Overview

A mathematical model, as it is understood here, consists of the assembled equations that describe a physical system or process. The act of setting up these equations and, by extension, solving them, is referred to as mathematical modeling. It is practiced with uncommon frequency in all physical and engineering sciences, including biology and medicine, and beyond those areas in economics and finance. Modeling has even reached into the so-called soft disciplines such as the social and behavioral sciences. Its aim in all these subject areas is to quantify a system in mathematical language and to provide quantitative answers to underlying problems. Modeling can thus be viewed, at its most elementary level, as a problem-solving tool. Although it also has much more profound uses, such as analysis and prediction, and although we shall also give analysis its due, our principal concern here will be with the elementary problem-solving aspects of modeling.

Students who have struggled with problem solving going back to their high school days will recall the uneasiness, even dread, upon being confronted with physical problems, particularly those that fall into the category of "word problems." They will recall, in particular, the difficulty of getting started, of overcoming that first hurdle of defining underlying principles and relations that will serve as building blocks for the model. This process is often an art, which requires physical insight as well as an understanding of physical principles, and the skill, usually attributed to engineers, of making suitable simplifying assumptions.

Being at least...