Mathematical Modeling for the Scientific Method

This book is nothing like I thought it would be. Rather than a series of exercises of the form:

• Posed problem based on the physical world
• Construction of the mathematical model of the problem
• Solution of the mathematical model

The book is almost exclusively a brief description of the mathematics needed to perform basic modeling.

Chapter 1 is a review of algebra from the basics of set theory up through the introduction of abstract algebraic structures such as groups and rings. Chapter 2 begins with covering relations and functions and concludes with the most commonly used probability functions. Chapter 3 covers the basics of statistics, from descriptive to inferential. Chapter 4 is a survey of linear algebra, chapter 5 covers differential and integral calculus with series and chapter 6 covers vector calculus. All of this is done in 380 pages.

Chapter 7 has the title “Mathematical Modeling” but it is still more theory than practice. Topics include “First Order Differential Equations” and “Systems of Nonlinear Differential Equations” and all of this is done in approximately 40 pages. Chapter 8 has the title “The Scientific Method” and is a strange one. A wide variety of topics are covered, but all very superficially. For example there are the following topics with the number of allocated pages:

• Quarks and Groups, two pages
• A Small Ising Model of Magnetism, two pages
• The Shape of DNA, one page
• Cellular homeostasis, two pages
• Thermodynamics of Oceans and Atmosphere, four pages
• Quantum Cosmology and the Big-Bang Theory, six pages

Exercises are given at the end of each section, further reducing the amount devoted to the topics, but no solutions are provided.

Given that this is a survey of most of the mathematics used in basic modeling, one questions what the assumed background of the reader is. With 380 pages devoted to the first six chapters, there is no real depth of coverage of any single subject. In my opinion; this is not enough for the student being exposed to the topic for the first time.

Charles Ashbacher splits his time between consulting with industry in projects involving math and computers, teaching college classes and co-editing The Journal of Recreational Mathematics. In his spare time, he reads about these things and helps his daughter in her lawn care business.