Math Math 378.2: Lecture Notes and Homework Assignments
Lecture notes will be posted to this site after each lecture is given
Day 01: 17 February
The definition of exterior differential system (EDS),
elementary examples of PDE as EDS, the EDS for
linear Weingarten surfaces, Jorgen's Theorem via Liouville's Theorem.
Lecture Notes for Day 1
Day 02: 22 February
The Frobenius Theorem and examples: PDE, the Maurer-Cartan equation,
and surface theory in 3-space.
Lecture Notes for Day 2
Day 03: 24 February
Cartan-Kahler Theory: The Grassmannian of tangent n-planes,
integral elements, polar spaces, regularity, the Cartan-Kahler
theorem.
Lecture Notes for Day 3
Day 04: 01 March
Cartan-Kahler Theory II: Examples,
the Cauchy-Kowalewski Theorem, more examples
Lecture Notes for Day 4
Day 05: 03 March
More examples. Cartan's Test for regular flags.
Lecture Notes for Day 5 (Same as lecture notes for Day 4)
Day 06: 08 March
Examples: Harmonic equations on 3-manifolds, orthogonal
coordinates, the existence of local Lie groups.
Lecture Notes for Day 6
Day 07: 10 March
When Cartan-Kahler does not apply: Prolongation and
the Cartan-Kuranishi Prolongation Theorem. Examples.
Lecture Notes for Day 7
Spring Break
Day 08: 22 March
This lecture was mainly taken up with the
computer algebra demonstration of prolongation
for solving the problem of classifying the surfaces
with each principal curvature constant along each
opposite principal curve. There were no separate
lecture notes
Maple File for the cyclides computation/prolongation
Day 09: 24 March
Examples: Orthogonal coordinates in general dimension,
isometric immersion with prescribed mean curvature.
Lecture Notes for Day 9