Math 549: Differentiable Manifolds I
University of Illinois at Chicago
|Boy's surface: An immersion of the projective plane into R3.
J. M. Lee, Introduction to smooth manifolds, 2ed, Springer GTM, 2012.
(Free ebook access through UIC library)
W. M. Boothby, An Introduction to
Differentiable Manifolds and Riemannian
Geometry, 2ed, Academic Press, 1986.
F. W. Warner, Foundations of Differentiable Manifolds and
Lie Groups, Springer GTM, 1983.
MWF 9:00am in Taft 305
Optional make-up lecture: Dec 11, 9:00am, in Taft 305
Monday 2-3 in SEO 503
Wednesday 11-12 in MSLC (SEO 430)
or by appointment
About the final exam
The final exam will be held on Tuesday December 12, 10:30am-12:30pm
in Taft Hall 305
(our usual classroom).
The exam will have a total of six problems, of which you will be required to complete any four.
The exam is cumulative. There will be a slight emphasis on material covered after the midterm.
The exam problems will be shorter than most of the homework problems, but similar in overall difficulty.
Calendar & Grading
This is a condensed summary of some important dates and the weights used to compute your course grade. The course syllabus is the definitive reference for course policies.
||Course grade fraction
||Most Mondays (See list below.)
||Due Mon Oct 16 at 9:00am in class
||Tue Dec 12, 10:30am-12:30pm
Homework should be submitted directly to the mailbox of the grader, Victor Jatoba
, in the third floor mailroom of SEO by 1:00pm
on the date indicated.
Note that the submission procedure for the midterm exam is different; it will be collected by the instructor at the beginning of lecture on October 16.
Typeset solutions are encouraged but not required. If writing solutions by hand, please make sure they are legible. Staple homework if it spans several sheets of paper.
Many of the homework problems are assigned directly from the primary textbook. Some other problems are modified versions of exercises in the primary or secondary textbooks (as indicated by a tag with the author's name) or are adapted from the following sources:
- Additional suggested problems from the main textbook (Lee):
- Chapter 1:
- Chapter 2: 3,9,11; after reading Theorem 2.29, also problem 14
- Chapter 5: 1,4,6,10,19
- Chapter 7: 1,3,6,23
- Chapter 8: 20,24,30
- Chapter 9: 7
- Chapter 11: 1,4,9,10
- Chapter 13: 10, 11, 21
- Chapter 15: 3
- Chapter 16: 4, 5, 11, 13
- Chapter 17: 2
- Chapter 20: 6
- All problems from the textbook that have been assigned so far:
- Chapter 1: 7,9
- Chapter 2: 8
- Chapter 3: 4,5,8
- Chapter 4: 6,10
- Chapter 5: 7
- Chapter 7: 2,4,13,16
- Chapter 8: 3,10,11,13,18,29
- Chapter 9: 2,3,16
- Chapter 11: 7
- Chapter 12: 7
- Chapter 13: 4
- Chapter 14: 5,6
- Chapter 15: 1,5
- Chapter 16: 2,10
- Chapter 17: 1
- Chapter 20: 3,5
- Math 518 (Differentiable Manifolds I), Fall 2014, at UIUC. Instructor: Nathan Dunfield
- The lecture notes on this site are very nice, and are based on our textbook.
- Fundamental theorem of ordinary differential equations
- P. Hartman, Ordinary Differential Equations, Wiley, 1964.
- E. Hille, Lectures on Ordinary Differential Equations, Addison-Wesley, 1969.
- Includes Osgood's criterion (unique solutions with low regularity)
- Calculus by Gilbert Strang, Wellesley-Cambridge Press, 1991. (Homework 10 refers to statements from a multivariable calculus text. This is provided as an example.)
- The History of Stokes' Theorem by Victor Katz, Mathematics Magazine 52(3), 1979.
Some pathologies and counterexamples
- A function with partial derivatives of every order, at every point, but which is not continuous (hence not differentiable).
- Space-filling curves
- Vector fields that are not uniquely integrable sometimes arise naturally in geometry, for example when considering osculating circles of plane curves (or similar osculation constructions). See e.g.
Students who are learning TeX / LaTeX and who want to typeset their homework may find this template helpful: