Volume 2 No.3                                                                                                                                   May 1999

From the Book Shelf . . .

In continuation of the reviews of textbooks prescribed at IIT Kanpur for the undergraduate core courses, DIRECTIONS brings to you a review of the text used for the Mathematics 101 course.

Calculus and Analytic Geometry, 6th edition

by George B. Thomas and Ross L. Finney

1985, Narosa Indian Student Edition, New Delhi

ISBN: 81-85015-52-X

Price: Rs. 325/-

The first Mathematics course encountered by science and engineering students at IIT Kanpur is a course on Calculus. In this course, students are taught limits, continuity, differentiation and integration of functions of one and several variables, along with applications of these topics. The textbook prescribed for this course is Calculus and Analytic Geometry by Thomas and Finney. This book maintains a careful balance between mathematical rigour and applications.

The balance between rigour and applications is important on many counts. For example, most students come across the concept of limit for the first time in a calculus course. So it is no wonder that students face all kinds of difficulties in understanding the language of epsilon and delta . This seems to be a universal phenomenon, and has invited a lot of pedagogic debate; it appears, however, that epsilon-delta is indispensable. Apart from this, both elementary and advanced calculus are taught in one semester, which makes it difficult for an instructor to cover all important topics and expose the subject as both interesting and useful.

There is no doubt that applications of calculus are important for scientists and engineers, but at the same time, one cannot ignore basic results and their proofs for a correct understanding. It must also be emphasised that a significant component of education is the training of the mind for precise thinking. In this respect, the book by Thomas and Finney is excellent.

Although a large number of books on calculus are available, this particular book is a good choice also because of its other merits. The book is designed to be used in a first course on calculus consisting of standard topics. It can be used by students who want additional explanations; we do not want a text book which is merely a collection of results, examples and exercises. The authors provide a lot of motivation for deeper study throughout, by describing fundamental concepts in a simple language. Results, methods and concepts are explained and illustrated with numerous examples and geometric figures. There are also a large number of problems, Geometric interpretations of various results and methods such as Rolle's theorem, Mean value theorem, Newton's method, Picard's method, Lagrange multipliers method... are provided whenever possible; this is very important for students to visualise these ideas. Two chapters are completely devoted to applications of derivatives and of definite integrals.

The book does not include proofs of some important results, such as the existence of maxima and minima of continuous functions on a closed bounded interval, and the existence of integrals of continuous functions (such results depend on the properties of a closed bounded interval, which are not included in this text). However, such results are stated clearly and used in proving other results. Of course, it is not possible to include everything in a single book. Thomas and Finney already contains more than a thousand pages. For mathematically inclined students, a suitable additional text might be the book, Calculus, Vol. 1 and 2, by T. M. Apostol.

P Shunmugaraj

Department of Mathematics

IIT Kanpur

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