The day-to-day of curves and fluxes

All but the last of these five American-authored texts are new editions of previously successful books and provide excellent coverage of differential and integral calculus and its applications. All are well illustrated, contain copious worked examples and problems and are wide ranging in their material.

Deborah Hughes-Hallett et al 's two texts are the work of consortia funded by the National Science Foundation of America. The first text is the more extensive of the two, covering both partial and ordinary differentiation, optimisation of functions of several variables, multiple integrals, line and flux integrals and the calculus of vector fields. The second restricts itself to functions of one variable, apart from a chapter on optimising functions of several variables. Both expect students to be familiar with the tools of mathematical technology and reference is often made to exploiting the power of graphical calculators, Computer Algebra Systems and spreadsheets. However, the development of material is not dependent on such aids. There is more than enough material here for a two-semester calculus course for mathematicians and others who need to use mathematics in their subject.

Robert Smith and Roland Minton's text covers a wide range of material, comparable to that of the first text. Each chapter is introduced with a brief application related to the mathematical concepts being developed.

Material is developed in several ways, through formal definitions and proofs, examples, use of graphs and tables, numerical work, end-of-section exercises and so on. When discussing curve sketching, the text is particularly good in pointing out that graphical calculators and CASs are no substitute for careful analysis and clear thinking. Access to Calculus: An Interactive Text , a CD, is included with each copy. This program takes key examples and figures from the text and puts them into an interactive format for practice.

Howard Anton et al 's text limits itself to functions of one variable but coverage of material is comprehensive and rigorous within that limit, particularly chapters on "Applications of the definite integral", and "Analytical geometry in calculus". The authors' aim is "to link calculus to the real world and the student's own experience". New sections on mathematical modelling have been added, and selected chapters end with modules called "Expanding the calculus horizon". The intention of these is to "take the student a step beyond the traditional calculus text". A CD containing the program Graphing Advantage Plus accompanies the text. This enables the user to draw graphs, calculate roots and intersections, integrate numerically, and fit curves by least squares.

Ken Binmore and Joan Davies' text is pitched at a higher level than the previous four. Much of the work presented on functions of one variable and their derivatives is designated as revision material, as is some introductory material on matrices. The authors soon concentrate on functions of several variables, and much of their applications are in economics and statistics - for example, in differentiation, the calculation of shadow prices as part of a constrained optimisation problem, and in integration joint and marginal probability distributions, and multivariate normal distributions. Matrix notation is used extensively in some areas of the text. The book ends with a discussion of linear first and second-order differential and difference equations, again with applications primarily to economics. The colour illustrations will help students to visualise complex mathematical objects. This is a book principally for students of mathematical economics.

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