Tom Apostol - Calculus vol.1 - One-variable Calculus, with an Introduction to Linear Algebra (1975).pdf

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Tom M. Apostol
CALCULUS
VOLUME 1
One-Variable Calculus, with an
Introduction to Linear Algebra
SECOND EDITION
New York
l
John Wiley & Sons, Inc.
Santa Barbara
l
London
l
Sydney
l
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 01967
by John WiJey
& Sons, Inc.
First Edition copyright 0 1961 by Xerox Corporation.
Al1 rights reserved. Permission in writing must be obtained
from the publisher before any part of this publication may
be
reproduced 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 1
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
T
O
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
calculus and analytic geometry. 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
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
a11 the important theorems are presented as an essential part of the growth of mathematical
ideas; the proofs are often preceded by a geometric or intuitive discussion to give the
student 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 proofs
are 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 before
differentiation. Although to some this may seem unusual, it is historically correct and
pedagogically sound. Moreover, it is the best way to make meaningful the true connection
between the integral and the derivative.
The concept of the integral is defined first for step functions.
Since the integral of a step
function is merely a finite sum, integration theory in this case is extremely simple. As the
student learns the properties of the integral for step functions, he gains experience in the
use of the summation notation and at the same time becomes familiar with the notation
for integrals. This sets the stage
SO
that the transition from step functions to more general
functions seems easy and natural.
vii

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