One of the goals of scientific inquiry is to understand the processes of nature on a quantitative basis. In pursuit of this goal, mathematicians create models to represent the order they observe, and in turn devise mathematical methods for interpreting and solving these models. This program will provide a thorough and engaging introduction to such mathematical methods and the associated techniques of model building. Differential equations will be an important component of the program. We will study both the derivation of these equations from physical and biological models and their solution using analytical, qualitative and computational methods. In addition, we will cover linear algebra and multivariable calculus and their various applications in physics and economics. In winter quarter we will consider non-linear systems and their role in cyclical, chaotic and self-organizing behavior. There will also be an introduction to the calculus of variations with applications to finding optimal curves and surfaces. In addition to the theoretical work, we will also discuss questions of a more philosophical and historical nature. Is mathematics discovered or created? What role do mathematical models play in representing reality, and who were the people behind the important developments in calculus.
Students will attend weekly lectures, workshops, seminars and computer labs and will be expected to give two oral presentations each quarter and write one research paper.
Fall quarter topics include include ordinary differential equations, multivariable calculus, linear algebra, and computer modeling, philosophy of math. Winter quarter topics include partial differential equations, vector calculus calculus of variations, nonlinear dynamics, history of mathematics.
Up to 28 of the 32 credits may be awarded as upper-division science credit, but this is contingent on upper-division performance.