Table Of ContentC A L C U L U S
T H E E LEME M E N T S
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Euclid
Allez en avant, la foi vous viendra.
d’Alembert
C A L C U L U S
THE ELEMENTS
MICHAEL COMENETZ
wORLD sCIENTIFIC
nEW jERSEY . lONDON . sINGAPORE . hONG KONG
Published by
World Scientific Publishing Co. Pte. Ltd.
P O Box 128, Farrer Road, Singapore 912805
USA office: Suite 1B, 1060 Main Street, River Edge, NJ 07661
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-Publication Data
Comenetz, Michael.
Calculus--the elements / Michael Comenetz.
p. cm.
Includes index.
ISBN 9810249039 (alk. paper) -- ISBN 9810249047 (pbk. : alk. paper)
1. Calculus. I. Title
QA303.2.C66 2002
515--dc21 2002069008
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Figures by the author and Monotype Composition Co.
Cover art by Charles Jones.
Copyright © 2002 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
electronic or mechanical, including photocopying, recording or any information storage and retrieval
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For photocopying of material in this volume, please pay a copying fee through the Copyright
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Printed in Singapore.
Preface
Thisbook isforanyoneacquaintedwiththe rudimentsofanalyticgeometry
whowishestolearnthe fundamentalideasofcalculusandafew oftheirappli-
cations. Itaimstopresenttheelementsclearlyandfullyenoughthattheymay
be understood. Tothis endconceptsareeverywhereemphasized,andphysical
interpretations of mathematicalideas are givena prominentrole; but no prior
study of physical science is expected of a reader. A feature of the exposition
(which I note here for experts) is that infinitesimals are cautiously introduced
and employedin parallelwith limits, so that the lucid symbols of calculus can
be read as they were meant to be, and as most scientists and engineers have
always read them. The plan of the book is sketched at the beginning of the
first chapter.
The “questions”(exercises,problems)accompanyingthe text supplement it
and provide the practice in calculation and problem-solving that is needed to
testone’sunderstanding. Intheinterestofactivereading,manyquestionscall
for arguments and details omitted from the discussion. There are answers in
the back, most of them including solutions full enough to be of service to the
learner studying alone.
TheReader’sGuidewhichfollowsthisprefaceexplainsthe referencesystem
and provides other information and advice. For translations of the epigraphs
that precede Chapter 1 see Appendix 2.
In both matter and form the book owes much to the classic lectures Dif-
ferential and Integral Calculus of R. Courant (who would have condemned
the unregenerate infinitesimals). Its writing was partially supported by St.
John’s College and the Beneficial-Hodson Trust. I thank Christopher Colby,
ThomasSlakey,andStewartUmphreyfortheirgeneroushelp,andIremember
with gratitude George Comenetz, my father, whose reading of the original
manuscript led to many improvements. My wife Sandy and sons Joshua and
Aaron gave me heart for the work.
Michael Comenetz
v
TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
Reader’s Guide
Reference system. Each chapter is divided into articles, which are di-
vided into sections. Article 5.3 is the third article of Chapter 5, and §5.3.6 is
the sixth section of that article. The reference §§5.3.6–8 is short for §§5.3.6–
5.3.8.
Some equations,propositions,etc., are labeled with numbers in parentheses
at the left margin. Numbering begins at (1) with each article. Reference to
such an item is by the number in parentheses, or if necessary by section and
number: §1.6.8(2) is the item labeled (2) in §1.6.8.
Some sections are sets of questions, which are numbered Q1, Q2, etc. A
referenceinthetextsuchasQ4istotheindicatedquestioninthenextfollowing
set. Other references to questions specify the section: §2.9.5.Q4 is the fourth
question in §2.9.5.
Figure 2-5 is the fifth figure in Chapter 2; Fig. A-5 is the fifth figure in the
Answers. Similarly for tables.
How to find things. Look in the Contents, the Topical Summary that
follows Chapter 8, and the Index, and use the cross-referencesin the text.
