Calculus: An Intuitive and Physical Approach (Second Edition)

Book Preface

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 images 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