Computer Algebra Recipes: An Introductory Guide to the Mathematical Models of Scienceis an eclectic collection of application stories together with associated Maple computer algebra worksheets or recipes. The text is aimed at mid-level undergraduates and assumes familiarity with calculus and linear algebra as well as some acquaintance with linear ordinary differential equations. No previous experience with the Maple software is assumed. A companion volume — due to be published soon and aimed at more advanced students — deals with more sophisticated differential equation models, ordinary and partial, linear and nonlinear.
The authors have designed this book to show how a computer algebra system such as Maple can be a valuable tool for computer modeling in a variety of ways: deriving, solving, plotting and exploring “what if” questions in engineering and in the physical, biological, and social sciences. A computer algebra system enables the instructor to explore more realistic applications with far fewer simplifying assumptions than is possible in traditional modeling courses.
The cookbook metaphor extends at least as far as the titles for the sections. “The Appetizers” discusses plotting, the presentation of data, and derivation of model equations. Each chapter here, as in the rest of the book, has at least two applications to illustrate and exercise the ideas that are presented. Moreover, every chapter has a generous number of problems that pursue additional aspects of the applications in the text or explore other applications with similar techniques. The “Entrees” section is the heart of the book. It includes discussion of algebraic models (scalar, vector and matrix), models using linear ordinary differential equations, and linear difference equation models. The final section “Desserts” focuses on Monte Carlo methods and fractal patterns.
This is a very much a “dig in and explore” book. There is not much general exposition about modeling or modeling techniques. Instead, there are more than seventy-five application stories, each with a Maple recipe. Associated with each recipe is an important scientific model or method. “What Was the Heart Rate of a Brachiosaur?”, for example, graphically examines data for heart rate versus body mass for several warm-blooded animals, looks at a least squares fit on a log-log scale, and asks whether a brachiosaur fits the pattern. “The Great Pyramid at Cheops” estimates energy requirements for the construction of a pyramid. “Fermi-Pasta-Ulam Is Not a Spaghetti Western” studies the normal modes of oscillation for a one-dimensional atomic lattice.
Although some of the applications are rather contrived, most of them seem to work well. Any student who works through this text will develop a very good sense of what mathematical modeling is about, and will build a solid repertoire of tools in the process. This text would work well for self-study or as a source of material for projects in a mathematical modeling class.
Bill Satzer (email@example.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.