CDS 202, Winter 2009
CDS 202. Geometry of Nonlinear Systems. 9 units (3-0-6); second term. Prerequisites: CDS 201 or AM 125 a. Basic diﬀerential geometry, oriented toward applications in control and dynamical systems. Topics include smooth manifolds and mappings, tangent and normal bundles. Vector ﬁelds and ﬂows. Distributions and Frobeniuss theorem. Matrix Lie groups and Lie algebras. Exterior diﬀerential forms, Stokes theorem.
|Week||Lec 1||Lec 2||Topic||Reading||Homework|
|1||N/A||8 Jan||Course introduction||None||None|
|2||13 Jan||15 Jan||Point set topology||MTA Ch 1||HW #1 (solns)|
|3||20 Jan||22 Jan||Inverse fcn thm, immersions, submersions||MTA Ch 2||HW #2 (solns)|
|4||27 Jan||29 Jan||Manifolds, mappings, tangent space||MTA Ch 3||HW #3 (solns)|
|5||3 Feb||5 Feb||Tangent bundle, vector fields||MTA Ch 4||HW #4 (solns)|
|6||10 Feb||12 Feb||Distributions, Frobenius theorem||MTA Ch 4||HW #5 (solns)|
|7||17 Feb||19 Feb||Lie groups and Lie algebras||MTA Ch 5||HW #6 (solns)|
|8||24 Feb||26 Feb||Tensor fields||MTA Ch 6||HW #7 (solns)|
|9||3 Mar||5 Mar||Differential forms||MTA Ch 7||HW #8 (solns)|
|10||10 Mar||12 Mar||Integration on manifolds||MTA Ch 8||None|
The primary course text is the third edition of Manifolds, Tensor Analysis, and Applications:
- Marsden, Ratiu and Abraham, Manifolds, Tensor Analysis, and Applications (if you are registered for the course, send e-mail to Richard Murray [murray@cds] for the password).
In addition, students may find the following textbooks useful:
- Boothby, An Introduction to Differential Manifolds and Riemannian Geometry, Revised second edition, 2002.
The ﬁnal grade will be based on homework and a ﬁnal exam.
- Homework: 75%
- There will be 8 one-week problem sets, due in class. Late homework will not be accepted without prior permission from the instructor.
- Final exam: 25%
- The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided.
If your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.
No collaboration is allowed on the ﬁnal exam.