In the latest edition of the *Mathematical Gazette*, there is an article by Thurston **[1]** with the title ‘*Should we reform the teaching of calculus?’* and several such things have been written in the past. In one such article, Munroe **[2]** said:

‘There are many conversations, committee meetings, etc., today about the modernisation of the undergraduate calculus course; but all too often the attack on the problem falls short of being comprehensive. Calculus has been in cold storage for over fifty years now.’

This appeared in AMM in 1958 and Munroe based it upon observations about the way in which modern differential geometry has produced a definition of the differential that is quite satisfactory for the purposes of integration and differentiation and which has been divorced from the notion of approximate increment. Earlier than that (in 1952), Fort **[3]** argued along similar lines by also showing the applicability of the Frechét differential to the calculus of single variable real functions.

Thurston begins by saying that *x*, *y* in expressions like *f'*(*x*) = *dy/dx* are functions for which *y* = *f* ^{o} *x* and *dy/dx* = *y'/x'*. Also, *dy*_{a}(*h*) = *f'*(*a*) *dx*_{a}(*h*) = *f'*(*a*)*h* and therefore (*dx*_{a}/dy_{a})(*h*)= *f'*(*a*) is a constant mapping.

This avoids the usual ambiguities of Leibniz notation. On the one hand, *dy*/*dx* is said to be an immutable entity that should not be thought of as the ratio of the ‘meaningless’ objects *dy* to *dx*. On the other hand, a bit later on, in the context of things such as integration by substitution, students are presented with expressions like *dy* = cos*x* *dx*

By such criteria, very few (if any) introductory calculus books are up-to-date because the matter of differentials and Leibniz notation may be touched upon but is never satisfactorily resolved. In this book, however, the definition of differentiability for functions *f*: **R**^{2} → **R** (p. 735) is so near to the Frechét version that only minor adjustments to the text would be sufficient to satisfy the requirements of the likes of Munroe, Fort and Thurston.

These considerations aside, it must be said that this book as an extremely impressive achievement and it forms the most comprehensive introduction to calculus that I have so far encountered. Moreover, it also seems to address many of the concerns raised by Bressoud **[4]** in two recent FOCUS articles. So let’s have a closer look at what it contains.

Weighing in at almost 6 lbs, it is the 4th edition of a book produced by the Calculus Consortium. There are thirteen main authors and six assistant authors (coordinated by E. J. Marks). Acknowledgement is also given to hundreds of named persons, who have presumably provided suggestions for improvement based upon their experience of using earlier versions. In effect, this book has gradually evolved from the first edition, so errors and omissions that may have appeared in earlier editions have been almost completely eradicated (the only errors I could detect were misprints on the top of p. 288 and lower down on p. 284)

Many of the changes from the 3^{rd} edition are organisational, but there is sufficient new material to warrant the claim that this volume is significantly distinct from earlier versions. For example, the chain rule is now applied to inverse functions, the material on exponential functions has been rewritten, the section of power series has been expanded and there is more material on area, arc-length and gradient of curves in polar coordinates and there is also greater emphasis on differential equations. But a complete itinerary of amendments is provided in the preface.

One of the main characteristics of this publication is that it serves as a simultaneous introduction to single variable calculus, real analysis, multivariable calculus and applied mathematics, which accounts for a thickness of nearly 1100 clearly presented (large) pages. Another feature is the way in which various themes develop from chapter to chapter. Real analysis, for example, makes its first appearance in chapter 1 (A Library of Functions). At this stage, there is first mention of continuity and this is followed by an introduction to the idea limit, which is then formalised by the use of ε-δ techniques and a summary of the standard limit theorems. In later chapters, there is cogent discussion of differentiability, l’Hôpital’s rule, Riemann sum, improper integrals and a chapter on sequences and series and tests of convergence etc. In fact, analytic ideas are prevalent in most of the chapters. Yet, despite such aspects of analysis being spread over many hundreds of pages, the treatment always maintains its coherence.

Of course, the danger in writing introductory texts in an accessibly intuitive style is that readability is often achieved at the expense of rigour — but definitely not in this case. Concepts and techniques are introduced according to what the authors describe as ‘The Rule of Four’, meaning that presentation of ideas is achieved by graphical, numerical, verbal and symbolic means. Key ideas are carefully highlighted and the text is frequently interspersed with clearly written examples that are not mere templates for the solution of the practice exercises.

