Chance to get a really good look at curves

Elementary Differential Geometry. First edition - Symmetries. First edition - Vector Analysis. First edition - Geometry. First edition - Understanding Analysis. First edition
November 29, 2002

   Roger Fenn's Geometry intends to inspire readers. The book was written in recognition of the diminished role of geometry in mathematics curricula, and it aims to have wide appeal. This is a very readable account, intended for undergraduates, although some of the material is suitable for advanced school pupils. The first chapter is on the geometry of numbers, motivated by the meaning of "geo-metry" as world measurement. Readers will never think of the number 3 in the same way again! The results for ∑ N n=1 n and ∑ N n=1 n² are derived here using square and triangular numbers.

Coordinate geometry is covered by an introductory chapter, where skills in curve sketching are identified as essential, and a later one on conics and quadratic surfaces. Also included is the deduction of planetary motion from Kepler's laws. Chapters on the geometry of Euclidean and complex planes cover basic material, while a chapter on vectorial-based solid geometry moves the reader into three dimensions.

Within the treatment of projective geometry, only linear properties of the projective plane are considered, but the reader should be inspired to read further. The chapter on spherical geometry deals with the ideas essential to navigation on a spherical world, and with the celestial sphere, providing an introduction to astronomy. The closing chapter on quaternions and octonions explains that Hamilton's extension of complex numbers has modern application in computer games. Answers to selected questions are given at the end of each chapter, supporting independent study.

Andrew Pressley's Elementary Differential Geometry is designed to render the study of curves and surfaces accessible to a wide audience. The target is higher-level undergraduates in mathematics, but the book could also appeal to sectors of engineering. Minimum prerequisite knowledge is required: calculus, vectors and linear algebra. As a result, the methods chosen are not always those that generalise; but this is appropriate.

The starting point is a consideration of what a curve is, and a patient introduction to basic ideas follows. The first two chapters discuss types of curves in the plane and in space and their local properties. They are followed by a short chapter on the global properties of simple closed curves in the plane. A chapter on surfaces in three dimensions introduces the ideas of surface patches. Excellent figures supplement a good account, sprinkled with illustrative examples. The need to measure distance between two points of a surface leads to the first fundamental form. The second fundamental form and normal and principal curvatures are the subject of the next chapter. Gaussian curvature and the Gaussian map are covered and, after chapters on geodesics and minimal surfaces, Gauss's theorem egregium and the Gauss-Bonnet theorem are considered. An excellent feature of the text is that as the sophistication of the material increases, the author maintains his habit of giving derivations, which are written out in full. Hints are given to many of the exercises, with full solutions at the end of the book.

D. L. Johnson's Symmetries is based on lecture courses for mathematics students at Nottingham University and at the University of the West Indies, and is written in a friendly style. Some sections will also be of interest to physicists and crystallographers. Prerequisite knowledge is limited to a good understanding of sets and functions, although some mathematical maturity is needed.

The book opens with a discussion of metric spaces and their groups. The Euclidean group is introduced, with isometries as its elements, and ideas are developed using isometries of the real line. Isometries of the plane, in particular the three basic types - translations, rotations and reflections - are considered. Group theory is needed and a crash course is provided. Products of reflections are considered as an alternative to the use of the normal form theorem in understanding the structure of the Euclidean group. Discrete subgroups of this group and their classification occupy the next chapters. These consider in turn: friezes and the OP and OR cases of the plane crystallographic groups, tessellations of the plane and sphere, triangular tessellations of elliptic and hyperbolic space. The book ends with an introduction to polytopes. Each chapter has a number of exercises.

Real analysis has also seen less emphasis recently. In writing Understanding Analysis, Stephen Abbott appreciates that modern readers may be unfamiliar with axiomatic arguments. Thus, many courses in analysis have been made easier by making them less interesting. The alternative is to make advanced topics accessible and focus on questions that give real analysis its value. Motivation is needed and is achieved by a discussion section at the start of each chapter.

The opening chapter on the real numbers is motivated by a consideration of √2. The preliminary discussion deals with the nature of proof, and introduces sets and functions before the axiom of completeness and its consequences. The project section, a feature of each chapter, is on Cantor's theorem. Rearrangement of infinite series serves to introduce the material on sequences and series and the topology of the real numbers is motivated through discussion of the Cantor set. Functions of a real variable lead to the ideas of limits and continuity as key concepts, paving the way for study of the derivative, while the probability of the extinction of surnames is the introduction to power series. Integration is considered, based on the Riemann integral and the Lesbegue criterion, and Lesbegue integration is introduced as the "standard integral in advanced mathematics". Recalling that there are functions whose improper Riemann integrals exist but that are not Lesbegue integrable, it is essential to read the section on "Generalised Riemann integrability".

Prerequisite knowledge has been kept to a minimum: an understanding of single variable calculus. The text is written with introductory students in mind and will certainly be persuasive in establishing the value of analysis.

The preface to the first edition of Klaus Janich's Vector Analysis begins: "An elegant author says in two lines what takes another a full page. But if a reader has to mull over those two lines for an hour, while he could have read the page in five minutes, then... it was probably not the right kind of elegance."

This text was written for second-year undergraduates in Germany and is suitable for those with a good understanding of calculus of several variables together with basic topology. Preferring time-saving to line-saving elegance, the pace is slower than that of many other texts, and this is helpful for those studying the subject independently.

Nevertheless, Janich provides an introduction to de Rahm co-homology, Riemann manifolds and tensor calculus. Throughout, the style is informal without being condescending.

The author ties together different approaches to the tangent space of a manifold coming from germs of real-valued functions, smooth curves, and that commonly adopted in physics literature based on Ricci calculus. This enables the reader to move between sources with little difficulty. Each chapter contains a test consisting of multiple-choice questions. The answers are provided and the reader is warned that some of the questions are so obviously simple that a healthy scare will result when they prove not to be so.

Nigel Steele is professor of mathematics, University of Coventry.


Elementary Differential Geometry. First edition

Author - Andrew Pressley
ISBN - 1 85233 152 6
Publisher - Springer
Price - £18.95
Pages - 332

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