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Front page > Course > SIF5034
MA3402 Analysis on Manifolds
Fall 2004
EKSTRA MEETING (last supper): thursday 9/12, at 16.00 - room 446 (4 fl.Matematikk)
(This will be an informal meeting, where you may pose questions )
About the course | Lecturer | Textbook and Curriculum | Exam | Lectures | Practice session | Exercises | About this page
| About the course |
|
|---|---|
| Lecturer | Professor Eldar Straume |
| Textbook: Spivak's book Supplementary notes: 1) Tensor Calculus- a brief overview 2) tensorfields 3) Handwritten lecture notes from Chap. 8. |
1. M.Spivak: A Comprehensive Introduction to Differential Geometry, Volume I , Publish or Perish, Inc, Houston,Texas 1999. 2.Tensor Calculus- a brief overview A somewhat popularized form, which may be useful for you. Most books on differential topology cover more or less our topics in the curriculum, see e.g. the web page of the spring 2004 course TMA4190. In particular, it is useful to have the text book of TMA4190: 3. James R.Munkres : Analysis on manifolds. Kap 6-Kap.9. The book is somewhat expensive and not so easily available. But Chap.6 is particularly nice for our course. 4. Per Holm: Differensialformer og Mangfoldigheter (in Norwegian) Some of the chapters discuss vector bundles, differential forms etc. Curriculum (from Spivak): Kap. 4 :Tensors Kap. 7 : Differential forms Kap. 8 : Integrasjon (except the proofs of Theorem 10 and 11) Kap. 9 : Riemannian metrics (page 301-316) Kap. 11: Excursion in the realm of algebraic topology: side 419-425, s.430- 431 . The Exercise sets are also part of curriculum. In addition, the student need some basic knowledge of differential topology, for example from Spivak Chap.1-3. About manifolds and vector bundles. |
| About
the exam. Time : Tuesday , Dec. 14 (cf. letter by mail) |
Candidates : Bertelsen, Nornes, Probst, Ravnås, Segtnan, Sydnes, Vidal, Lohne |
| Lectures Plans and progress Remarks |
Monday 15.15-17.00 Room F4 Friday 12.15-14.00 Room F4 Week 35 : We start with a brief review of basic material about manifolds and vector bundles. See e.g. Spivak, Chap.1-3 Week 36 : read about vector bundles, various functorial constructions with vector spaces (dual, direct sum, tensor product, hom..) and multilinear algebra. We are now into Chap.4 of Spivak, and will need the mentioned constructions as basic tools. Week 37 : Chap.4 and all thesee tensor bundles and tensor fields. See also the Tensor Calculus note. Week 38 : Still Chap.4 and tensor calculus. Week 39 : general tensor bundles and tensor fields, contraction of tensor fields.See last part of chap.4.Then we continue with chap.7 Week 40 : 1)Study the note tensorfields 2) Read Spivak, Chap.7 (The text on p. 217-219 about "Frobenius integrability theorem " is excluded). Week 41: Work with home project Week 42: Chap. 7, including Poincare lemma, but not Frobenius integrability teorem. Week 43: Chap.7, about differential forms, closed and exact ones, the Poincare lemma etc. Week 44: We start up with Chap.8 : Integration Week 45 : Integration, Stokes theorem (2 versions), cohomology, etc.. Week 46 : Still Chap. 8 , and start looking at Chap.11 (Mayer-Vietoris) Week 47 : Still Chap.8 +Chap.11 : the pages about Mayer-Vietoris Week 48 : End of. 8, + some geometry (from Chap.9 and lecture). Some handwritten lecture pages will be handed out. See also web sides soon available i connection with exercises Spivak Chap.8 : 3, 4, 7, 13, 14, 17. Here is also some background material, in order to understand and solve the problems |
| Homework/exercises : Exercise 1 Exercise set 2 Exercise set 3 mid semester project Exercise set 4 (solutions ) Additional :Exercise set 5 (a simple exercise with Riemannian metric) |
Week 34 :
Investigate the exercise Ex.1 Week 35 : Ex.2 Week 37 : See Ex.1 and Ex.2 p.127 in Spivak. (In Ex. 2 there is an error, N should be the real numbers R). Week 38 : Same as previous week, plus Ex.3 and 4. (find out what is an orientation of a vector bundle) Week 39-40 : Ex. 8, p.131, but first of all Exercise set 3 Week 41: Midsemester home project Week 42: Midsemester project, and the above Exercise set 3. Week 43: Last chance to digest Exercise set 3, and a chance to think more about the problems in the Midsemester project. Week 47-48: Exercise set 4: Spivak Chap.8 : 3, 4, 7, 13, 14, 17, 31 See also the single exercise on Riemannian metric given in Exercise set 5. |
| Office hours | Until further notice : any time |
| About this page |
The address of this page is
http://www.math.ntnu.no/~eldars/MA3402eng.htm |
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Ansvarlig for innhold: Informasjonsdirektøren, NTNU Teknisk ansvarlig: Webmaster Oppdatert: 20. aug. 1999
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