Discussing calculus

The first three of these texts form part of Arnold's Modular Mathematics series, designed primarily for students on mathematics or mathematics-related degree programmes based on semester-long modules.

Calculus and ODEs is based on a first-year undergraduate mathematics module 12 weeks long. The text begins with an introduction to functions, limits and rates of change, followed by a discussion of continuity and differentiability. The author avoids the full rigour of formal analysis, but nevertheless offers the reader mathematically convincing arguments. Further chapters cover the essentials of one-variable calculus, reaching as far as differential equations, dealing with solution techniques and addressing issues of uniqueness and existence of solution. Throughout the text the emphasis is on the mathematical process and no real attempt is made to discuss applications.

Vectors in 2 and 3 Dimensions discusses vectors from a wholly geometric viewpoint. Starting from scratch, the author covers the topics one would expect in an introductory text, up to triple scalar and vector products, emphasising their applications in two and three-dimensional geometry. Then come vector spaces, linear combinations and bases, followed by linear transformations and matrices, eigenvalues and eigenvectors from a geometric viewpoint. There is some discussion of vector calculus as a means of studying curves and surfaces, culminating in the Serret-Frenet formulae.

Vector Calculus seeks to cover the basic principles and methods of multivariable calculus from partial differentiation to vector calculus, including the integral theorems of Green, Gauss and Stokes. It is intended for second-semester mathematics students but, says the author, is accessible to engineers who "can omit the theoretical chapters without much loss". There is a lot of material here, arguably too much. The author acknowledges he has "sacrificed some depth and detail in places" and lack of detail sometimes obscures the discussion of topics. As with the first two volumes, there is a good supply of exercises.

Calculus: A New Horizon , the sixth edition of a popular text, is intended for students planning careers in engineering or science. Starting from a discussion of elementary functions and culminating in multivariable calculus, the text covers most of the calculus and related topics such students need. Calculus: Single Variable is the second edition of a text produced by a consortium based at Harvard University. It aims to provide a solid foundation in one-variable calculus for subsequent courses in mathematics and other disciplines. Thus functions, limits, continuity, derivatives and integrals and differential equations are explored, both formally and informally. Their applications are demonstrated in modelling examples and exercises, from the physical and life sciences, as well as economics. The chapter on using the definite integral is particularly good, and there are exercises for readers.

John Stone is principal lecturer in mathematics, Sheffield Hallam University.