This course is about the essence of science -- understanding the world in which we live. We use mathematics as a language to help us describe and understand our world. Because the purpose of Mathematical Modeling is to "talk about" our world, the most important part of this course are the applications -- our mathematical discussions about real world phenomena. In this first chapter we have looked at the following applications.
Applications of Modeling
These applications along with the other applications we develop in this course are the foundation of our discussion of mathematical modeling. The best way to learn any subject is by using it and the only way to learn mathematical modeling is by building, analyzing, and interpreting models.
Kinds of Mathematical Models
We have looked at four broad kinds of mathematical models.
The rest of this course is divided into four "chapters" -- organized by the mathematics used for these four kinds of models. But these four kinds of models are not disjoint. Many models combine elements from all four chapters.
Three laboratories
We have also discussed three different "realities" or "laboratories" that we use to study our world.
"Real" reality is the basis for all our work because we are interested in our own real world. Gedanken reality -- the models we build in our heads -- is ultimately the only way we can think about our world. The physicists developing quantum mechanics spoke often about gedanken, or thought, experiments. Virtual reality is in one sense nothing more than the same kinds of simulations we have done for years but they are becoming more and more "realistic" as we immerse ourselves in virtual sight, sound, feel, and even smell and taste.
These three realities or laboratories work as a team. For example, aircraft wings are often designed using the equations of gedanken reality and tested first by computer simulation, then in wind tunnels, and only finally by a test pilot in a real aircraft. Virtual reality is an artist's rendering of gedanken reality which in turn is based on our perception of real reality.
Mathematics -- the language of modeling
Like other languages, the essence of mathematics is the way it enables us to express, communicate, and reason about ideas and, especially, ideas about our world. The word "red" in English is important because it describes the color below. Without seeing this color one misses a great deal about the word "red."
Working with real problems requires the full power of mathematics -- the ability to work with symbols, with graphics, and with numerical calculations. For this reason computer algebra systems like MathCad, Maple, Mathematica, and the CAS built into the TI-92 are an integral part of our tool kit. They give us powerful environments for doing mathematics. And together with a browser, like Netscape, some cables, and equipment like the TI-CBL they give us the ability to use the full power of mathematics with real data from the real world.
We have already used the TI-CBL to collect data for the following phenomena.
The most important ingredient in this course -- YOU
Everything we have discussed above -- the content of the course -- the tools and the technology -- would be useless without you. Indeed, without you there would be no purpose. The purpose of mathematical modeling is to enable people like you and me to learn about our world, to form mental pictures of how it works and how we can make it a bit better. Mathematical modeling requires your active participation -- thinking, working with your computer algebra system, with old-fashioned paper and pencil, exploring the world with the TI-CBL, rubber bands, and TinkerToys, and exchanging ideas with friends and colleagues.