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Differential and Integral Calculus Differential and Integral Calculus DIFFERENTIAL AND INTEGRAL CALCULUS DIFFERENTIAL AND INTEGRAL CALCULUS BY R. COURANT Professor of Mathematics in New York University TRANSLATED BY E. J. McSHANE Professor of Mathematics in the University of Virginia VOLUME I SECOND EDITION Wiley Classics Library Edition Published 1988 WILEY INTERSCIENCE PUBLISHERS A DIVISION OF JOHN WILEY & SONS, INC. First published 1934 Second edition 1937 Reprinted 1940, 1941, 1942 (.twice) 1943, 1944, 194S, 1946, 1947, 1948 1949, 1950, 1951, 19S2, 19S3, J9S4 1)55, 1956 (twice). 1957 (twice) 1958, 1959, 1960, 1961, 1962, 1993 1964, 1965, 1966, 1967,1968,1970 10 ISBN 0 471 17820 9 ISBN 0-471-60842-4 (pbk.). Printed in the United States of America PREFACE TO THE FIRST GERMAN EDITION Although there is no lack of textbooks on the differential and integral calculus, the beginner will have difficulty in finding a book that leads him straight to the heart of the subject and gives him the power to apply it intelligently. He refuses to be bored by diffuseness and general statements which convey nothing to him, and will not tolerate a pedantry which makes no distinction between the essential and the non-essential, and which, for the sake of a systematic set of axioms, deliberately conceals the facts to which the growth of the subject is due. True, it is easier to perceive defects than to remedy them. I make no claim to have presented the beginner with the ideal textbook. Yet I do not consider the publication of my lectures superfluous. In order and choice of material, in fundamental aim, and perhaps also in mode of presentation, they differ con- siderably from the current literature. The reader will notice especially the complete break away from the out-of-date tradition of treating the differential calculus and the integral calculus separately. This separation, a mere result of historical accident, with no good foundation either in theory or in practical convenience in teaching, hinders the student from grasping the central point of the calculus, namely, the connexion between definite integral, indefinite integral, and derivative. With the backing of Felix Klein and others, the simultaneous treatment of differential calculus and integral calculus has steadily gained ground in lecture courses. I here attempt to give it a place in the literature. This first volume deals mainly with the integral and differential calculus for func- tions of one variable; a second volume will be devoted to functions of several variables and some other extensions of the calculus. vi PREFACE TO THE FIRST GERMAN EDITION My aim is to exhibit the close connexion between analysis and its applications and, without loss of rigour and precision, to give due credit to intuition as the source of mathematical truth. The presentation of analysis as a closed system of truths without reference to their origin and purpose has, it is true, an aesthetic charm and satisfies a deep philosophical need. But the attitude of those who consider analysis solely as an abstractly logical, introverted science is not only highly unsuitable for beginners but endangers the future of the subject; for to pursue mathe- matical analysis while at the same time turning one's back on its applications and on intuition is to condemn it to hopeless atrophy. To me it seems extremely important that the student should be warned from the very beginning against a smug and presumptu- ous purism; this is not the least of my purposes in writing this book. The book is intended for anyone who, having passed through an ordinary course of school mathematics, wishes to apply him- self to the study of mathematics or its applications to science and engineering, no matter whether he is a student of a univer- sity or technical college, a teacher, or an engineer. I do not promise to save the reader the trouble of thinking, but I do seek to lead the way straight to useful knowledge, and aim at making the subject easier to grasp, not only by giving proofs step by step, but also by throwing light on the interconnexions and purposes of the whole. The beginner should note that I have avoided blocking the entrance to the concrete facts of the differential and integral calculus by discussions of fundamental matters, for which he is not yet ready. Instead, these are collected in appendices to the chapters, and the student whose main purpose is to acquire the facts rapidly or to proceed to practical applications may post- pone reading these until he feels the need for them. The appen- dices also contain some additions to the subject-matter; they have been made relatively concise. The reader will notice, too, that the general style of presentation, at first detailed, is more condensed towards the end of the book. He should not, however, let himself be disheartened by isolated difficulties which he may find in the concluding