Phys/Eng Students Math group

Current subject: Introduction to manifolds
Current room/time: Wendt 129, Wednesdays 3-5 pm.

Meeting minutes

1. 6/9/10. Discussed the formal definition of "directional derivatives", the identification of vectors with "derivations at a point", vector fields which assign vectors to an entire open subset ("derivations"), germs of functions, linear maps, dual space, and the dual basis.
2. 6/16/10. More basics of finite dimensional vector spaces (Ch.3 of Tu). Discussed covectors, multilinear functions, k-tensors, permutations, alternating and symmetric k-tensors. Introduced the tensor product and representation of tensor product as a matrix.
3. 6/23/10. Review of past sessions, and review of hoardes of definitions on finite dimensional vector spaces.... multilinearity, k-tensors (e.g., covectors), k-tensor properties (symmetry, antisymmetry), permutations, tensor product. Also introduced the famous "wedge product" (or infamous), anticommutativity of wedge product.
4. 7/7/10. Anticommutativity and associativity of wedge product, and the basis set for alternating k-linear functions. Discussed hyperplanes and wedges of odd k-tensors with themselves being zero.
5. 7/21/10. Problems from chapter 3.
6. 9/22/10. Started chapter 4.
7. 10/20/10. More chapter 4, introduced the exterior derivative.

Some study materials

1. Introduction to manifolds by Loring Tu. This book is rather concise, elementary, and so it makes for a good starter book. On the downside it doesn't motivate the abstract algebra very well and is not always very intuitive. Table of contents has a good rough "traveller's guide". I will upload this later.
2. A professor in Australia named Mike Alder has some brilliant, wonderful notes online, covering a lot of different math. You can find sources at his webpage here . The link under "Geometric topology" (about half way down) is a very good, wordy introduction into general topology. The link under "Introduction to differential geometry" is somewhat more advanced but also probably very helpful, haven't looked at it yet.
3. Wikipedia entries for vector space, dual space, and linear map are interesting and useful. Good information on fundamentals.
4. The book Foundation of mechanics by Abraham and Marsden is supposed to be a classic, and the first chapter is available online. More advanced than Tu or Adler. The beginning has a concise introduction to topology and abstract math, but it is very dense and probably impossible to get through if you haven't looked at a more casual treatment first (like say Adler, above.)