Speaking of exercises, about 40% of this book consists of sets of well-structured problems that are graded for a variety of purposes. Some are for the practice of techniques, whilst others consolidate formation of key concepts. But, in most exercise sets, there are longer, open-ended problems designed to enhance modelling skills or extend mathematical understanding. The sheer number (and variety) of exercises ensures that this is an ideal teaching text as well as a book that can adequately support individual study. If mathematical knowledge is acquired by ‘doing’, then much will be gained by completing only a fraction of the activities contained herein.

As for the contents, the first 600 pages are concerned with single variable calculus and associated topics. Chapters 2, 3 and 4 focus upon differential calculus and the first of these develops the key concept of derivative, whilst chapter 3 discusses techniques of differentiation together with some theorems about differentiable functions. Chapter 4 looks at further uses of the derivative, such as optimisation and a variety of modelling problems. But use of Leibniz notation is delayed until page 89 and due caution is given as to the meaning of *dy/dx* = *f'*(*x*) [since we are then bound to explain — or explain away — statements like *dy*/*d*3 = *f'*(3)]. As previously mentioned, there is no real clarification of the concept of differentials — we get nearer to it in the later chapters on multivariable calculus.

Chapter 5 is the first of four chapters on integration and it concerns the ‘key concept’ of the definite integral. The topic is introduced in the context of velocity-time graphs and area problems from which the notion of Riemann sum is defined and formalised.

The first fundamental theorem of calculus is obviously one of the main results, but the approach is always exploratory, with the inclusion of many interesting allied ideas.

In chapter 6, anti-derivatives are treated graphically, numerically and analytically (in accordance with the ‘Rule of Four’). Differential equations are introduced for the first time and the second fundamental theorem is also discussed. Techniques of integration are covered in chapter 7, together with some revision of algebraic and trigonometric identities. This is followed by methods of approximation, including Simpson’s rule and the chapter concludes with comparison of improper integrals.

Chapter 8 considers various uses of the definite integral, including geometrical applications, applications to mechanics, economics, statistics and probability.

As mentioned, what makes the work on real analysis digestible is the fact that the various ideas are evenly disseminated throughout the book and they are always shown to have immediate relevance to the mathematics under discussion. In this vein, the chapter on sequences and series is followed by work on Taylor and Fourier series (chapter 10)

The theme of single variable calculus concludes with an excellent chapter on differential equations, which concentrates mainly on variables separable and simple second order equations. The ‘Rule of Four’ is evoked by use of flow fields (visual), the general form of a variables separable eqn, (symbolic), families of solutions, (graphical) and many modelling activites (verbal) and students would not be overwhelmed by being introduced to too many different sorts of equations in one go.

The chapter also lays heavy emphasis on modelling a wide range of physical and biological situations and it presents many methods for interpreting equations and their solutions, including a very interesting section on phase plane analysis.

What of the 400 pages devoted to ‘multivariable calculus’? For a start, this is generally confined to real valued functions on **R**^{2} or **R**^{3}, although there are several chapters on vector calculus. The treatment is such that students’ ability to visual surfaces and curves will be greatly enhanced. There are dozens of illustrations and much attention is paid to the importance of contour diagrams, flow lines, flow fields etc. The chapter headings are as follows:

- Functions of several variables.
- Vectors.
- Differentiating functions of several variables.
- Optimization.
- Integrating functions of several variables.
- Parameterization and vector fields.
- Line integrals.
- Flux integrals.
- Calculus of vector fields.

In other words, the material covered is what one would expect in a ‘first course’ in multivariable calculus. The aim seems to be to provide clear understanding of the basic ideas, rather than the pursuit of full analytic rigour. For example, there is no reference to Fubini’s theorem and Stokes’ theorem is given an outline proof only.

Conversely, these chapters do provide very good understanding of the basic concepts and techniques and the abundance of illustrations, examples and exercises make it an ideal introduction to advanced calculus.

Overall, the book is extremely well organized and it ends with a useful Ready Reference section that summarises all the main ideas and rules followed by the main index. Given the amount of time necessary to work through the mass of material contained in this book, the absence of a bibliography is perhaps is something of a relief!

**References**

**[1]** H. Thurston, *Should we reform the teaching of calculus?* Mathematical Gazette, no. 515 (2005), pp.233-234.

**[2]** M.E. Munroe, *Bringing Calculus Up-to-Date,* AMM, vol. 65 (1958), pp81-90.

**[3]** M.K. Fort, *Differentials,* AMM, vol.59 (1952), pp. 392-395.

**[4]** D. Bressoud, *The Changing Face of Caculus: First-Semester Calculus as a High School Course,* vol. 24 (2004) issue 6, pp6-9 and issue 8, pp. 14-17

Peter Ruane ([email protected]) is retired from university teaching, where his interests lay predominantly within the field of mathematics education.