Table Of ContentCALCULUS
An Intuitive and Physical Approach
Morris Kline
SECOND EDITION
DOVER PUBLICATIONS, INC.
Mineola, New York
Copyright
Copyright © 1967, 1977 by John Wiley & Sons, Inc.
All rights reserved.
Bibliographical Note
This Dover edition, first published in 1998, is an unabridged republication of the work originally
published in 1977 by John Wiley and Sons, Inc., New York.
Readers who would like to receive the solutions to the exercises may request them from the publisher at
the following e-mail address:
[email protected]
Library of Congress Cataloging-in-Publication Data
Kline, Morris, 1908–
Calculus : an intuitive and physical approach, second edition / Morris Kline.
p. cm.
“An unabridged republication of the work originally published in 1977 by John Wiley and Sons, Inc.,
New York”—T.p. verso.
Includes index.
ISBN-13: 978-0-486-40453-0
ISBN-10: 0-486-40453-6
1. Calculus. I. Title.
98-36211
QA303.K68 1998
CIP
515—dc21
Manufactured in the United States by Courier Corporation
40453609
www.doverpublications.com
Dedicated to the memory of
Morris Kline
1908–1992
Mathematician, Educator
PREFACE TO
THE SECOND EDITION
The basic features of the first edition have been retained, such as the intuitive
approach and real applications. As to the approach, the last chapter introduces a
rigorous treatment. Though this chapter could be used in conjunction with the
opening chapters of the book, I do not recommend doing so; the rigorous
presentation is difficult to grasp and obscures the understanding. Rigor
undoubtedly refines the intuition but does not supplant it. The rigorous approach
should be reserved for a course in advanced calculus for mathematicians.
Most of the applications still belong with the physical sciences; however, no
knowledge of physics is presupposed. Several sections on physical applications
have been dropped and applications to the social and biological sciences have
been added instead. These applications are vital. The theory and technique of the
calculus are, in themselves, meaningless. Moreover, since most students who
take calculus will be scientists or engineers, they will be highly motivated by the
applications. Many calculus texts dispose of applications by asking students to
calculate centers of gravity and moments of inertia. But since students have no
idea of how these quantities are used, the only consequence is that the gravity of
the problems produces inertia in the students.
In addition to extending the scope of the applications, I have made a
number of other improvements. More drill exercises have been added, the
exercises have been carefully graded as to difficulty, and there are more
illustrative examples. Students doing homework exercises acquire the habit of
searching for an illustrative example which they can imitate and thereby do the
exercises without thinking. To counter this many illustrative examples are
incorporated into the text instead of being set apart formally. Hopefully the
students will read the text instead of mechanically following the technique of the
usual illustrative example.
A chapter on differential equations and a section on such numerical
methods as Simpson’s rule and the trapezoidal rule have also been added. More
use has been made of vector analysis, particularly in treating velocity and
acceleration.
Chapter 9, “The Definite Integral,” has been moved forward so that
students using the mathematics in another course taken simultaneously can
utilize the concept sooner.
Some changes in notation are unquestionably an improvement. The
notation has been restricted to the customary situation where time is the
independent variable; in other situations y′, dy / dx and f′(x) are used. The
notation ∫y dx has been introduced early to denote antidifferentiation. The use of
dy / dx from the outset is the result of numerous requests. Admittedly, a useful
notation in showing the variables involved, it also suggests that the derivative is
a quotient, whereas we must take great pains to convince the student that it is
not.
The pace of the first few chapters has been speeded up somewhat. However,
it does seem desirable, because students are weak in algebra, to keep the algebra
simple at the outset while students are acquiring the concepts of the calculus.
Two other changes may be helpful. Since not all classes go at the same pace
or have the same objectives the two volumes of the first edition have been
replaced by the present single volume. Thereby instructors are freer to choose
the topics they deem most appropriate. Second, since time may not permit the
inclusion of all the applications, those sections that can be omitted without
disrupting the continuity have been starred.
Some figures have been improved; where the precise shape is significant
the computer has been employed to achieve accuracy.
An instructor’s manual which contains full solutions of all the exercises,
suggestions for teaching, and additional material for advanced students is
available to instructors on request to the publisher.
