We all use math every day;
to predict weather, to tell time, to handle money.
Math is more than formulas or equations;
it's logic, it's rationality,
it's using your mind to solve the biggest mysteries we know.
— Numb3rs
This course is about the representation of physical, economic or biological phenomena by mathematical "models" to better understand the phenomena and to make predictions regarding them. It is about the processes we go through to develop models, and to understand the models we have derived. Modeling is thus the quintessential aspect of applied mathematics.
This class will begin with a survey of what modeling is and a discussion of the modeling process. A short review of the mathematical background to be used in the course will then be conducted. After that point, the course will be structured as a number of modules. In each module, the basic principles will be presented and then models will be derived from them. Some of the models will be used to predict physical behavior; others will be used as part of a design process. For example, in the mechanics section, Newton's laws form the fundamental concepts. An example of a simple mechanical model will be to design a bathroom scale. An attempt will also be made to look at modeling from the point of view of "how do things work?" For example, in the section on electronics - after already having done mechanics - we will see how a capacitance microphone works.
I. Introduction
i. What is Modeling? Why Do Modeling?
ii. The Modeling Process
iii. Dimensional Analysis, scales and scaling
II. Some Mathematical Background/Review
A. Systems of ODE's
B. Dynamical Systems and the Phase Plane
i. Preditor-Prey
II. Warfare Strategy
C. Other Solution Techniques
III. Modules
A. Mechanics
i. Basic Concepts: Newton's Laws, Hooke's Law
ii. Models: Grandfather Clock, Fish Scale, Automobile suspension system,...
B. Thermodynamics
i. Mathematical Description of Heat Transfer
ii. Models: Concrete block vs Frame construction, Multiglazed windows, Wine cellar placement...
C. Electronics
i. Basic Concepts: Kirchoff's Laws
ii. Models: Speaker crossover network, Filters, Condenser mike,...
D. Biological Models
i. Basic Concepts: Growth and Decay
ii. Models: Plant growth, pollution cleanup, competing species,
Why are there no large mammalian predators in Australia? ...
E. Finance
i. Basic Concepts: Present/future value, growth (and decay, sigh...)
ii. Models: Investments, bond pricing, retirement plans,...
F. Traffic Flow
i. Basic Concepts: Wave propagation, hyperbolic equation, shock waves
ii. Models: Single lane highway/tunnels, traffic light, crashes,...