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Student's Guide to J. Marsden's and A. Weinstein's 'Calculus'

is omitted). The exponential function is defined as the inverse of the logarithm, and the hyperbolic functions follow in the usual way. It is, of course, convenient to have all these functions at one's disposal during the subsequent development, but I believe that the price to be paid in time, labour and lack of rigour is too high. Chapter 2 is devoted to limits (almost entirely of functions) and continuity. Sequences are used only to provide an alternative characterization of a functional limit. Infima and suprema and the completeness axiom for the real number system are introduced during the discussion of properties of continuous functions. It is shown that the completeness axiom is equivalent to the nested interval principle, but there is no attempt to explain the status or to assess the importance of either. In particular, no mention is made of the fact that the set of rational numbers does not have the completeness property. The next two chapters deal with differentiation and its applications. The treatment is very thorough, but a reader meeting some of the less elementary topics (e.g. Darboux's or Taylor's theorems) could well face unnecessary difficulties owing to the absence of any introductory remarks. In fact, very often in this book all explanation of and motivation for a theorem appear only after the proof. On the other hand, there is an excellent section on Newton's method for approximating to roots of equations. Also a general feature in this book, very evident in chapter 4, is the provision of many interesting examples. Chapter 5 gives a good and careful account of Riemann integration and contains a long section on numerical integration. An almost encyclopaedic guide to systematic integration is found in chapter 6, together with many exercises. However, there is one section which begins with two pages of formulae for arc length, volume and surface area without any explanation or justification. The subject matter of the last chapter, of over 100 pages, is sequences and series. Although limits of sequences have been used earlier in the book, a new start is made now and the development takes in points of accumulation and upper and lower limits. Cauchy sequences are also defined and shown to converge (but without comment). Stirling's and Wallis's formulae feature among the numerous applications and examples. The account of series includes a whole battery of convergence tests. A long section on uniform convergence contains an example of a continuous nowhere differentiable function; and the final topic of the book is that of power series. Though the book has considerable virtues, I do not think that it can fulfil its avowed purpose of smoothing the novice mathematician's path from calculus to analysis; for it fails to maintain a consistent standard of rigour, it does not provide adequate explanation at several crucial points, and its 500 pages are apt to overwhelm an inexperienced reader with detail. However, the professional analyst will find much of value in this book. He will be particularly grateful for the vast collection of often fascinating exercises, but he will also be glad to have at hand the several exhaustive sections mentioned in the descriptions of chapters 5-7. In addition, the book contains some interesting historical material.