Week |
Monday |
Wednesday |
Thursday |
Friday |
| Sep 12-16 | Geometry of R^n. Inner product and norm. Handout 1 |
Topology of R^n. Open and closed sets. |
Tutorial. | Compactness. |
| Sep 19-23 | Heine-Borel Theorem.
Handout 2 |
Heine-Borel theorem in R^n. | Tutorial. | Continuous functions.
Assignment 1 due. |
| Sep 26-30 | Continuity and compactness.
Handout 3 |
The derivative. | Tutorial. | Basic properties. |
| Oct 3-7 | Directional derivatives.
Handout 4 |
Partial derivatives.
The Jacobian matrix. | Tutorial. |
The Inverse Function Theorem.
Assignment 2 due. |
| Oct 10-14 | Thanksgiving.
Handout 5 |
The Inverse Function Theorem, cont'd. | Tutorial. | Visualizing f: R^n --> R^m.
Gradients and level sets. |
| Oct 17-21 | The Implicit Function Theorem.
Handout 6 |
The Implicit Fcn Thm and
Submanifolds of R^n. Assignment 3 due. | Tutorial. | Lagrange multipliers. |
| Oct 24-28 | Higher order derivatives.
Handout 7 |
Taylor's Theorem. | Tutorial. | the Riemann integral. |
|
Oct 31 -Nov 4 |
Measure zero and content zero.
Handout 8 |
Exam review. | Exam 1 | Continuity and oscillation. |
| Nov 7-11 | Fall Break
Handout 9 |
Integrable functions. | Tutorial. | Integrable functions cont'd. |
| Nov 14-18 | Fubini's Theorem.
Handout 10 |
Fubini's Theorem cont'd. Assignment 4 due. | Tutorial. | Partitions of unity. |
| Nov 21-25 | Extended integrals.
Handout 11 |
Change of variables, statement. | Tutorial. | Change of variables, proof. |
|
Nov 28 -Dec 2 |
Change of variables, proof.
Handout 12 |
Change of variables, examples. Assignment 5 due. | Tutorial. | Sard's Theorem. |
| Dec 5-9 | k-volume in R^n. Handout 13 |
Integrals over k-surfaces in R^n . |
Study period. | Exams begin. |
| Dec 12 -Jan 6 |
Exam period and Winter break. | Exam period and Winter break. | Exam period and Winter break. | Exam period and Winter break. |
| Jan 9-13 | Sard's Theorem.
Handout 14 |
Coordinate systems for a manifold. | Tutorial. | Integral of scalar functio over a manifold. |
| Jan 16-20 | Preview of Stokes' Theorem.
Handout 15 Assignment 6 due. |
Tensors and tensor products. | Tutorial. | Basic properties of tensors. |
| Jan 23-27 | Alternating tensors. | Alternating tensors cont'd | Exam 2. | Wedge product, Basis for A^k(V) |
|
Jan 30 -Feb 3 |
Some formulas. | Volume form and orientation. | Tutorial. | Vector fields in R^n. |
| Feb 6-10 | Differential forms in R^n.
Assignment 7 due. |
Pullback and differential.
| Tutorial. | Exterior derivative, div and curl in R^3. |
| Feb 13-17 | Closed and exact forms. | The Poincaré lemma. | Tutorial. | The Poincaré lemma, proof. |
| Feb 20-24 | Reading Week. | Reading Week. | Reading Week. | Reading Week. |
|
Feb 27 -Mar 2 |
Singular cubes and chains. Assignment 8 due. | Boundary of a chain, homotopy of chains. | Tutorial. |
Stokes' Theorem on chains. |
| Mar 5-9 | Examples and applications. | More examples and applications. | Tutorial. | Manifolds with boundary. |
| Mar 12-16 | Tangent space to a manifold. Assignment 9 due. | Vector fields and forms on manifolds. | Tutorial. |
Orientation, induced orientation. |
| Mar 19-23 | Stokes' Theorem on Manifolds. | Examples and applications. | Tutorial. Exam 3 | More examples and applications. |
| Mar 26-30 | Volume element revisited. |
Classical theorems of vector calculus. | Tutorial. | Classical theorems cont'd. |
| Apr 2-6 | Additional topics. Assignment 10 due. |
Additional topics. | Tutorial. last day of classes. |
Good Friday |