Morris Kline
New York, N. Y.
August, 1976
PREFACE TO
THE FIRST EDITION
Anyone who adds to the plethora of introductory calculus texts owes an
explanation, if not an apology, to the mathematical community. I believe that an
introductory course in the calculus should present the subject intuitively and
should relate it as closely as possible to science. A text for such an approach is
not available.
In this book the justification of the theorems and techniques is consistently
intuitive; that is, geometrical, physical, and heuristic arguments and
generalizations from concrete cases are employed to convince. The approach is
especially suitable for the calculus because the subject grew out of physical and
geometrical problems. These problems tell us what functions we should take up,
what concepts we want to formulate, and what techniques we should develop. In
view of the fact that the human mind learns intuitively and that time does not
permit both an intuitive and a rigorous presentation in elementary calculus, it
seems to me that the approach adopted is the correct one.
The intuitive approach is explained to the student so that he will know what
kind of evidence is being used to support arguments. Thus he is told that a graph
of a typical function may not represent all functions. On the other hand, he is
also told that the elementary functions are well behaved except at isolated points
and that he can usually trust his intuition. As he works with the ideas of the
calculus, he will sharpen his intuition. If he continues with mathematics, he will
learn the analytical foundations and proofs that guard against the failings of
intuition.
The use of an intuitive approach in the first treatment of a subject is not an
innovation. Arithmetic is learned intuitively in elementary school and then the
logic of it is learned gradually through the work in algebra. Geometry is learned
intuitively in junior high school and then the formal deductive approach is
presented in senior high school. A difficult subject such as the calculus,
therefore, should certainly be introduced by an intuitive approach.
This approach has many merits beyond that of being the only feasible one.
Every pedagogue today champions discovery, but few teach it. How does one
discover in mathematics? By thinking in physical and geometrical terms, by
conjecturing or guessing, by formulating hypotheses and testing them, and by
generalizing on specific cases. Physical problems that call for the creation of
mathematics set the stage for discovery. That the intuitive approach may lead to
errors is granted, but “truth emerges more readily from error than from
confusion.” The student must be allowed to make mistakes, for if he makes no
mistakes, he will not progress.
After the basic material has been covered, the need for rigor is motivated,
and the last two chapters of Part 2 of this book do offer an introduction to precise
language and proof. These chapters are intended as a transition to advanced
calculus.
One alternative to an intuitive approach is a rigorous treatment. In my
opinion, a rigorous first course in the calculus is ill advised for numerous
reasons. First, it is too difficult for the students. Beginners are asked to learn a
mass of concepts so subtle that they defied the best mathematicians for two
hundred years. Even Cauchy, the founder of rigor, gave formulations that are
crude compared to what the current rigorous presentations ask students to
absorb. And Cauchy, despite his concern for rigor, missed the distinctions
between continuity and differentiability and between convergence and uniform
convergence. Before one can appreciate a precise formulation of a concept or
theorem, he must know what idea is being formulated and what exceptions or
pitfalls the wording is trying to avoid. Hence he must be able to call upon a
wealth of experience acquired before tackling the rigorous formulation.
Furthermore, having students master a polished deductive organization does not
teach them how to think and how to do mathematics, for thinking and doing are
not deductive processes. How can discovery take place when students are asked
to work with ideas that are already overladen with sophistication and
refinement? Finally, the rigorous approach is misleading. Because the
introductory calculus course is the student’s first contact with higher
mathematics, he obtains the impression that real mathematics is deductive and
that good mathematicians think deductively.
Rigor has its place in mathematics education. It is a check on the creations
and it permits an aesthetic (as well as an anaesthetic) presentation. But it is also
to some extent gilt on the lily and an interdiction against the inclusion of
functions which rarely occur in practice and which must even be invented with
Weierstrassian ingenuity. A rigorous first course in calculus reminds one of the
words of Samuel Johnson; “I have found you an argument but I am not obliged
to find you an understanding.” Even if the rigorous material is understood, its
value is limited. As Henri Lebesgue pointed out: “Logic makes us reject certain
arguments but it cannot make us believe any argument.”
