Calculus on Manifolds

Calculus on Manifolds

Math 4B03/6B03, Winter 2012
Dr. Ben Mares

$$\int_M d\omega = \int_{\partial M} \omega.$$

(Credit: Abstruse Goose)


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  • Some of you may be interested in the undergraduate summer school at Notre Dame. See the link at the bottom of the page.
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  • Office hours: Tu Th 1-2pm, or by appointment.

Final evaluation


This is suggested reading for the upcoming lectures. Be warned that I sometimes change my mind, but I will do my best to keep this up-to-date.

Tuesday, Apr 3

Topics: Cohomology of $S^n$ using Mayer-Vietoris, Outline of Mayer-Vietoris, closing perspectives

References: M&T Chapters 4&5



The big picture

Where have we been?
  • Intro (M&T Chapter 1)
  • Tensor algebra and tensor calculus (Hitchin Chapter 2, Section 5)
  • Alternating algebra (M&T Chapter 2, excluding characteristic polynomials (2.16))
  • de Rham cohomology and Poincaré lemma (M&T Chapter 3)
  • Homotopy (M&T Chapter 6, only 6.7, 6.8, 6.10)
  • Smooth manifolds (M&T Chapter 8)
  • Differential forms on manifolds (M&T Chapter 9, excluding 9.16 and after).
  • Integration on manifolds (M&T Chapter 10)
  • Chain complexes and cohomology (M&T Chapter 4)
  • Mayer-Vietoris sequence (M&T Chapter 5)
Where are we going?
  • On to do exciting new things!

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