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Each oral exam with start with one of the problems from the list of starting questions. All students are advised to prepare 5 to 10 minute answers to those questions.
I proved the degree formula for a smooth map of connected, closed, oriented n-manifolds, discussed the existence of unit vector fields on spheres, and proved Brouwer's fixed-point theorem. Precise information on the syllabus and the form of the oral exam will be posted by Thursday June 6th.
My sample solutions are now available at http://www.uio.no/studier/emner/matnat/math/MAT4520/v13/oblig/sol.pdf.
There will be no lectures May 20th and 22nd. The final lectures will be Monday May 27th and Wednesday May 29th. The exam dates will be Thursday June 20th and Friday June 21st.
I lectured on integration of n-forms over oriented n-manifolds, and started the proof of Stokes' theorem in this generality.
I proved Stokes' theorem for the integration of an exact k-form over a k-chain in an n-manifold.
I will continue chapter 8, with integration of k-forms along k-chains, leading to Stokes' theorem.
I finished chapter 7 about homotopy invariance of pullbacks, and started chapter 8 with a discussion of line integrals.
The mandatory assignments have been graded. I have sent an email to one student, asking for a revision within two weeks. All the other answers were satisfactory or better. You can pick up your copy in my office (B627), or in class (May 6th).
I discussed closed and exact forms on R^n and R^n - {0}, for n=2 and n=3. Thereafter I talked about smooth homotopies, and led up to the theorem that closed forms are exact on smoothly contractible manifolds.
Chapter 7, problems 1, 2, 3, 5 and 6.
I continued chapter 7, with the wedge product of alternating forms.
Chapter 5, Problems 9, 10, 11, 12 and 13.
I skipped chapter 6, and started chapter 7 by discussing alternating forms on a vector space.
I will finish chapter 5, with the relation between commuting 1-parameter groups and vanishing Lie brackets.
Chapter 5, Problems 1, 2, 3, 4 and 6.
The mandatory assignment is now available at /studier/emner/matnat/math/MAT4520/v13/oblig/oblig.pdf . It is due by April 24th.
I discussed Lie derivatives of functions, cotangent fields and vector fields.
There will be no class on Monday March 25th, Wednesday March 27th or Monday April 1st.
Due to the Abel prize announcement, there will be no class from 12:15 to 13:00. From 13:15 to 14:00 we look at the following exercises: Chapter 4, problems 1, 2 (where N should be R), 3 (where df(v) should be df(p)(v)), and 4.
I discussed the 1-parameter group of diffeomorphisms generated by a (compactly supported) vector field.
I started chapter 5 on vector fields and differential equations.
I characterized covariant tensors of order k by their linearity over the smooth functions, and mentioned contravariant and mixed tensors.