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Mathematical Modeling In The Environment

Charles R. Hadlock
Mathematical Association of America
Publication Date: 
Number of Pages: 
Classroom Resource Materials
[Reviewed by
Leah Edelstein-Keshet
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The author of this book is particularly well suited to writing about the subject. Starting off as a mathematics professor, he spent 13 years as an environmental consultant before returning to the classroom. Thus, many of the examples, experiences, and insights in the book are realistic and convincing.

The book is divided into two parts: Part 1 contains elementary treatment with minimal mathematics and is relatively self-contained. Part 2 follows up with somewhat more mathematical ideas. Even this "advanced treatment" will still be readily accessible to an undergraduate with some exposure to elementary differential equations, since, by and large, the mathematics is underplayed.

Each of the two parts of the book is divided into three topics: ground water, air pollution, and the treatment of hazardous materials. The treatment is multi-faceted, combining physical principles, chemistry, dimensional considerations, common sense, and some mathematical analysis. There are numerous "case studies", some of which could have been directly drawn from the type of problems that Hadlock encountered as a consultant. Most of the material is accessible to any reader with a minimum of mathematical skill, even those who are "rusty" in calculus. The book comes with a disk containing several DOS programs that complement the exercises on hazardous material.

I found the text and the examples quite fascinating: "Aha! so this is what people do in environmental modeling!" I found myself thinking. Many of these problems were quite novel, or were presented in a way that is not often encountered in a mathematics department. The part that makes the treatment particularly interesting is that actual specific properties of materials (e.g. boiling points, molecular weights, etc, of hazardous chemicals) enter into consideration and are used in the exercises.

The book comes with a detailed solution manual complete with a lot of advice about how to teach the course, something that many instructors would find appealing and helpful if they are non-experts in the area of environmental modeling.

For me the only down side of the book was the feeling that perhaps a bit more of the mathematical aspect of the problems was missing. Although the author probably meant to avoid intimidating non-mathematical readers, the resulting text sometimes gets wordy, and it can be difficult for a mathematical reader to cut to the heart of the matter and decide "what this is all about." I think a way to avoid this might have been to include some optional material summarizing the salient mathematical points of the chapter, to help orient a mathematics professor to the direction of the mathematics. I found myself getting impatient with some of the longer descriptions since I was always trying to assess: "What is the main mathematical lesson we can be teaching here?" (One would expect that a monograph published by the MAA can aim to show off a bit of the mathematics, at least as optional "asides".) Another suggestion might be to include some ideas for further study by mathematicians, at a more concise and deeper mathematical level than what is available in Part 2.

I find myself asking how I might use this book. Certainly, it makes interesting recreational reading and expands one's horizon. To integrate it directly into mathematics curriculum may be a challenge, because in a mathematics department we may be pressed to teach a more mathematical course. Judicious use of a few examples in the undergraduate courses such as Calculus and Differential Equations may work. (There is stiff competition, however, in many of the shorter "modules" published by UMAP, which may be easier to pick up and use.) The text would be ideal as reading material for undergraduate projects or possibly an honors thesis, as it is well written, clear in its explanations, and can be read by any student interested in seeing some "real world applications." I would even suggest that advanced high school students could enjoy this book and reap a benefit from seeing these applications. All in all, this is an interesting book and I am glad I had the opportunity to review it.

Leah Edelstein-Keshet is professor of mathematics at the University of British Columbia, and author of Mathematical Models in Biology, a text suitable for undergraduates.