The reasons often given for a rigorous presentation—that students must
learn what a real proof is or that students should not be asked to unlearn later
what they have already been taught—are hollow. One cannot give the whole
truth at once in any subject. Even we, as teachers, do not face the whole truth for,
in fact, the whole question of what is rigorous in mathematics was never so
much up in the air as it is now. At any rate, what may seem ideally right and
efficient is pedagogically intolerable.
There is also the alternative of compromising on rigor by offering precise
definitions and proofs in some portions of the text and intuitive or
pseudorigorous arguments in others. This alternative seems to me to have almost
all of the disadvantages of a rigorous presentation and the additional one of
confusing the student about what proof really is.
The second essential respect in which this book differs from current ones is
that the relationship of mathematics to science is taken seriously. The present
trend to separate mathematics from science is tragic. There are chapters of
mathematics that have value in and for themselves. However, the calculus
divorced from applications is meaningless. We should also keep in mind that
most of the students taking calculus will be scientists and engineers, and these
students must learn how to use mathematics. But the step from mathematics to
its applications is not simple and straightforward and it creates difficulties for the
student from the time he is called upon to solve verbal problems in algebra. The
mathematics courses fail to teach students how to formulate physical problems
mathematically. The science and engineering courses, on the other hand, assume
that students know how to translate physical problems into mathematical
language and how to make satisfactory idealizations. The gap between
mathematics and science instruction must be filled, and we can do so to our own
advantage because thereby we give meaning and motivation to the calculus.
In this book real problems are used to motivate the mathematics, and the
latter, once developed, is applied to genuine physical problems—the
magnificent, impressive, and even beautiful problems tendered by nature. I have
selected those that do not require a background in physics. However, if the
student is to think properly even about familiar concepts, such as weight, force,
velocity, acceleration, light, and work, the book must say a little about them and
about the physical laws that are involved. It does. To relegate physical problems
only to the exercises is, in my opinion, ineffective in teaching students how to
apply mathematics. If the student is not informed about the physics of a problem,
all he can do in tackling an exercise is to apply a formula mechanically or guess
what technical process is called for, do it, and check his answer. If the book does
not supply an answer and if the student’s result is off by a factor of 1000, he will
not realize it because he has no judgment about what to expect.
I have tried to incorporate several other features that may contribute to the
pedagogy. In many instances I have deliberately made false starts so as to have
the student realize that correct methods and correct proofs are almost always
preceded by groping and to have students appreciate why we finally take one
course rather than another. I wish to dispel the impression that good
mathematicians are able to proceed directly to the right conclusion because strict
logic or a God-given insight guides them. I have given full details on the
mathematical steps in the hope that the book will be readable. The style is
informal.
The pace of the first few chapters of Part 1 has been deliberately made slow.
It is well known that calculus students are still struggling with elementary
algebra. By keeping the opening algebra simple, I trust that the calculus ideas
will stand forth. In later chapters the algebra becomes more difficult.
The order of the mathematical concepts and topics was chosen so that
applications could be introduced almost from the beginning. However, no
artificial or complicated arrangement was necessary. Differentiation and
antidifferentiation are the key concepts in the first thirteen chapters and then
summation is introduced. The early introduction of applications should permit
good correlation with physics courses if these are taken concurrently. The
concept of differentials was deliberately delayed until Chapter 12 of Part 1. Most
books do stress that the derivative is a limit of a quotient, but often present
differentials early. No matter how much the student is cautioned that the
derivative is not a quotient, if he is allowed to work almost at once with the
quotient of the differentials, he tends to forget the true meaning of the derivative.
Beyond the matter of the proper approach, there are a few other points that
may warrant comment. Despite my own preference for a course in analytic
geometry which precedes the calculus, I have included the former. This is a
concession to the current fad. Some work with vectors is included, but I have not
done much with the differentiation and integration of vector functions. I believe
that this topic requires a new way of thinking and that we should not try to teach
two major classes of techniques at the same time. The calculus of scalar
functions is the basic one, and this should be mastered first. Apropos of the
inclusion of vector analysis, the present trend to include this topic, linear
