A close examination of the complex and varied world around us reveals a high degree of underlying order. Our goal as scientists is to understand and explain this order and we do this most precisely using the language of mathematics. Indeed, the degree to which the universe lends itself to a mathematical description is remarkable. The goal of this advanced program is to develop the mathematical language needed to describe and create physical models of our world. To that end, we will examine a number of key physical theories and systematically develop the mathematical tools that we need to understand them.
We will begin, in fall quarter, with a detailed study of classical mechanics - the mathematical description of the clockwork universe envisioned by Newton and others who followed him. We will focus initially on linear approximations for which analytical solutions are possible. The mathematical methods we will learn for this purpose include differential equations, vector calculus and linear algebra. In winter quarter we will move beyond linear approximations and study non-linear systems and chaos and the implications of these ideas for the determinism implied by classical mechanics. We will also extend the Newtonian synthesis to the realm of the very fast and very massive by considering Einstein's theories of special and general relativity. Mathematical topics associated with these ideas include the geometry of spacetime, tensor calculus and variational calculus.