Introduction to Applied Mathematics for Environmental Science

For many years, first as a student and later as a teacher, I have ob­ served graduate students in ecology and other environmental sci­ ences who had been required as undergraduates to take calculus courses. Those courses have often emphasized how to prove theo­ rems about the beautiful, logical structure of calculus, but have ne­ glected applications. Most of the time, the students have come out of such courses with little or no appreciation of how to apply calculus in their own work. Based on these observations, I developed a course de­ signed in part to re-teach calculus as an everyday tool in ecology and other environmental sciences. I emphasized derivations―working with story problems (sometimes quite complex ones)―in that course, and now in this book. The present textbook has developed out of my notes for that course. Its basic purpose is to describe various types of mathemati­ cal structures and how they can be apphed in environmental science. Thus, linear and non-linear algebraic equations, derivatives and in­ tegrals, and ordinary and partial differential equations are the basic kinds of structures, or types of mathematical models, discussed. For each, the discussion follows a pattern something like this: 1. An example of the type of structure, as apphed to environmental science, is given. 2. Next, a description of the structure is presented. 3. Usually, this is followed by other examples of how the structure arises in environmental science. 4. The analytic methods of solving and learning from the structure are discussed.

Introduction to Mathematics for Environmental Science evolved from the author’s 30 years’ experience teaching mathematics to graduate and advanced undergraduate students in the environmental sciences. Its basic purpose is to teach various types of mathematical structures and how they can be applied in a broad range of environmental science subfields. Derivatives and integrals, ordinary and partial differential equations, and linear and non-linear algebraic equations are the basic kinds of structures (types of mathematical models) discussed.

For each mathematical topic, the book emphasizes derivation of mathematical models from "story problems" first, and solution of equations second. It suggests finding analytic solutions when possible, but discusses methods for finding numerical solutions to problems that can’t be solved analytically. For most topics, examples of using Matlab software to solve and explore the structures are also included.

This text assumes that its readers will already have been introduced to the basic ideas of differential and integral calculus, so it is not an introduction to calculus. It does, however, include three early chapters to review basic algebra, derivatives, and integrals.

Professor Parkhurst earned his B.S. degree in applied mathematics from the College of Engineering, University of Colorado, Boulder, in 1965, and his Ph.D. in Botany (Plant Ecology) from the University of Wisconsin, Madison, in 1970. He then worked three years at the CSIRO Division of Atmospheric Science in Aspendale, Victoria, Australia, before joining the Environmental Science faculty of the School of Public and Environmental Affairs, Indiana University, Bloomington, in 1973. He held a joint appointment in the Biology Department there as well. He retired from that position in December, 2005, after 32 years. His research areas include mathematical modelling of adaptations of plant leaves to environment, environmental riskanalysis, and applications of statistics in environmental and public health areas.