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Front page > Course > SIF5034
MA3402 Analysis on Manifolds
Fall 2005
About the course | Lecturer | Textbook and Curriculum | Exam | Lectures | Practice session | Exercises | About this page
| About the course |
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|---|---|
| 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 in Spivak |
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): not updated yet Chap. 4 : Chap. 5 : Chap. 7 : Chap. 8 : Integrasjon (except the proofs of Theorem 10 and 11) Chap 9: Riemannian metrics (page 301-316) Chap. 11: Excursion in the realm of algebraic topology: side 419-425, s.430- 431 . The Exercise sets may also be useful for the final exam, and they give you some experience. 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 : Saturday, December 17 starts at 9.00, room 546 |
Final Grades (6 students, listed in the order of examination): C, B, A, D, B, E |
| Lectures Plans and progress Remarks |
Monday 14.15-16.00 Room R80 (Realfag building) Thursday 8.15-10.00 Room R53 (Realfag building) Week 34 : We start with a brief review of basic material about manifolds and vector bundles. Week 35: Read about the tangent bundle and (the dual) cotangent bundle, and various algebraic constructions (duality, multilinear lagebra etc) needed to construct the various tensor bundles. Week 36 . Study Chapter 4 and the Supplementary notes. The basic objects are the tangent bundle and cotangent bundle. Their sections are called vector fields and covector fields (or 1-forms) respectively. From this one can construct all possible tensorfields by considering sections of the various vector bundles obtained by taking tensor products (and exterior or symmetric tensor products) of the two bundles. Week 37 : After you have got some overview of tensors and tensorfields in chap.4, you should start going through chap. 5 about the connection between vector fields and diff. equations. We shall go through the main themes there, but we shall not go through the existence and uniqueness proofs since they are time consuming. Week 38 : Chapter 5 : vector fields, dynamical systems etc. Week 39 : Chapter 5, the rest. Week 40 : We start with chap. 7 : differential forms Week 43: We complete Chap.7 and start with Chapter 8 :about integration. Week 45: Chapter 8 : Week 46-47 : The remaining topics: from Chap. 8, Chap.11 (Mayer-Vietoris) and a little from Chap.9 (Riemannian metric) Final meeting : The "grill fest" on Tuesday, 13.12, at 14.15 in Room 546 (SBII). |
| Homework/Exercises |
Week 35 :Exercise
set 1 Week 37 : Exercise set 2 Week 39 : Spivak Chap. 5.11a,b, 5.12, 5.13 Week 40 : Exercise set 3 Week 41 : Do the midsemester project. Week 43 : Spivak Chap.7: 13, 14a,b, 27 Week 45-46 : Spivak Chap.8 : 3, 4,7,13,14,17 (see the note "Notes/hints to Chap.8" ,but try yourself as much as possible) Week 47 : Exercise set 5. This is an exercise concerning Riemannian metric |
| Office hours | Until further notice : any time |
| About this page |
The address of this page is
http://www.math.ntnu.no/~eldars/MA3402eng05.htm
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Ansvarlig for innhold: Informasjonsdirektøren, NTNU Teknisk ansvarlig: Webmaster Oppdatert: 20. aug. 1999
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