How traffic jams spring up on clear roads

Mathematical modelling is at the very heart of all of applied mathematics and most of quantitative physical science. It exposes the relative importance of different elements of a problem and it may, occasionally, even permit useful prediction, but most of all it orders our thoughts and imposes rigour on our thinking. In this advanced text, A. C. Fowler leads the reader through a host of applications of continuum mathematical models across a wide range of disciplines in engineering and science. The book is difficult, Fowler says, "unforgiving", and is written for the professional or advanced student.

In the introduction the main steps in the development of continuum models are outlined. This section gives a foretaste of the diversity of application that is the hallmark of the book. Examples range from an explanation of those irritating traffic jams that spring out of nowhere when the road ahead is clear to a description of convection in the earth's mantle, the driving force for plate tectonics. The second part is devoted to the methods that will be prerequisites for an understanding of the main text. Key techniques are introduced and again illustrated with a wide range of examples. Classical problems, heat transfer, viscous flow, solid mechanics and electromagnetism are covered in the third part and serve not only to review these important successes for continuum mathematics, but also to develop general classes of equations that will resurface repeatedly throughout the book.

Those of us who teach undergraduate courses in mathematical modelling, particularly in interdisciplinary degrees such as environmental sciences, are constantly on the look-out for examples to broaden our teaching into areas outside our own specialism, and it is in the breadth and depth of such examples that the main strength of this book lies.

The presented case studies are almost self-contained and span an incredible breadth of applied science and engineering. Enzyme kinetics, complex chemical reactions, parasite infestations and ground and surface water flow are all dealt with at a uniform and advanced level.

The notes are particularly useful, often giving unexpected snippets of interesting information, including, for example, recipies for different variants of the Belousov-Zhabotinski reaction that could be cooked up in any university science faculty and would give new life to a lecture on oscillatory chemical reactions.

The book is too difficult to be a course text, but is a must for university libraries. Research students who are not applied mathematicians but who are prepared to wrestle with the difficulty of the content will be rewarded by the insight the models, or personalised variants of them, will give to their work. I would therefore recommend the book to anyone who is excited by the power of mathematics in describing the world. It might even become a classic of the genre.

John McCloskey is senior lecturer in environmental science, University of Ulster, Coleraine.