[T.Apostol] Calculus Volume (1), Analiza matematyczna
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//-->Tom M. ApostolCALCULUSVOLUME 1One-Variable Calculus, with anIntroduction to Linear AlgebraSECOND EDITIONNew YorklJohn Wiley & Sons, Inc.Santa BarbaralLondonlSydneylTorontoC O N S U L T I N GEDITORGeorge Springer,Indiana UniversityXEROX@is a trademark of Xerox Corporation.Second Edition Copyright 01967by John WiJey& Sons, Inc.First Edition copyright 0 1961 by Xerox Corporation.Al1 rights reserved. Permission in writing must be obtainedfrom the publisher before any part of this publication maybereproduced or transmitted in any form or by any means,electronic or mechanical, including photocopy, recording,or any information storage or retrieval system.ISBN 0 471 00005 1Library of Congress Catalog Card Number: 67-14605Printed in the United States of America.1 0 9 8 7 6 5 4 3 2TOJane and StephenPREFACEExcerpts from the Preface to the First EditionThere seems to be no general agreement as to what should constitute a first course incalculus and analytic geometry. Some people insist that the only way to really understandcalculus is to start off with a thorough treatment of the real-number system and developthe subject step by step in a logical and rigorous fashion. Others argue that calculus isprimarily a tool for engineers and physicists; they believe the course should stress applica-tions of the calculus by appeal to intuition and by extensive drill on problems which developmanipulative skills. There is much that is sound in both these points of view. Calculus isa deductive science and a branch of pure mathematics. At the same time, it is very impor-tant to remember that calculus has strong roots in physical problems and that it derivesmuch of its power and beauty from the variety of its applications. It is possible to combinea strong theoretical development with sound training in technique; this book representsan attempt to strike a sensible balance between the two. While treating the calculus as adeductive science, the book does not neglect applications to physical problems. Proofs ofa11 the important theorems are presented as an essential part of the growth of mathematicalideas; the proofs are often preceded by a geometric or intuitive discussion to give thestudent some insight into why they take a particular form. Although these intuitive dis-cussions Will satisfy readers who are not interested in detailed proofs, the complete proofsare also included for those who prefer a more rigorous presentation.The approach in this book has been suggested by the historical and philosophical develop-ment of calculus and analytic geometry. For example, integration is treated beforedifferentiation. Although to some this may seem unusual, it is historically correct andpedagogically sound. Moreover, it is the best way to make meaningful the true connectionbetween the integral and the derivative.The concept of the integral is defined first for step functions.Since the integral of a stepfunction is merely a finite sum, integration theory in this case is extremely simple. As thestudent learns the properties of the integral for step functions, he gains experience in theuse of the summation notation and at the same time becomes familiar with the notationfor integrals. This sets the stageSOthat the transition from step functions to more generalfunctions seems easy and natural.vii
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