Methods of Applied Mathematics with a Software Overview

The title suggests two questions one might ask about this text: what topics in applied mathematics are covered, and what is the “software overview.” The mathematics consists largely of transform methods such as those of Laplace and Fourier. The “software overview” introduces the reader to three software tools.

The first edition of this text relied on a single commercial software product, MATLAB™. This began life as a collection of routines for doing matrix computations rapidly and accurately, but has since evolved into a more general computational tool. In many situations it has replaced Fortran.

The edition at hand drops MATLAB in favor of three free software tools. Octave is roughly a clone of MATLAB and runs most MATLAB code. The code cited in this book is mainly in Octave. Python is a general purpose programming language that is used for all the many graphical images in this book, but fewer examples of Python code are given. Finally, Maxima is a computer algebra system with roots going back farther than most CASs. It is used sparingly. Appendices provide brief introductions to all three tools. These seem excellent choices at even better prices. All require the user to type in strings of text commands and it is recommended that students have some programming experience, though no particular language is suggested.

Minimum prerequisites for the text are two years of calculus, including differential equations, and a course in linear algebra. (There is an appendix on linear algebra, but it is a very brief review, and would not be appropriate for learning the subject.) Inverting some of the transforms involves functions of a complex variable, and all the student needs to know for that purpose is included in a single 65-page chapter. The book is advertised as suitable for a one or two semester course.

The target audience appears to be engineers and applied mathematicians. The transform approach to differential equations seems to be especially popular with electrical engineers, and indeed we find many diagrams of electrical circuits here. The book is well written as English prose but many engineers may find it more mathematical than they might like. The book opens rather abruptly with an analogy between Fourier and Taylor series and then immediately begins determining coefficients for the former without any clue as to why one might want to do so. Discussions of anything an engineering major might consider an application are few and far between. The title might better include the phrase “applicable mathematics.”

This book seems a better match to applied mathematicians who want to learn these particular topics, and less suited to engineers who want detailed instruction on how to apply these methods in their own field. In any case, the clear prose and the “software overview” are valuable assets.

After a few years in industry, Robert W. Hayden (bob@statland.org) taught mathematics at colleges and universities for 32 years and statistics for 20 years. In 2005 he retired from full-time classroom work. He contributed the chapter on evaluating introductory statistics textbooks to the MAA’s Teaching Statistics.