Sean Dineen's * Multivariate Calculus and Geometry * - part of the Springer Undergraduate Mathematics series (Sums), as are all the books reviewed here - is based on his lecture course at University College, Dublin. It is an excellent and readable text for higher level students in mathematics or the mathematical sciences.

As the author states, multivariate calculus probably brings together, for the first time in the student experience, four important areas: analysis, linear algebra, geometry and differential calculus.

While prerequisite knowledge is assumed in algebra and calculus, there is no requirement for a detailed background in analysis. However, as the reader progresses through the material, the vital role of analysis in mathematics becomes clear, if it has not already done. Naturally, geometry permeates the entire text.

Each chapter begins with a concise summary of what it covers and the author gives a continuous commentary throughout the text, a distinctive feature of many books in the Sums series. This will be extremely helpful to students, especially those who are covering the material by some form of self-study.

The author's experience as a teacher comes through in the quality of the explanations and the detailed attention to vital underlying ideas, which are often (wrongly) taken for granted elsewhere.

All the material one would expect is present, including extrema of real-valued functions on Rn, the Frenet-Serret equations and line and surface integrals. The integral theorems of vector analysis are well presented and the book concludes with chapters on Gaussian and geodesic curvature. Solutions are given for all end-of-chapter problems.

David Wallace's * Groups, Rings and Fields * is written for students meeting abstract algebra for the first time, probably in the first year of a degree course in mathematics. The aim is to provide stimulating material in a form suitable both for those who take just a single course in the field, as well as for those who wish to pursue further study.

The opening of the book is gentle, as it should be, with the only stated - advantageous, not necessary - prerequisite being a nodding acquaintance with matrices and complex numbers. Opening with a chapter on sets and mappings, the pace builds up gradually and, by page 25, the reader encounters an unproven conjecture! Successive chapters deal with the integers, and first meetings with rings and groups.

More advanced treatments follow in the last two chapters, and a good level of coverage is achieved by the end of the book, including a discussion of the Sylow p-subgroups of a finite group.

Each chapter starts with a helpful and interesting summary and, throughout, topics are put into their historical, and sometimes behavioural context. I was intrigued to learn of the examples of longevity associated with an interest in number theory, and also that matters in the affairs of sets resemble affairs of the heart - in that neither are as straightforward as they seem.

The examples are well chosen and enlightening. Again, both the level and quality of explanation mean that this book is most suitable for self-study. There are plenty of end-of-chapter exercises, with hints to their solution.

* Measure, Integral and Probability * , by Marek Capinski and Ekkehard Kopp is an account of measures and integrals that is accessible to current senior undergraduates in mathematics. It has been written from a number of years' experience in teaching the material, which is apparent in the attention paid to basic ideas in analysis. Many students will particularly value this inclusion, and it will certainly aid self-study.

The main application area for the material is probability theory and this forms the second part of each chapter. It provides a good source of illustrative and readily appreciable examples throughout the development.

The level of explanation is excellent, and great care has gone into providing motivation for the study of all aspects of the material. Indeed the book opens with a clear and convincing explanation of why the Riemann integral will not suffice in advanced applications. At the core of the book - which covers all the basic concepts and ideas - material is developed fairly gently, with helpful guidance on the method of proof of the various theorems.

Chapter two gives a detailed construction of Lebesgue measure and its properties. The axioms of probability space are abstracted in its second half, and in chapter three random variables and their induced probability distributions are considered. The Lebesgue integral is defined and discussed in chapter four in a particularly readable manner.

After chapters on spaces of integrable functions and product measures, the pace increases, but not unrealistically, to the final chapter, on limit theorems.

Overall, this is an excellent and interesting text. My one carp is that the index needs a little attention. Full solutions to the exercises are provided at the end of the book.

Also from the University of Hull stable is Zdzislaw Brzezniak and Tomasz Zastawniak's * Basic Stochastic Processes * . Again, this is a text for final-year undergraduates in mathematics and, as such, requires a good knowledge of probability and calculus, including analysis.

The opening chapter gives a review of probability theory and a detailed treatment of conditional expectation is contained in chapter two. As would be expected, martingales in discrete time and Markov chains are discussed in further chapters.

Finally, a very good and well structured introduction to stochastic processes in continuous time paves the way for a development of Ito stochastic calculus in the final chapter.

The underlying motivation throughout the book is the application of the ideas to financial modelling. However, this is not developed in particular depth, and some pointers to further reading would have been useful.

This book fulfils its aim of providing good and interesting material for an advanced undergraduate study.

In conjunction with Capinski and Kopp's book, the texts would also provide an excellent programme for postgraduate students and others needing an introduction to the important field of probability-based financial modelling in the face of uncertainty.

An excellent feature and a real aid to self-study is the provision of fully worked solutions to all the exercises, conveniently located at the end of the chapter in which they occur.

Turning to the foundations of modern mathematics, * Topologies and Uniformities * by Ioan James is based on his lectures to senior undergraduate and postgraduate students at Oxford University.

It is a partly rewritten version of his 1987 book, * Topological and Uniform Spaces * , with revised exercises and the addition of worked solutions.

Although the material has been selected with the non-specialist in mind, this comprehensive account will be of particular interest and use to the intending specialist.

The book opens with an historical essay describing the evolution of mathematics, placing in context the work of Cantor and Dedekind in the last century through to that of Cartan and Weil in 1937.

Here the book's set aim is to discuss the ideas developed during this period, taking advantage of more recent developments in the fields of topological and uniform spaces. The material is presented in three parts, with the first six chapters describing topological theory.

The author's preference for an earlier than usual treatment of compactness emerges as the major difference between this work and others in the field. It follows on directly from the discussion of open and closed functions, preceding consideration of the separation conditions.

The middle part of the book is on the uniform theory and the last four chapters cover the areas of connectedness, countability and completeness.

Very full and helpful answers to the exercises are provided and there is also a select bibliography.

All in all, these are five very good additions to the Sums series.

Nigel Steele is head of mathematics, University of Coventry.

## Groups, Rings and Fields. First Edition

Author - David Wallace

ISBN - 3 540 76177 2

Publisher - Springer

Price - £17.95

Pages - 248

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