957 by Fizmatgiz, Moscow MADE IN GREAT BRITAIN INTRODUCTION THIS is the first volume of a five-volume course of higher mathematics which has been studied by Soviet mathematicians, physicists and engineers for forty years. In the first two editions (1924, 1927), which were practically identical, this first volume was written jointly by J. D. Tamarkin and V. I. Smirnov, but on the title page of later editions, prepared without the late Professor Tamarkin's cognizance and deviating from the two earlier editions in many respects, Professor Smirnov's name appears alone. Professor Tamarkin's career and his contributions to both Russian and American mathematics are well known to British and American readers, but the achievements of Professor Smirnov are known to a more restricted circle. Vladimir Ivanovitch Smirnov, who was born in 1887, has had a distinguished career in research and teaching which fits him ideally for the writing of a comprehensive work of extensive proportions. His research has been mainly in the theory of functions and of differential equations but he has made valuable contributions to applied mathematics and, in particular, to theoretical seismology and all his work has been characterized by a broad scientific outlook and he has done more than any other Soviet mathematician to main- tain and strengthen the connections between mathematics and physics. His pupils, among whom are numbered S. L. Sobolev, N. E. Köchin and I. A. Lappo-Danilevskii, have maintained this tradition of work- ing in both pure and applied mathematics, a tradition which Smirnov inherited from his teacher V. A. Steklov. Professor Smirnov's teaching experience in the old Institute of Transport, in a technical high school, in the Physics Department of the Mathematics and Physics Faculty of the University of Lenin- grad, and as Director of the Theoretical Section of the Institute of Seismology, Moscow, led him to study the design of a special course of higher mathematics for physicists and engineers, a project in the course of which he received the counsel of his many physicist friends particularly V. A. Fock and T. V. Kravets. The five-volume set of which the present volume is the first is the outcome of that study. It is, of course, designed as a first course for pure mathematicians in the xl xii INTRODUCTION topics considered as well as for students and research workers whose main interest lies in the applications of mathematics. The whole work is notable not only for the wealth of the illustrations it draws from physics and technology to illuminate points in pure mathematics, but also for the clarity of the exposition. This has al- ready been recognized in the Soviet Union by the esteem by which the author's work is held by academic teachers, by the award in 1947 of the State Prize (previously called the Stalin Prize) to the author for this work, and it is to be hoped that through Mr. Brown's translation its merits will become just as well known in the English-speaking world. The present volume is an introduction to calculus and to the prin- ciples of mathematical analysis including some introductory material on functions of several variables as well as on functions of a single variable. As well as providing the material necessary for the under- standing of the methods of mathematical physics it is an excellent introduction to these subjects for students of pure mathematics. I. N. SNEDDON PREFACE TO THE EIGHTH RUSSIAN EDITION THE present edition differs very considerably from the last. The material relating to analytic geometry has been excluded, and the remaining material has been rearranged as a result. In particular, applications of the differential calculus to geometry are now to be found collected in § 7 (Chapter II). A chapter has been added which was previously the first of Volume II, dealing with complex numbers, the basic properties of integral polynomials, and the systematic integration of functions. Further substantial additions must be mentioned, apart from the various minor additions and modifications to the text. In view of the fact that quite subtle and difficult problems of higher analysis are encountered in later volumes, it was thought useful to give the theory of irrational numbers, and its use in proving tests for the existence of limits and the properties of continuous functions, at the end of § 2 (Chapter I) after the theory of limits. A rigorous definition and study of the properties of the elementary function is also to be found there. The proof of the existence of implicit functions is included in Chapter V, dealing with functions of several variables. The text is arranged so that the large type can be read independently. The small type sections contain examples, some additional particular problems, all the theoretical material referred to above, and the final section of Chapter IV, which deals with theory of a more difficult kind. My sincere thanks are due to Professor G. M. Fikhtengol'ts for a number of valuable suggestions regarding the text, which I have incorporated during the final revision of the book. PREFACE TO THE SIXTEENTH RUSSIAN EDITION THE basic text and plan of the book have remained unchanged in the present edition, though there are a number of alterations due to the requirements of accuracy and completeness. This refers especially to applications of the differential and integral calculus to geometry. V. SMIRNOV xiii PREFACE TO THE EIGHTH RUSSIAN EDITION THE present edition differs very considerably from the last. The material relating to analytic geometry has been excluded, and the remaining material has been rearranged as a result. In particular, applications of the differential calculus to geometry are now to be found collected in § 7 (Chapter II). A chapter has been added