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 **