Timeline for What should be taught in a 1st course on smooth manifolds?
Current License: CC BY-SA 2.5
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| Mar 25, 2011 at 17:38 | history | edited | John Sidles | CC BY-SA 2.5 |
Spivak's new book
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| Mar 21, 2011 at 17:23 | comment | added | John Sidles | On further consideration of Deane Yang's (excellent) question "What would be the goal", a very useful final two weeks of of lectures might survey the topic "Some origins of metric and symplectic structures in mathematics, science, and engineering." And yet, an entire course surely could be devoted to this topic alone. | |
| Mar 21, 2011 at 17:06 | comment | added | John Sidles | What would be the goal? Hmmmm ... that would depend upon the class. For a class of engineers and/or scientists, I would suggest ... hmmm ... thermostatic flow s(Liouville's Theorem), with applications in synthetic biology (because that's where the jobs are). For mathematicians, maybe ... hmmm .... de Rham cohomology? Definitely, the pedagogic challenge here is not too few good options for continued study, but rather, far too many of them. | |
| Mar 21, 2011 at 16:55 | comment | added | Deane Yang | I don't have a problem with a course introducing symplectic geometry via physics. It seems to me that a first course should choose one focused topic and goal and do just enough to achieve the goal. Guillemin and Pollack chose the Gauss-Bonnet theorem and do a beautiful job of staying focused on that. Another possibility is Hamiltonian mechanics. But what would be the goal (analogous to Gauss-Bonnet)? | |
| Mar 21, 2011 at 16:38 | history | answered | John Sidles | CC BY-SA 2.5 |