which was previously the first of Volume II, dealing with complex numbers, the basic properties of integral polynomials, and the systematic integration of functions. Further substantial additions must be mentioned, apart from the various minor additions and modifications to the text. In view of the fact that quite subtle and difficult problems of higher analysis are encountered in later volumes, it was thought useful to give the theory of irrational numbers, and its use in proving tests for the existence of limits and the properties of continuous functions, at the end of § 2 (Chapter I) after the theory of limits. A rigorous definition and study of the properties of the elementary function is also to be found there. The proof of the existence of implicit functions is included in Chapter V, dealing with functions of several variables. The text is arranged so that the large type can be read independently. The small type sections contain examples, some additional particular problems, all the theoretical material referred to above, and the final section of Chapter IV, which deals with theory of a more difficult kind. My sincere thanks are due to Professor G. M. Fikhtengol'ts for a number of valuable suggestions regarding the text, which I have incorporated during the final revision of the book. PREFACE TO THE SIXTEENTH RUSSIAN EDITION THE basic text and plan of the book have remained unchanged in the present edition, though there are a number of alterations due to the requirements of accuracy and completeness. This refers especially to applications of the differential and integral calculus to geometry. V. SMIRNOV xiii CHAPTER I FUNCTIONAL RELATIONSHIPS AND THE THEORY OF LIMITS § 1. Variables 1· Magnitude and its measurement· Mathematical analysis has a fundamental importance for exact science; unlike the other sciences, each of which has an interest only in some limited aspect of the world around us, mathematics is concerned with the most general properties inherent in all phenomena that are open to scientific investigation. One of the fundamental concepts is that of magnitude and its measurement. It is characteristic of a magnitude that it can be meas- ured, i.e. it can be compared in one way or another with some specific magnitude of the sort which is accepted as the unit of measurement. The process of comparison itself depends on the nature of the magnitude in question and is called measurement. Measurement results in an abstract number being obtained, expressing the ratio of the observed magnitude to the magnitude accepted as the unit of measurement. Every law of nature gives us a correlation between magnitudes, or more exactly, between numbers expressing these magnitudes. It is precisely the object of mathematics to study numbers and the various correlations between them, independently of the concrete nature of the magnitudes and laws which lead us to these numbers and correlations. Thus, every magnitude is related by its measurement to an abstract number. This number depends essentially, however, on the unit assumed for the measurement, or on the scale. On increasing this unit, the number measuring a given magnitude decreases, and con- versely, the number increases on decreasing the unit. The choice of scale is governed by the character of the magnitude concerned and by the circumstances in which the measurements are carried out. The size of the scale used for measuring one and the same 1 2 FUNCTIONAL EBLATIONSHIPS AND THE THEORY OF LIMITS [2 magnitude can vary within the widest possible limits — for instance, in measuring length in accurate optical studies the accepted unit of length is an Angstrom (one ten-millionth of a millimetre, 10~10m); whereas use is made in astronomy of a unit of length called a light- year, i.e. the distance travelled by light in the course of a year (light travels approximately 300,000 km in one second). 2. Number. The number which is obtained as a result of measurement may be integral (if the unit goes an integral number of times into the magnitude concerned), fractional (if another unit exists, which goes an integral number of times both into the measured magnitude and into the unit previously chosen — or in short, when the measured magni- tude is commensurable with the unit of measurement) and finally, irrational (when no such common measure exists, i.e. the given magni- tude proves incommensurable with the unit of measurement). It is shown in elementary geometry, for instance, that the diagonal of a square is incommensurable with its side, so that, if we measure the diagonal of a square using the length of side as unit, the number |^2 obtained by measurement is irrational. The number π is similarly irrational, obtained on measuring the circumference of a circle, the diameter of which is taken as unit. Reference can usefully be made to decimal fractions, in order to understand the idea of irrational numbers. As is known from arith- metic, every rational number can be represented in the form of either a finite or an infinite decimal fraction, the infinite fraction being peri- odic in the latter case (simple periodic or compound periodic). For instance, on carrying out division of the numerator by the denominator in accordance with the rule for division into decimal fractions, we obtain: -^- = 0.151515... =0.1(5), -^ = 0.2777... = 0.2 (7). Conversely, as is known from arithmetic, every periodic decimal fraction expresses a rational number. In measuring a magnitude, incommensurable with the unit taken, we can first reckon how many times a full unit goes into the measured magnitude, then how many times a tenth of a unit goes into the re- mainder obtained, then how many times a hundredth of a unit goes into the new remainder and so on. Measurement of a magnitude, incommen-