| Date |
Lecture # |
Topics Covered |
| 08/26/2008 |
1 |
|
| 08/28/2008 |
2 |
| Vectors and Vector Fields in $\mathbb{R}^n$ |
| Inner Products and Norms |
| Tensors in a Vector Space |
|
| 09/02/2008 |
3 |
| Alternating Tensors |
| Wedge Product |
|
|
| 09/04/2008 |
4 |
| Volume and Orientation |
| Inner Product of Forms |
|
| 09/09/2008 |
|
|
| 09/11/2008 |
5 |
| Inner Product of Forms (Continued) |
| Hodge Star |
|
| 09/16/2008 |
6 |
| Differential Forms in $\mathbb{R}^n$ |
| The Exterior Derivative |
| Relationship to Vector Calculus |
|
| 09/18/2008 |
7 |
| Basic Point-Set Topology |
| Manifolds |
| Smooth Functions |
|
| 09/23/2008 |
8 |
| Smooth Functions (Continued) |
| Tangent Spaces |
| Derivative of a Function |
|
| 09/25/2008 |
9 |
| Vector Bundles |
| Differential Forms on a Manifold |
| Exterior Calculus on a Manifold |
|
| 09/30/2008 |
10 |
| Orientable Riemannian Manifolds |
| The Codifferential and Laplacian on Forms |
|
| 10/02/2008 |
11 |
| Orientation of Ordered Bases |
| Manifolds with Boundary |
| Integration on Manifolds |
|
| 10/03/2008 |
12 |
| (MAKE-UP CLASS for 09/09/2008) |
| Simplicial Complex |
| Group Theory |
|
| 10/07/2008 |
13 |
| Group Theory (Continued) |
| The Group of Oriented p-Chains |
|
| 10/09/2008 |
|
|
| 10/14/2008 |
14 |
|
| 10/16/2008 |
15 |
| Chains |
| Boundary Operator for Chains |
|
| 10/21/2008 |
16 |
|
| 10/23/2008 |
17 |
| Orientable Manifold Complex |
| Representing Chains on a Computer |
|
| 10/28/2008 |
18 |
| Cochains, Cocycles, and Coboundaries |
|
| 10/30/2008 |
19 |
| Relative Chains and Cochains |
|
| 11/04/2008 |
20 |
| Discretization of Differential Forms |
|
| 11/06/2008 |
21 |
| Interpolation of Cochains |
| Properties of Whitney Forms (Part 1) |
|
| 11/11/2008 |
22 |
| More on Barycentric Coordinates |
| Properties of Whitney Forms (Part 2) |
|
| 11/13/2008 |
23 |
| Inclusions, Restrictions, and Pullbacks |
| Properties of Whitney Forms (Part 3) |
|
| 11/18/2008 |
24 |
| Properties of Whitney Forms (Part 4) |
|
| 11/20/2008 |
25 |
| Properties of Whitney Forms (Part 5) |
|
| 11/20/2008 |
26 |
| EXTRA CLASS (Room 2405, 3:00 p.m. to 4:15 p.m.) |
| Computing Whitney Forms |
| Consequences of $RW = id_{C(K; \R)}$ |
|
| 11/25/2008 |
|
|
| 11/27/2008 |
|
|
| 12/02/2008 |
27 |
| Linear Operators on Normed Spaced |
|
| 12/04/2008 |
28 |
| Riesz Representation Theorem |
|
| 12/08/2008 |
29 |
| EXTRA CLASS (Room 4102, 11:00 a.m. to 12:15 p.m.) |
| Coercive and Positive Definite Bilinear Forms |
| Variational Formulation |
|
| 12/09/2008 |
30 |
| Examples of Variational Formulation |
| Finite Element Method |
| Computing Inner Product of Whitney Forms |
| Matrices for Discrete Codifferential and Laplacian |
|
| 12/17/2008 |
|
| Final Exam (2 p.m. to 5 p.m.) |
|