What is Calculus? From Simple Algebra to Deep Analysis

The approach to differentiation in this book begins with Descartes’s thoughts on ‘double points on algebraic curves’ (which he used to construct normals and their associated tangents). This technique leads to a definition of differentiability that avoids limits of the type \(0/0\). In the simplest case, the tangent to the graph of a polynomial \(P\) at the point \((a, P(a))\) is a line that intersects the graph of \(P\) with multiplicity greater than or equal to 2. The unique line satisfying this condition has slope \(q(a)\), where \(q\) is the polynomial determined by the factorisation \(P(x) - P(a) = q(x)(x - a)\).

This idea pervades all material in opening chapter, which is called ‘Prelude to Calculus’. In the context of algebraic functions (mainly polynomials), the notion of derivative clearly emerges, and the basic rules of differentiation are initially derived within this limited context.

Ideas on real analysis first appear in the following chapter, where number systems are studied with respect to cardinality and completeness. Transcendental functions are revised and concepts of local and global boundedness are introduced.

Great emphasis is subsequently placed upon techniques for measuring ‘rates of change’ so that, half way through the book, readers will be ready to digest Carathéodory’s more general definition of differentiability:

A function \(f\) defined in a neighbourhood of a point \(a \in \mathbb{R}\) is differentiable at \(a\) if there exists a factorisation \[f(x) - f(a) = q(x)(x - a),\] where \(q\) is continuous at \(x=a\).

The value \(q(a)\) is called the derivative of \(f\) at \(a\) and is denoted by \(D(f)(a)\) or \(f'(a)\).

In addition to bypassing limits of the sort \(0/0\), other claimed advantages of this formulation are that it provides easier proofs of the chain rule and the inverse function rule — and it is also said to generalize more naturally to multivariable calculus. However, there is no use of Leibniz notation and the concept of ‘differential’ isn’t crystallised within these pages.

On the other hand, local numerical linear approximation is the context in which differentiability, limits and continuity are more rigorously examined. Throughout the book, real analysis is seen to serve the needs of calculus which, in turn, is shown to have many real-world applications. There are many examples, illustrations and useful exercises (no solutions provided), so this makes the book an ideal text for its stated purpose.

As for the book’s ‘stated purpose’, Michael Range says that:

…it could be used as a text for an honors class with well motivated students, where the instructor has flexibility in adjusting course content.

…could also be used as a text in a first course on real analysis.

Having enjoyed reading this book, I feel that it may not be first choice for use with students who are new to calculus; but it certainly would provide excellent a corrective revision of calculus for those who have been taught it simplistically. It also represents an imaginative (and less rarefied) introduction, to real analysis, because ideas on real analysis only appear when they illuminate particular aspects of calculus under discussion.

Another attractive feature of the book is its historical element, which includes reference to the algebraic/geometric method of Apollonius for finding tangents; the work of Galileo on falling bodies; the methods of Newton, Leibniz and Cantor’s notions of cardinality. This, together with the above-mentioned features, make this book a uniquely imaginative introduction to real analysis alongside a cogent account of the principles, and applications, of differentiation and the Riemann integral.

Peter Ruane’s reading on calculus began with Teach Yourself Calculus (P. Abbot, 1940) and continued in numerous stages to Spivak’s Calculus on Manifolds and it is now rounded off nicely by the book under this review.