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On Wednesday 16.05 we shall start reviewing the early parts of Spivak. I am open for suggestions; but one possibility is tangent vectors, vector bundles and flows, + a little more about foliations. The lecture on Friday 18.05 is cancelled! The revewing continues on Wednesday 23.05 and Friday 25.05.
Wednesday 09.05 we covered the rest of Spivak Ch VIII, and shall not introduce any new material.On Friday 11.05 , we shall solve Ma 252-exam '99, problem nr.2, and go on reviewing integral manifolds and foliations.
In the lecture on Wednesday 02.04 we covered Ch.VIII Spivak up to 13. Theorem on p. 277. On Friday 04.05 we shall solve the Ma 252 problems '99 nr. 3 and '00 nr. 2 (related to the notion of degree), and also '97, nr.2 if we have time. In the next 2 weeks, we shall finish Ch. VIII, and maybe give a brief outline of Ch.XI. Then we shall review the material we have covered.
On Friday 27.04 we shall solve the MA 152 exam. problems '97, nr.1 and '99 nr.1.
In today's lecture, we covered Stokes' Theorem, both for manifolds with boundaries and for singular chains. Next week, we shall go on with the material on pp.255-56 and 264-67, and then continue the study of De Rham Cohomology. On Friday, we shall solve the exam problems for Ma 252, '01 nr.3 and '02 nr. 4, Ch.III nr. 17, and perhaps Ch.VII nr.4.
The lecture tomorrow, Friday 30.03, is cancelled. The lectures in the week after Easter, Wednesday 11.04 and Friday 13.04, are given as usual. We shall round off Ch.VII, Differential forms,and start on Ch.VIII, Integration, preparing for the proof of the general Stokes' Theorem. On Friday, we shall review manifolds with boundary, and solve problems II.14, and III.15, and maybe also III.17. Relevant exam problems: '02: nr.2 and 4, and '01:nr. 3.
On Wednesday 21.03, we covered differential forms and their differential (=exterior derivative), roughly pp. 207-213 in Spivak. On Friday 23.03, we'll do the MA 252 exam problems: Fall 2002 nr.2 and Fall 2001 nr.1. We shall comment on Ch.IV, problems 1d, 2, 4, and 6. Next Wednesday, 28.03, we shall continue with the Poincare' lemma for the differential.
On Wednesday 14.03, we finished our discussion of Foliations, Ch. VI, and covered the Multilinear algebra part of Ch. VII,roughly pp. 201-207. On Friday 16.03, we solved Problems 4, 5, and 11 of Ch. V.On Wednesday 21.03,we shall introduce Differential forms and their exterior derivative (or differential.) On Friday 23.03, we shall solve problems, trying to follow wishes of the audience.Some relevant MA 252 exam problems: (i):Fall 2002, nr.2. ( f from M to N is submersive at the points where rank(df)=dimN ). (ii): Fall 2001, nr.1 and Fall 2000, nr. 1. They are similar, and are related to Lie derivatives.
Last week, we covered most of Ch. V, and shall finish this material on Wednesday 07-03. Then, we shall start a quick, cursory treatment of Ch. VI, Integral Manifolds.Suggestion of new problems:Ch. IV, nr.1d, 2 (Obs. misprint: R instead of N),4, and 6. Ch.V, nr.4, 5, 11, 13, and 15. We have then covered a lot of material already, and what remains are the two most substansial chapters, VII and VIII, + a few applications from Ch. XI.Hence,we may take some time reviewing the material we have covered and solving problems, and I invite your suggestions on the best way of doing this.
Tomorrow, Wednesday 21/2, we quickly shall round off the chapter on Tensors, and then discuss material related to Vector Fields and Differential Equations , p. 135-148 in Spivak. In the last message I forgot and wrote in Norwegian. I recommended the problems Ch.III, nr.5,7,12-14,16,19,22, and 23 (Difficult?):
I morgen, fredag 16/2, regner med ? bli ferdig med det jeg skal gjennomg? fra kapittel IV, Tensors.Noen aktuelle oppgaver fra kap. III: nr. 5, 7, 12-14, 16, 19, 22, og 23(vanskelig?)Neste uke begynner vi gjennomgang av kap.V , og skal ogs? gjennomg? noen oppgaver. Vi er kommet s?pass langt allerede, at vi n? kan gi oss litt bedre tid.
On Friday 9/2, we shall finish the discussion of orientation,and solve/comment on the problems: Ch.I, nr. 3, 4a-c,13 and 15, and Ch.II nr.7 and 11,12. (Shall also say something about manifolds with boundary.)
Last week we finished Ch.II, except the notion of manifold with boundary. We also introduced tangent spaces, the general concept of vector bundle, and the tangent bundle of a smooth manifold. Tomorrow (7/2-'07), we'll hopefully cover the rest of Ch.III.
So far, we have covered material corresponding to (most of) Spivak: Ch.I, p.1-15 and Ch.II, p.27-45 and 50-52. Tomorrow (26-01) we shall finish differential calculus, explain immersions and imbeddings(p.46-49 and 52-53), and present Steiner's Roman surface (p. 15-17 and Ch.II, problem 29), as an exemple. Some relevant problems: Ch.I, nr.2-4,13,15, and Ch.II, nr.7,9,10-12, and32.( More demanding problems?: Ch.II, nr.14,15,17,and 21 ?) Some of the problems will be solved at lectures or informal meetings. Then we shall continue with the tangent- and other vector bundles (Ch.III).
Fikk nettopp melding om at Spivak's bok vil v?re tilgjengelig p? Akademika imorgen (fredag 19/1) etter kl. 13.
F?rste forelesning blir onsdag 17/1 kl. 14:15-16 i rom B 62 i Abels hus. Etter r?d fra siste ?rs foreleser har vi skiftet l?rebok til M. Spivak: "A Comprehensive Introduction to Differential Geometry, Vol. 1", som forventes ? v?re p? plass p? Akademika n?r undervisningen starter. Vi g?r bort fra Barden og Thomas' bok: " An Introduction to Differental Manifolds", som har v?rt brukt de to siste ?rene, fordi noen studenter fant den for knapp,og fordi der er noen forvirrende feil. Den fins imidlertid i bokhandelen, og kan v?re nyttig som st?ttelitteratur.