CDS 202, Winter 2009

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This is the homepage for CDS 202 (Geometry of Nonlinear Systems) for Winter 2009.


  • Richard Murray (, 109 Steele

Lectures and course mailing list:

Teaching Assistant:

  • Paul Skerritt

Office hours/recitations:

  • Office hours: Tue 4-5pm, Wed 4-5pm, 110 Steele
  • Problem solving sessions: Wed, 6-7pm, 214 Steele

Course Description

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.

Course Schedule

Week Lec 1 Lec 2 Topic Reading Homework
1 6 Jan N/A Course introduction and scheduling Murray (1994) None
2 8 Jan 13 Jan Point set topology MTA Ch 1 HW #1 (solns)
3 15 Jan 20 Jan Manifolds, maps, tangent spaces MTA Ch 2.3-2.4, 3.1-3.3 HW #2 (solns)
4 22 Jan 27 Jan Immersions, submersions, inverse function theorem MTA Ch 2.5, 3.5 HW #3 (solns)
5 29 Jan 3 Feb Tangent bundle, vector fields, flows MTA Ch 3.3, 4.1-4.2 HW #4 (solns)
6 5 Feb 10 Feb Distributions, Frobenius theorem MTA Ch 4.2, 4.4 HW #5 (solns)
7 12 Feb 17 Feb Lie groups and Lie algebras MTA Ch 5.1-5.2 HW #6 (solns)
8 19 Feb 24 Feb Applications of Lie groups MTA Ch 5.3 + KM94 HW #7 (solns)
9 26 Feb 3 Mar Differential forms MTA Ch 6.1-6.2, 7.1-7.3 HW #8 (solns)
10 5 Mar 10 Mar Integration on manifolds, exterior derivative MTA Ch 7.4-7.5,8.1-8.3 HW #9 (solns)

Course Text

The primary course text is the third edition of Manifolds, Tensor Analysis, and Applications:

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 9 one-week problem sets, due in class one week after they are assigned. 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 (likely N hours over a 4-8N hour period).

The lowest homework score you receive will be dropped in computing your homework average. In addition, 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 Policy

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.