Heck has revised and expanded this book’s successful first edition [1]. One motivation for the new edition seems to have been the appearance of Maple V release 4. This new version of Maple incorporates a new worksheet interface with \LaTeX output, which improves the appearance of almost every page.

Heck has made many minor changes in and improvements to this book. However, this edition is far more than a simple update. There are 200 more pages than in the first edition, and the bibliography has nearly doubled in size, incorporating not only new material, but additional references to the mathematical and computational background as well. There are also many more exercises, and I think the book would now form an excellent basis for a self-study course in Maple. There are numerous other changes throughout, too many to be detailed in this review.

Chapter 1 compares the relative capabilities of computer algebra systems and integral tables. However (and this may be a philosophical difference), I wish the author had not said that “we may call computer algebra systems mathematical expert systems,” because the inner workings of computer algebra systems are very different from those of most mathematical experts. The information on algebraic numbers has been greatly expanded, and a confusing error has been corrected. There is new coverage of attributes and properties, notably Maple’s assume facility, which is also the subject of a new chapter later in the book. The assume facility, which is relatively new, allows users to declare that a variable should be integral, or positive, or odd, or real. It needs to be handled with some care, however, precisely because Maple is not an expert system.

Coverage of input/output facilities has been expanded and clarified. The chapter on Maple’s internal representation, which is very different from its printed representation and influences the behavior of operations such as substitution, has been updated and expanded. It also contains a new discussion of the algsubs command, which performs algebraic substitution rather than data structure substitution. The chapter concludes with a brief, but excellent, description of the problem of specialization, wherein a particular value is given to an indeterminate, which may render some previous calculations invalid. This is a major problem in all algebra, both manual and computer, and I recommend this discussion to users of all computer algebra systems.

The chapter on functions has been enlarged, with more material on the Lambert W function and on the problem of local variable capture. The chapter includes new material on piecewise defined functions, which are one of the areas in which Maple has made much progress recently. More examples of the use of anonymous functions are given.

The chapters on differentiation and integration have been updated, especially the explanation that definite integration is more than just substituting into an indefinite integral; this explanation was already good in the first edition. The material on integral transforms has been expanded to track the changes in Maple, both in terms of new built-in transforms and in terms of the use of addtable to add new definitions to transforms. The discussion of summation has also been enhanced, and the author provides an excellent warning about the use of acceleration techniques in summation. There are also more examples and explanations in chapter 11, “Series, Approximation, and Limits,” in which Heck covers Taylor series and Chebyshev, Padé, and Chebyshev-Padé approximations.

Chapter 12, “Composite Data Types,” has been expanded; the new material includes a section on tables and one on the somewhat arcane rules that apply to evaluation tables. The chapter on simplification has been updated, but still correctly warns the reader of the dangers of some automatic simplification and of some of the user-callable simplifications, such as expand and combine. The discussions of combine and simplify have been greatly expanded, as has the material on trigonometric simplification.

The chapter on graphics has been substantially revised to take account of changes in Maple and in graphics interfaces, notably the change from PostScript to Encapsulated PostScript and the addition of many new options. The new plottools and geometry packages are described well, as are the facilities for plotting statistical and other data, and there is a useful list of options for graphics.

Chapter 16, “Solving Equations,” is expanded and revised. For some problems, Maple does not employ guaranteed algorithms, often because no such algorithm is known. In a section of the chapter called “Some Difficulties,” Heck explains how to check Maple’s results on such problems. I wish more tutorial information were as helpful as this.

Chapter 17, “Differential Equations,” includes new coverage of the useful DEtools package and of partial differential equations, and several examples demonstrate new facilities of Maple. Again, the author shows how to help Maple solve problems that it cannot solve by itself and how to check its results. Several new tables on the different methods of solving differential equations have been added, though it is unfortunate that one table indicates that Euler is the default method, while the text gives the RKF45 method as the default.

The last chapters, on linear algebra, have been almost totally rewritten, prompted by both the changes in Maple and the revisions of the previous chapters. Several tables have been added to describe the functionality available.

This is a good piece of work made even better, as well as more current, by an extremely thorough revision.