James M. Henle - Infinitesimal Calculus

INFINITESIMAL
CALCULUS JAMES M. HENLE
Department of Mathematics
Smith College EUGENE M. KLEINBERG
Department of Mathematics
State University of New York at Buffalo DOVER PUBLICATIONS, INC.
Mineola, New York Copyright Copyright 1979 by The Massachusetts Institute of Technology All rights reserved. Bibliographical Note This Dover edition, first published in 2003, is an unabridged republication of the work published by The MIT Press, Cambridge, MA, 1979. The publisher would like to thank The MIT Press for their assistance in the preparation of this edition. Library of Congress Cataloging-in-Publication Data Henle, James M.

Infinitesimal calculus / James M. Henle, Eugene M. Kleinberg.
p. cm. Originally published: Cambridge, Mass.: MIT Press, cl979. eISBN-13: 978-0-486-15101-4 1. Calculus. I. I.

Kleinberg, Eugene M. II. Title. QA303.2.H45 2003 515dc21 2003043991 Manufactured in the United States by Courier Corporation
42886904 2013
www.doverpublications.com to our parents ... we shall discover much Emptiness, Darkness and Confusion; nay, if I mistake not, direct impossibilities and contradictions. Whether this be the case or no, every thinking reader is entreated to examine and judge for himself...

George Berkeley, Bishop of Cloyne (16851753)

Calculus has always been a difficult subject to learn well. In the last half century alone there have been literally scores of calculus books published, each trying harder than the next to simplify the subject. At the turn of the century it was popular to teach calculus by using so-called infinitesimals. This approach had the advantage of making the basic theory quite intuitive and easy to understand, but mathematicians lacked a rigorous definition of just what infinitesimals were, and so anyone advancing beyond the basics quickly became lost. Thus we were led to the / approach to calculus, an approach that, although totally precise and rigorous, was a disaster for students to learn and teachers to teach. Most recently s and s have been shelved along with all other attempts at teaching the basics of calculus.

Instead we have settled into teaching specific methods for applying calculus in specific situations. The problem here, of course, is that even though individual methods might be fairly easy to master, there exist very many, seemingly distinct, methods to be learned, and even then most courses leave students hopelessly short. This active evolution in the teaching of calculus was always prompted by continual dissatisfaction with earlier approaches and had nothing to do with new mathematical insights into calculus itself. Indeed there were no such new insights; that is, there were none until just recently. In the early 1960s the mathematician Abraham Robinson pioneered a body of work known as Nonstandard Analysis which makes precise and mathematically rigorous the intuitively pleasing concept of infinitesimal. Originally Robinsons field was reserved as an advanced graduate subject, but the ideas are both simple and important, and in recent years everyone from standard mathematicians to economists, physicists, and social scientists have been using his methods with stunning success.

A most natural place for Robinsons insight is as a next (and possibly final) point in the evolution of the teaching of calculus. We can now develop calculus using infinitesimals and enjoy all of their simplicity and intuitive power, yet at the same time work in a mathematically precise and rigorous atmosphere. This approach, although quite new, has been used at a number of universities with remarkable success. This book presents a rigorous development of calculus using infinitesimals in the style of Robinson. It does not make any attempt to cover the assorted methods of calculus for applications, but rather it concentrates on theory, the area which previously was so difficult. We feel that with this new approach, basic theory is now quite accessible to studentseven those who are interested in calculus solely for its applications.

Indeed a knowledge of basic theory lets one dispense with learning many of the canned methods in favor of attacking problems directly and formulating ones own methods. The only prerequisite assumed for this book is a good foundation in high school mathematics. Our early chapters deal with the field of mathematical logic. Some of this is necessary for an understanding of infinitesimal calculus, but much of it is not. That which can be skipped is indicated in the text. We received help from a great many people during the preparation of this manuscript, much of it from students taking preliminary versions of the course.

Special thanks are also due to Professors Frank Wattenberg and David Schaffer, and to Mitchell Spector. Finally, Denise Borsuk typed countless versions of the text and displayed patience, good humor, and skill. To her we are most grateful. J. M. M. M.

Kleinberg

God made the integers, all else is the work of man. Leopold Kronecker (18231891) God created infinity, and man, unable to understand infinity, had to invent finite sets. Gian-Carlo Rota (1932- ) The history of modern mathematics is to an astonishing degree the history of the calculus. The calculus was the first great achievement of mathematics since the Greeks and it dominated mathematical exploration for centuries. The questions it answered and the questions it raised lay at the heart of mans understanding of not only geometry and number, but also space and time and mathematical truth. It began with the surprising unification of two rather different geometrical problems, and almost immediately its ideas bore fruit in dozens of seemingly unrelated areas.

The methods it developed gave the physical sciences an impetus without parallel in history, for through them natural science was born, and without them physics could not have progressed much further than the mystical vortices of Descartes. In the beginning there were two calculi, the differential and the integral. The first had been developed to determine the slopes of tangents to certain curves, the second to determine the areas of certain regions bounded by curves. Algebra, geometry, and trigonometry were simply insufficient to solve general problems of this sort, and prior to the late seventeenth century mathematicians could at best handle only special cases. The general idea of the calculus, its fundamental theorem, and its first applications to the outstanding problems of mathematics and the natural sciences are due independently to Isaac Newton (16421727) and Gottfried Leibniz (16461716). Their work was certainly built on foundations laid by others, but their penetrating insights represented what is easily the most significant mathematical breakthrough since the Greeks.

Remarkably, the powerful methods developed by these two men solved the same class of problems and proved many of the same theorems yet were based on different theories. Newton thought in terms of limits whereas Leibniz thought in terms of infinitesimals, and although Newtons theory was formalized long before Leibnizs, it is far easier to work with Leibnizs techniques. The approach to the calculus we shall employ is based on Leibnizs ideas as formalized by Abraham Robinson in 1961 under the name of nonstandard analysis. Simply stated, our approach will involve expanding the real number system by introducing new numbers called infinitesimals. These new numbers will have the property that although different from 0, each is smaller than every positive real number and larger than every negative real number. Of course our infinitesimals cannot themselves be real numbers, but so what? This sort of expansion of a number system through the introduction of new numbers which themselves correspond to nothing in the real world is common in mathematics.

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