As stated in the introduction, the purpose of this book is to demonstrate how a programming language like Maple can serve as “a valuable adjunct tool in easily deriving, solving, plotting, and exploring interesting, modern scientific models chosen from a wide variety of disciplines”. Many mathematicians may give a different meaning to the definition of ‘Computer Algebra,’ so I want to clarify immediately that this text is not about elimination theory, resultants or Gröbner bases. What you will find in it, is a number of Maple worksheets (‘recipes’), illustrating models used in physics, chemistry, and biology. Each recipe begins with a clear explanation of the concrete situation to be modelled. Then the mathematical model is introduced, usually in the form of a differential system which is successively studied and plotted in the phase plane.
This is not a theoretical book on dynamical systems; it is not intended to explain in a rigorous way mathematical issues such as stability theorems, numerical solutions or chaotic systems. Nevertheless, it approaches these notions (for example, the Lyapunov exponent in chapter 8), with an utilitarian point of view. So this book will be especially suitable to undergraduate students in science and engineering. As explained by the authors, the book presupposes that the reader has learned or is about to learn about scientific models and it is intended to improve the understanding of the modelling techniques.
You will find a great variety of models, touching, for example, on the following topics: wave equations, solitons, variational calculus, Laplace transforms, Poincaré sections, bifurcation theory, Bessel equations. Probabilistic models (such as Brownian motion or Black-Scholes equations) are not included.
The book is written in an amusing style: the recipes are classified as appetizers, entrees and desserts and each section begins with a funny citation. The result is a very pleasant reading.
Even though this is not a Maple handbook, no prior knowledge of the language is assumed and all the relevant commands are indexed at the end of the book. All the recipes are included on the CD-ROM enclosed with the book. Many problems are proposed to the reader, allowing him to further explore the techniques given in the text. The book may be used for self-study, or as the basis of an online course.
Fabio Mainardi earned a PhD in Mathematics at the University of Paris 13. His research interests are Iwasawa theory, p-adic L-functions and the arithmetic of automorphic forms. He may be reached at firstname.lastname@example.org.