Mathematical magic explained

Linear Equations and Matrices is one of a series of texts designed to build into a mathematics toolkit for engineers. It is written by an experienced teacher and author, and the style reflects this throughout the book. The coverage is thorough and the material is presented at an appropriate rate for first-year engineering undergraduates. Applications of the material are largely dealt with in two chapters, one on circuit analysis and the other on structural analysis, aimed at electrical and civil engineers respectively.

The brief final chapter discusses iterative methods for the solution of linear equations. The role of diagonal dominance is correctly highlighted. However, the problem of ill-conditioning, which can seriously affect the accuracy of computer-generated solutions, is not. This topic is discussed in an earlier chapter, but as the use of computer routines and packages is now the norm in applied linear algebra, a greater emphasis here on "what can go wrong" would have been appropriate.

The book provides some secure foundations for further study and it will be appreciated by engineering undergraduates for its clarity and level of explanation. It is very well written and contains numerous worked examples and exercises (with answers).

Mastery of the material contained in Bolton's text is an essential step towards the understanding of mathematics at higher levels. In such a book it would, however, be difficult to convey the sense of enthusiasm and excitement evident throughout Chiang Mei's book, Mathematical Analysis in Engineering.

The author of this first-class book is an engineer who is highly competent in mathematics, who enjoys doing mathematics and evidently enjoys writing about it. It is an advanced text, and, according to the claim on the back cover, the book does not follow the usual pattern of first stating the mathematical principles to be adopted and then finding the application. Instead, the paradigm is said to be - start with a problem, find the mathematics which suits it, and end with a mathematical analysis of the physics. Especially for students, whose "feel" for what mathematics might be needed is limited, it can be rather depressing when, in the middle of solving a problem or case-study, the desperately needed mathematics is produced "rabbit-like" from a hat. Fortunately though, the claim is bogus!

The book opens with a chapter on the mathematical formulation of physical problems and draws on examples from the field of civil engineering. The governing partial differential equations for the vibration of strings and elastic rods are obtained, and, in addition, the derivation of those for the flow of traffic and for seepage flow in a porous medium are given. The linearisation process for the nonlinear partial differential equations governing shallow-water waves is clearly described.

Having set partial differential equations at centre stage, a short second chapter deals with the classification of such equations, emphasising the implications of the existence, or otherwise, of characteristic curves. These ideas are developed in an interesting way in the third chapter where they are used to discuss the flow of traffic. Later chapters include very good accounts of the technique of separation of variables, of Fourier methods, of Green's functions, Bessel functions and of the theory of a function of a complex variable.

The chapter on perturbation methods is an exceptionally clear introduction to the subject.

From what has been said so far, this book might appear to be about applied mathematics, and indeed, it is a valuable contribution to the literature in that field. However, the real applications are there as well, notably to the modelling of the flow of traffic, the description of seepage flow and to the solution of foundation engineering problems.

The author does leave the reader in suspense over the problem of a vertically pressed flat footing, since the model developed produces a physically unacceptable solution. However, he does suggest a possible remedy, although its investigation would be a major research task.

The book contains a number of interesting end-of-chapter exercises. It is unfortunate perhaps that Macsyma rather than the currently more popular Maple has been used for the chapter demonstrating the value of computer algebra. Serious workers in the field will be able to overcome this, however. I consider this to be an outstanding book, attractive both in content and in the standard of the typesetting and general layout.

Nigel Steele is head of mathematics, Coventry University.