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 ) of dierential forms on M that is closed under exterior d
Lecture 3. Integral Elements and the Cartan-Khler Theorem
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The lecture notes for this section will mostly be denitions, some basic examples, and exercises. In
particular, I will not attempt to give the proofs of the various theorems that I state. The ful
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, as opposed to
dimension 3, as was discussed in Lecture 5
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 < n < N (which I will assume from now on), the set Gn (V
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 always the case, in which case
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other methods must be
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-Khler Theorem.
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5.1. Weingarten surfaces
Let x : N R3 be
Lecture 4. The Cartan-Khler Theorem: Ideas in the Proof
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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 a very special kind. You are probably familiar with th