Lecture 1. Basic Systems
1.1. What is an exterior differential system?
An exterior dierential system (EDS) is a pair (M, I) where M is a smooth manifold and I (M )
is a graded ideal in the ring (M ) o
Lecture 3. Integral Elements and the Cartan-Khler Theorem
a
The lecture notes for this section will mostly be denitions, some basic examples, and exercises. In
particular, I will not attempt to give t
Lecture 7. Applications 3: Geometric Systems Needing Prolongation
7.1. Orthogonal coordinates in dimension n.
In this example, I take up the question of orthogonal coordinates in general dimensions, a
Lecture 2. Applications 1: Scalar rst order PDE, Lie Groups
2.1. The contact system
For any vector space V of dimension N over R, let Gn (V ) denote the set of n-dimensional subspaces
of V . When 0 <
Lecture 6. Prolongation
Almost all of the previous examples have been carefully chosen so that there will exist regular ags, so
that the Cartan-Khler theorem can be applied. Unfortunately, this is not
Lecture 5. Applications 2: Weingarten Surfaces, etc.
This lecture will consist entirely of examples drawn from geometry, so that you can get some feel for
the variety of applications of the Cartan-Khl
Lecture 4. The Cartan-Khler Theorem: Ideas in the Proof
a
4.1. The Cauchy-Kowalewski Theorem
The basic PDE result that we will need is an existence and uniqueness theorem for initial value problems
of