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柯朗-微积分和数学分析引论-英文版-第一卷--高清扫描版-Courant, R., John, F. (1965). Introduction to Caculus and Analysis (Volume One.pdf
Introduction to CALCULUS AND ANALYSIS Volume one Richard Courant and fritz John Courant Institute of mathematical Sciences New York Universit Interscience publishers A Division of John Wiley and Sons, Inc New york· London· Sydney Copyright 1965 by Richard Courant All Rights Reserved. This book or any part thereof must not be reproduced in any form without the written permission of the publisher Library of Congress Catalog Card Number: 65-16403 Printed in the United states of americe Preface During the latter part of the seventeenth century the new mathe matical analysis emerged as the dominating force in mathematics. It is characterized by the amazingly successful operation with infinite processes or limits. Two of these processes, differentiation and inte gration, became the core of the systematic Differential and Integral Calculus, often simply called"Calculus, "basic for all of analysis The importance of the new discoveries and methods was immediatel y felt and d intellectual t. Yet, to gail of the powerful art appeared at first a formidable task, for the avail ible publications were scanty, unsystematic, and often lacking in clarity. Thus, it was fortunate indeed for mathematics and science in general that leaders in the new movement soon recognized the vital need for writing textbooks aimed at making the subject ac cessible to a public much larger than the very small intellectual elite of the early days. One of the greatest mathematicians of modern times Leonard Euler, established in introductory books a firm tradition and these books of the eighteenth century have remained sources of inspira tion until today, even though much progress has been made in the clarification and simplification of the material After Euler, one author after the other adhered to the separation of differential calculus from integral calculus, thereby obscuring a key point, the reciprocity between differentiation and integration. Only in 1927 when the first edition of R. Courants german vorlesungen uber Differential und Integralrechnung, appeared in the Springer-Verlag was this separation eliminated and the calculus presented as a unified subject From that German book and its subsequent editions the present york originated. With the cooperation of James and Virginia McShaue a greatly expanded and modified English edition of the " Calculus"wes prepared and published by Blackie and Sons in Glasgow since 1934, and vi Preface distributed in the United States in numerous reprintings by science-Wiley During the years it became apparent that the need of college and uni versity instruction in the United States made a rewriting of this work desirable. Yet, it seemed unwise to tamper with the original versions which have remained and still are viable Instead of trying to remodel the existing work it seemed preferable to supplement it by an essentially new book in many ways related to the European originals but more specifically directed at the needs of the present and future students in the United States. Such a plan became feasible when Fritz John, who had already greatly helped in the prepara tion of the first English edition, agreed to write the new book together with R. Courant While it differs markedly in form and content from the original, it is animated by the same intention: To lead the student directly to the heart of the subject and to prepare him for active application of his knowledge. It avoids the dogmatic style which conceals the motivation and the roots of the calculus in intuitive reality to exhibit the interac- tion between mathematical analysis and its various applications and to emphasize the role of intuition remains an important aim of this new book. Somewhat strengthened precision does not, as we hope, inter- fere with this aim Mathematics presented as a closed linearly ordered system of truths without reference to origin and purpose has its charm and satisfies a philosophical need But the attitude of introverted science is unsuitable for students who seek intellectual independence rather than indoctrina tion; disregard for applications and intuition leads to isolation and atrophy of mathematics. It extremely important that students and instructors should be protected from smug purism The book is addressed to students on various levels to mathema ticians, scientists, engineers. It does not pretend to make the subject easy by glossing over difficulties, but rather tries to help the genuinely interested reader by throwing light on the interconnections and purposes of the whole Instead of obstructing the access to the wealth of facts by length discussions of a fundamental nature we have sometimes postponed such discussions to appendices in the various chapters Numerous examples and problems are given at the end of various chapters. Some are challenging, some are even difficult; most of them supplement the material in the text. In an additional pamphlet more eface VIl problems and exercises of a routine character will be collected, and moreover, answers or hints for the solutions will be given Many colleagues and friends have been helpful. Albert A. Blank not only greatly contributed incisive and constructive criticism but he also played a major role in ordering, augmenting, and sifting of the problems and exercises, and moreover he assumed the main responsi- bility for the pamphlet. Alan Solomon helped most unselfishly and effectively in all phases of the preparation of the book. Thanks is also due to Charlotte John, Anneli Lax, R. Richtmyer, and other friends including James and virginia McShane The first volume is concerned primarily with functions of a single variable. whereas the second volume will discuss the more ramified theories of calculus for functions of several variables A final remark should be addressed to the student reader. It might prove frustrating to attempt mastery of the subject by studying such a book page by page following an even path. Only by selecting shortcuts first and returning time and again to the same questions and difficulties can one gradually attain a better understanding from a more elevated oint An attempt was made to assist users of the book by marking with an asterisk some passages which might impede the reader at his first at tempt. Also some of the more difficult problems are marked by an asterisk We hope that the work in the present new form will be useful to the young generation of scientists. We are aware of many imperfections and we sincerely invite critical comment which might be helpful for later Improvements Richard Courant fritz John June 1965 Contents Chapter Ⅰ ntroduction 1.1 The Continuum of numbers a. The System of Natural Numbers and Its Extension. Counting and Measuring, b Real numbers and Nested Intervals. 7 c. Decimal fractions Bases other Than Ten, 9 d Definition of Neighborhood, 12 Inequalities, 12 1.2 The Concept of Function 7 ping-Graph, 18 b. Definition of the Concept of Functions of a Continuous Variable. Domain and range of a function 21 c Graphical Representation. Monotonic Functions, 24 d Continuity, 31 e. The Intermediate Value Theorem. Inverse Functions, 44 1.3 The Elementary Functions Rational Functions, 47 b. Alge Functions, 49 c. Trigonometric Functions, 49 d. The Exponential Function and the Logarithm, 51 e Compound functions Symbolic products Inverse Functions, 52 14S 55 1.5 Mathematical Induction 57 x Contents 1.6 The Limit of a Sequence a c. a 1 f. Geometrical Illustration of the Limits of and VP, 65 g. The Geometric Series, 67 Vn.69 i n,69 1. Further Discussion of the Concept of limit 70 a. Definition of Convergence and Divergence, 70 b. Rational operations with limits, 71 c Intrinsic Convergence Tests. Monotone Sequences, 73 d. Infinite Series and the Summation Symbol, 75 e. The Number e, 77 f. The Number t as a Limit 80 1. 8 The Concept of limit for Functions of a Con- Variable a. Some Remarks about the Elementary Functions. 86 Supplements S1 Limits and the Number Concept 89 b Real Numbers determined by Nested Sequences of Rational intervals. 90 c Order. Limits and Arithmetic Operations for Real Numbers, 92 d Completeness of the Number Continuum Compactness of Closed Intervals. Convergence Criteria, 94 e Least Upper Bound and Greatest Lower Bound, 97 f Denumerability of the Rational Numbers 98 S2 Theorems on Continuous Functions 99 S 3 Polar Coordinates S4 Remarks on Complex Numbers 103 PROBLEMS 106


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