Mathematica for Scientists and Engineers: Using Mathematica to do Science

Rating:
Summary: Somewhat dated....but still useful
Review: Since this book was published, Mathematica has come out with version 4.1, which makes the book somewhat dated, since it is written in 3.0. However, it could still be of benefit to someone who wants to use Mathematica for more specialized tasks in science and enginnering. Some of the Mathematica code in the book is given for problems that are not usually discussed in books on Mathematica. It could serve as a supplement to a course in Mathematica if one is willing to put up with its being out of date, since there are many interesting exercises assigned at the end of each chapter. It would not be too difficult to update the book to Mathematica 4.1. Practicing scientists interested in using Mathematica for visualization could use the book as a handy reference.

Chapter 1 begins with a short review of how to use Mathematica and then the author jumps right into the Riemann zeta function. He gives a fairly lengthy discussion of this function, complete with graphics and Mathematica code. He uses both his own code for the function as well as the built-in function Zeta to illustrate the properties of the Riemann zeta function, particularly its zeros. He shows how the choice of grid spacing can hide the singularity structure of this function, if not chosen finely enough.

Chapter 2 is an overview of numerical methods in Mathematica. He begins with the problem of numerical integration by calculating the specific heat of a crystalline lattice using Nintegrate. The ability of Mathematica to integrate numerically nasty integrands, such as sin(1/x), is then investigated, with the problems with the singularity and convergence discussed in some detail. He also discusses numerical contour integration, which is not usually done in Mathematica books. The Duffing oscillator is treated as an example of solving differential equations numerically using Mathematica. Most importantly though the author shows how to solve partial differential equations numerically using Mathematica. Although performance issues in solving PDEs will appear in using Mathematica to do this, the author uses the built-in function NDSolve to show how one might gain insight into the behavior of solutions. He treats the case of sound waves in a pipe and the elastic string with fixed ends subjected to a constant transverse force. Then after a brief look at numerical sums and products, he treats the quantum mechanical problem of a particle in a one-dimensional well. The chapter ends with a consideration of the 3-body problem, including the restricted 3-body problem. The author's treatment is pretty good but he fails to discuss in detail the numerical instability of the orbits at large times.

Symbolic manipulation, the tour-de-force in Mathematica, is treated in detail in chapter 3. He gives as an example a very interesting simulation of a cyclotron, and shows how Mathematica can be used to efficiently do the algebra in this problem. He then illustrates how to use Mathematica to manipulate series, with series solutions of ordinary differential equations given the emphasis. Most helpful is his discussion on how to treat residues in complex analysis using Mathematica. The author also shows how a cavilier use of symbolic integration can cause problems with some integrands. Such a discussion is very useful in it flags the first-user of Mathematica or to the mathematics student who might be careless in using the built-in function Integrate. This is followed by a fairly detailed discussion of how to differentiate symbolically using Mathematica and how to do vector calculus. Anyone who has done the calculations in the solution of the Schroedinger equation for the hydrogen atom will appreciate the ability of Mathematica to make the arithmetic less painful, and the author shows this in this chapter. Then after a brief discussion of matrix algebra and simplification using Mathematica, the author moves on to a topic that is usually not treated at all in Mathematica books, namely integral transforms. He shows briefly how to deal with the Laplace and Fourier transforms. He also introduces the ability to do local scoping of variables using the Module command.

Chapter 4 covers Mathematica's powerful graphics capability. Most of this material can be found in other books, and in fact the absence of color in the book is somewhat disappointing given the author's heavy use of color directives when plotting functions in this chapter. He does however give a useful short discussion on how Mathematica samples a function to be plotted. A fun example of how to create a Smith chart using Mathematica is detailed by the author. In addition, he treats a physical problem of the spinning top. This is a problem that cries out for visualization when learning about it for the first time, and the author does a fine job of explaining its motion, including the Mathematica code for doing an animation. Unfortunately he does not discuss the solution of the spinning top using elliptic curves nor the case of the spinning top without a fixed point, which involves the use of hyperelliptic curves. Mathematica can be used effectively I have found to deal with these cases.

The last chapter is oriented more to matters in computer science, as the author treats data structures, namely lists, in Mathematica. He gives the reader hints on how to program in Mathematica efficiently, and how to use the built-in function Compile to improve program performance in Mathematica. He also illustrates the very powerful ability of Mathematica to do rule-based programming via an example of generating fractal images. Most importantly, he shows how to use Mathematica to solve partial differential equations using finite differences, another topic usually not done in other Mathematica books. A brief discussion of how to do tensor manipulation ends the chapter.