E-Book Overview
Perhaps my love of this text is because I was one of those students who only wanted to know how to use a derivative or integral to solve a physical problem when I took my first calculus class in 1973. Years later when I was visiting MIT looking for economic texts, I came across Calculus by Apostol and after briefly skimming the index, decided to purchase it. It was three years later that I actually began studying the material in the text and honestly, I felt as though blinders had been removed from my eyes. From the opening chapters on step functions and integration, the material on diffferentiation and limits and proceeding into the topics of differential equations, sequences and series, complex numbers, differential equations and linear algebra, I was excited to be studying mathematics and finally understanding the material versus memorizing formulas. This text motivated me to purchase Apostol\'s second and third books on multivariable calculus and analysis. Today I am studying Royden\'s real Analysis text to prepare me for my next topic - probability. Is this book for everyone? Probably not. But if you ever felt that you had shortchanged yourself in college mathematics and are in a career where you can benefit from starting over and learning calculus the right way, I recommend the Apostol text. (And I firmly believe that all of us can benefit in our jobs from studying the logic of mathematics!)
E-Book Content
Tom M. Apostol
CALCULUS VOLUME 1
One-Variable Calculus, with an Introduction to Linear Algebra SECOND
New York
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EDITION
John Wiley & Sons, Inc.
Santa Barbara
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London
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S ydney
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Toronto
C O N S U L T I N G
EDITOR
George Springer, Indiana University
XEROX® is a trademark of Xerox Corporation.
Second Edition Copyright ©!967 by John Wiley & Sons, Inc. First Edition copyright © 1961 by Xerox Corporation.
All rights reserved. Permission in writing must be obtained from the publisher before any part of this publication may be reproduced or transmitted in any fonn or by any means, electronic or mechanical, including photocopy, recording, or any infonnation storage or retrieval system. ISBN o 471 00005 I Library of Congress Catalog Card Number: 67-14605 Printed in the United States of America. 1 0 9 8 7 6 5 4 3 2
To
Jane and Stephen
PREFACE
Excerpts from the Preface to the First Edition There seems to be no general agreement as to what should constitute a first course in Some people insist that the only way to really understand calculus is to start off with a thorough treatment of the real-number system and develop the subject step by step in a logical and rigorous fashion. Others argue that calculus is calculus and analytic geometry.
primarily 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 develop
manipulative skills. There is much that is sound in both these points of view. Calculus is a 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 derives much of its power and beauty from the variety of its applications. It is possible to combine
a strong theoretical development with sound training in technique; this book represents an attempt to strike a sensible balance between the two. While treating the calculus as a deductive science, the book does not neglect applications to physical problems. Proofs of all the important theorems are presented as an essential part of the growth of mathematical
ideas; the proofs are often preceded by a geometric or intuitiv