chapters. Such gaps in understand- ing, if not too frequent, usually fill up of their own accord. PREFACE TO THE ENGLISH EDITION When American colleagues urged me to publish an English edition of my lectures on the differential and integral calculus, I at first hesitated. I felt that owing to the difference between the methods of teaching the calculus in Germany and in Britain and America a simple translation was out of the question, and that fundamental changes would be required in order to meet the needs of English-speaking students. My doubts were not laid to rest until I found a competent colleague in Professor E. J. McShane, of the University of Virginia, who was prepared not only to act as translator but also —after personal consultation with me—to make the improve- ments and alterations necessary for the English edition. Apart from many matters of detail the principal changes are these: (1) the English edition contains a large number of classified examples; (2) the division of material between the two volumes differs somewhat from that in the German text. In addition to a detailed account of the theory of functions of one variable, the present volume contains (in Chapter X) a sketch of the differentiation and integration of functions of several variables. The second volume deals in full with functions of several inde- pendent variables, and includes the elements of vector analysis. There is also a more systematic discussion of differential equa- tions, and an appendix on the foundations of the theory of real numbers. Thus the first volume contains the material for a course in elementary calculus, while the subject-matter of the second volume is more advanced. In the first volume, however, there is much which should be omitted from a first course. These sec- tions, intended for students wishing to penetrate more deeply into the theory, are collected in the appendices to the chapters, l* vii (E798) VI11 PREFACE TO THE ENGLISH EDITION so that beginners can study the book without inconvenience, omitting or postponing the reading of these appendices. The publication of this book in English has only been made possible by the generosity of my German publisher, Julius Springer, Berlin, to whom I wish to express my most cordial thanks. I have likewise to thank Blackie and Son, Ltd., who in spite of these difficult times have undertaken to publish this edition. My special thanks are due to the members of their technical staff for the excellent quality of their work, and to their mathematical editors, especially Miss W. M. Deans, who have relieved Prof. McShane and myself of much of the respon- sibility of preparing the manuscript for the press and reading the proofs. I am also indebted to many friends and colleagues, notably to Professor McClenon of Grinnell College, Iowa, to whose encouragement the English edition is due; and to Miss Margaret Kennedy, Newnham College, Cambridge, and Dr. Fritz John, who co-operated with the publisher's staff in the proof-reading. R. COURANT. CAMBBIDOE, ENGLAND, June, 1934. PREFACE TO THE SECOND ENGLISH EDITION This second edition differs from the first chiefly in the improvement and rearrangement of the examples, the addi- tion of many new examples at the end of the book, and the inclusion of some additional material on differential equations. R. COURANT. NEW ROCHBLLE, N.Y., June, 1937. CONTENTS Page INTRODUCTORY REMARKS 1 CHAPTER I INTRODUCTION 1. The Continuum of Numbers ....... 5 2. The Concept of Function 14 3. More Detailed Study of the Elementary Functions - 22 i. Functions of an Integral Variable. Sequences of Numbers 27 5. The Concept of the Limit of a Sequence 29 6. Further Discussion of the Concept of Limit .... 38 7. The Concept of Limit where the Variable is Continuous • • 46 8. The Concept of Continuity - - 49 APPENDIX 1 Preliminary Remarks ........ 56 1. The Principle of the Point of Accumulation and its Applications - 58 2. Theorems on Continuous Functions ------ 63 3. Some Remarks on the Elementary Functions .... 68 APPENDIX II 1. Polar Co-ordinates 71 2. Remarks on Complex Numbers 73 CHAPTER II THE FUNDAMENTAL IDEAS OF THE INTEGRAL AND DIFFERENTIAL CALCULUS 1. The Definite Integral 76 2. Examples 82 tx X CONTENTS Page 3. The Derivative 88 4. The Indefinite Integral, the Primitive Function, and the Funda- mental Theorems of the Differential and Integral Calculus - 109 5. Simple Methods of Graphical Integration - - - 119 6. Further Remarks on the Connexion between the Integral and the Derivative ......... 121 7. The Estimation of Integrals and the Mean Value Theorem of the Integral Calculus • • • • • - - 126 APPENDIX 1. The Existence of the Definite Integral of a Continuous Function 131 2. The Relation between the Mean Value Theorem of the Differential Calculus and the Mean Value Theorem of the Integral Cal- culus • 134 CHAPTER III DIFFERENTIATION AND INTEGRATION OF THE ELEMENTARY FUNCTIONS 1.
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