2009 February 13 / jj|d|a|v|i|s|@|c|a|r|l|e|t|o|n|.|e|d|u

Senior Seminar: Manifolds

Carleton College Math 395, Winter 2009, Prof. Joshua R. Davis


The first goal of this course is Stokes' theorem, which generalizes the fundamental theorem of calculus to manifolds of arbitrary dimension. Applications of Stokes' theorem can be found throughout mathematics and physics; I have even seen the theorem in geology papers. But what is truly impressive about this theorem is its elegance. The proof is not difficult, and the end result is the short but powerful formula

Of course, such a formula in isolation is meaningless; I have not stated the hypotheses or defined any of the terms, because the definitions are subtle and abstract. Indeed, Stokes' theorem is a good first example of the "big machinery" approach to mathematics, in which definitions are difficult, proofs are easy, and results seem natural and simple. Our primary text for this material is Calculus on Manifolds by Michael Spivak.

After we learn Stokes' theorem, we will investigate other topics as the students desire: more concepts in differential topology, applications to physics, connections to complex analysis or algebra, etc. If the students' interests vary greatly, then we may split into individual or group projects.


We meet in CMC 328 during period 5A (MW 1:50PM-3:00PM, F 2:20PM-3:20PM). Here's how you get in contact with me:

Dr. Joshua R. Davis (call me Josh if you like)
E-mail: jj|d|a|v|i|s|@|c|a|r|l|e|t|o|n|.|e|d|u
Office: CMC 327, x4482
Office hours: Sun 12-1, Tue 10-11, Wed 3-4, and Thu 3-4. You can also make an appointment; simply pick a free time from my weekly schedule and e-mail me. You can also talk to me after class.

This course is unique, and there are few students, so the usual statistical approach to grading is inappropriate. Students will be graded individually, based on the following criteria.

Around the middle of the term (and anytime that you ask for it) I will give you a progress report on your grade.


Please complete the Introductory Survey by Monday 11:59 PM.

M 1/0501forward, prefaceintroduction
W 1/07021.1, 1.2Euclidean space
F 1/09031.3, 2.1differentiabilityprove Chain Rule nicely
M 1/12042.2chain rule1.10, 2.1
W 1/14052.3partial derivatives
F 1/16062.4continuous differentiability
M 1/19072.5inverse function theorem
W 1/21082.6manifolds
F 1/23093.1, 3.2integration
M 1/26103.4Fubini's theorem3.28, 3.35
W 1/28113.3, 3.5integrability
F 1/30123.6partition of unity
M 2/02133.6change of variables3.28, 3.35
W 2/04143.6
F 2/06154.1Sard's theorem
W 2/11164.1multilinear algebra
F 2/13174.14.3, 4.5, 4.6, 4.11
M 2/09--4.1
W 2/11164.1
F 2/13174.1
M 2/16184.2differential forms
W 2/18194.2
F 2/20204.2homework
M 2/23214.2exterior differentiation
W 2/25224.2Poincare lemma
F 2/27234.3singular n-cubes
M 3/02244.4fundamental theorem
W 3/04255.1, 5.2manifolds
F 3/06265.3Stokes' theorem
M 3/02275.4, 5.5classical Stokes' theorems
W 3/0428de Rham cohomology