Structure of the book. The first four chapters form the basic sequence;
the remaining four are essentially independent of one another. (See the intro-
ductiontoChapter1.) Attheendofeachofthefirstfourchapterscalculushas
been portrayed as a whole, with progressively greater fullness and definition.
Alternate routes. A. What can be skipped. Passagesin smallprint,
including a few entire sections, are remarks that can be omitted at the first
readingofachapteroruntiltheyarecited. Manyothersectionscanbeskipped,
postponed,orreadinpart. Thesearesignaledbyfootnotes,whichgivefurther
guidance when needed. The most important cases in the early chapters are
Art. 1.4, on force, which can be postponed until Chapter 5, and Art. 3.7, on
the mean value theorems and the proof of the Fundamental Theorem, which
can be reduced to avoid theory and proof. As for the questions, answer as
many as you can. In each set the earlier ones are thought to be easier.
vii
viii READER’SGUIDE
B. Shortcuts to certain topics. When I say here that part Z of a se-
quence XYZ can follow part X, part Y being omitted, I do not mean that Z
containsnoreferencestoY,onlythatsuchreferencesareinessentialoroccurin
passagesthatcanbeomitted. Thisappliestothetext inZ,butnotnecessarily
to all the questions there, some of which may depend on Y. The following are
the chief options.
1. The chain rule, inverse functions, and partial derivatives (Chapter 4
through §4.5.6) can follow the arithmetic differentiation rules (Art. 3.4); then
the Fundamental Theorem (Art. 3.5) will be done next.
2. After the Fundamental Theorem (Art. 3.5) the theory of maxima and
minima (Art. 3.9) requires only the reduced versionof Art. 3.7, together with
§3.8.1.
3. Integration in the plane and in space (Art. 4.8) can follow §4.5.1.
4. AfterChapter3,thetopicofdifferential equations of motion (Chapter5)
requiresonlythechainrule(Art.4.1),inversefunctions(Art.4.3),andthelimit
of (sinh)/h (§4.4.2), if the value of an integralat the very end of Chapter 5 is
accepted without proof.
5. The differential equation of intrinsic (exponential) growth (Chapter 6)
requires only the chain rule (Art. 4.1) and inverse functions (Art. 4.3), except
for the second part of the proof in §6.2.2 (which uses §§4.6.2–3).
6. Articles 7.1–7.4, on arc length, curvature, local geometry, and parametric
representation, can be read after §4.6.3 (omitting Art. 4.5).
7. Chapter 8, on the Taylor series, can be read after Art. 4.4.
C. Ill-advised reduction. Overcome the temptation to reduce concepts
of calculus to their geometrical interpretations (area, slope, etc.). Calculus is
more than geometry.
D. Physical units. Appendix 1 shows how to convert from the Interna-
tional System used here to the older but still important cgs system.
E. Newton’s lemmas. Interesting versions of some basic propositions
of calculus by an originator of the subject are found at the beginning of
Book 1 of his Principia (1687). They can be read along with the present
book, which has notes relating them to arguments here (see the Index under
Newton, Principia).
F. The computer. If you can understand what is written in this book
you will be able to illustrate it and explore further with the aid of a graphing
calculator or the like. But there is no substitute for thinking, calculating, and
sketching on one’s own.
READER’SGUIDE ix
Abbreviations. Cf. = compare,e.g. = for example, i.e. = that is to say,
q.e.d. = which was to be proved.
The Greek alphabet.
A α alpha I ι iota P ρ rho
B β beta K κ kappa Σ σ sigma
Γ γ gamma Λ λ lambda T τ tau
∆ δ delta M µ mu Y υ upsilon
E ε epsilon N ν nu Φ φ phi
Z ζ zeta Ξ ξ xi X χ chi
H η eta O o omicron Ψ ψ psi
Θ θ theta Π π pi Ω ω omega
ix
