Advanced Problem Solving with Maple A First Course

The aim of this book is to teach problem solving skills using examples from applied mathematics together with appropriate computational tools. The authors want to cover the span that goes from problem formulation to model construction and on through to analysis, testing of assumptions, and evaluation of the sensitivity of results. They rely on Maple software as the primary computational tool.

 

Basic calculus is the primary prerequisite. Some linear algebra is used in the later chapters. The authors include more material than could be covered in one semester, so instructors have some flexibility in their selection of topics. The book could profitably be used for an introductory course, and perhaps for a capstone course with some selected advanced subjects.

 

After providing some basic background in Maple, the book goes on to concepts and techniques for modeling problems using first order ordinary differential equations. The applications are mostly standard ones and include Newton’s law of cooling, the chemistry of compound mixtures, and population dynamics. Numerical methods are introduced early and the authors take advantage of Maple’s graphing capabilities to introduce slope fields and plots of solution trajectories. Systems of differential equations follow naturally with a discussion of equilibrium states and with examples that include applications to economics, competition between species, models of diffusion, and electric circuits.

 

Problems in linear programming (and the related integer and mixed-integer programming) are perhaps the most unusual topic treated in a course at this level, and there are very nice examples with the location of ambulances for emergency services and transport of hazardous materials.

 

The final parts of the book take up regression, curve fitting, and some problems in probability and statistics. An introduction to reliability engineering is a part of this. There is also a particularly nice case study in airline overbooking. The last chapter has a brief discussion of simulation, mostly of Monte Carlo type.

 

Each chapter has a collection of exercises as well as a collection of potential projects. The authors rightly regard the latter as a very important part of a student’s learning experience.

 

This book aims to teach problem-solving, not modeling. Each exercise comes in the context of a single modeling method (differential equations, linear programming, statistics, or simulation). The more difficult questions of how to teach modeling from scratch – without a preexisting context - are not addressed. It just isn’t what the authors are trying to do here. They teach some important skills, and what they are doing is clear, thoughtful, and thorough. 

 

The authors’ introduction suggests that a second volume is planned that would treat more advanced topics including discrete dynamical systems, constrained and unconstrained optimization, and game theory.

 

Bill Satzer (bsatzer@gmail.com), now retired from 3M Company, spent most of his career as a mathematician working in industry on a variety of applications. He did his PhD work in dynamical systems and celestial mechanics.