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Infitisimal calculus (104031) Tom M Apostol

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Technion – Israel Institute of Technology

Infitisimal calculus (104031)

Tom M Apostol – Calculus vol. 1.pdf

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 applications 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 important 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 discussions 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 development 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.
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. . . WI Preface
Prefuce to the Second Edition
The second edition differs from the first in many respects. Linear algebra has been
incorporated, the mean-value theorems and routine applications of calculus are introduced
at an earlier stage, and many new and easier exercises have been added. A glance at the
table of contents reveals that the book has been divided into smaller chapters, each centering
on an important concept. Several sections have been rewritten and reorganized to provide
better motivation and to improve the flow of ideas.
As in the first edition, a historical introduction precedes each important new concept,
tracing its development from an early intuitive physical notion to its precise mathematical
formulation. The student is told something of the struggles of the past and of the triumphs
of the men who contributed most to the subject. Thus the student becomes an active
participant in the evolution of ideas rather than a passive observer of results.
The second edition, like the first, is divided into two volumes. The first two thirds of
Volume 1 deals with the calculus of functions of one variable, including infinite series and
an introduction to differential equations. The last third of Volume 1 introduces linear
algebra with applications to geometry and analysis. Much of this material leans heavily
on the calculus for examples that illustrate the general theory. It provides a natural
blending of algebra and analysis and helps pave the way for the transition from onevariable calculus to multivariable calculus, discussed in Volume II. Further development
of linear algebra Will occur as needed in the second edition of Volume II.
Once again 1 acknowledge with pleasure my debt to Professors H. F. Bohnenblust,
A. Erdélyi, F. B. Fuller, K. Hoffman, G. Springer, and H. S. Zuckerman. Their influence
on the first edition continued into the second. In preparing the second edition, 1 received
additional help from Professor Basil Gordon, who suggested many improvements. Thanks
are also due George Springer and William P. Ziemer, who read the final draft. The staff
of the Blaisdell Publishing Company has, as always, been helpful; 1 appreciate their sympathetic consideration of my wishes concerning format and typography.
Finally, it gives me special pleasure to express my gratitude to my wife for the many ways
she has contributed during the preparation of both editions. In grateful acknowledgment
1 happily dedicate this book to her.



Infitisimal calculus (104031) Tom M Apostol

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