18.101 — Analysis II
Fall 2007
Announcements Syllabus ExercisesHandouts

Announcements


Vedran has offered to hold office hours 4-6 on Friday in 2-312

Last year's final is posted in the handouts section. This years final will cover everything up to and including the Lie derivatives of densities handout. Compared to last year's final, you will need to know more about Lie derivatives of densities and vector fields, and a little more about flows of vector fields on manifolds.

Suggested courses that require 18.101: 18.952: Theory of Differential forms. Instructor: Victor Guillemin The theory of integrating differential forms on manifolds is algebraically nicer than integrating densities. This course will include applications topology and classical mechanics. 18.994: Seminar in Geometry Instructor: Brett Parker. This year, the seminar in Geometry will concentrate on intersection theory and transversality which are important tools in differential topology. 18.755: Introduction to Lie Groups Instructor: Sigur Helgason. If you liked the homework questions on Lie groups, you may be interested in this course.

Chapter 10 of Sternberg-Loomis Advanced Calculus is posted as a suppliment to lectures for integration of densities on manifolds. The full text is availible on Sternberg's website.

Thanks to our grader Vedran, we now have a complete set of solutions to the problem sets posted.

Syllabus

Lectures

    Lectures will take place at 11am MWF in 2-143

Instructor

The instructor for this course is Brett Parker. Office: 2-175, Email: (his last name)@math.mit.edu

Office hours

Office hours will be at 4 on Tuesdays, and 5 on Wednesdays, in 2-175,
Vedran Sohinger will hold office hours on Wednesdays from 1-2 in 2-312.

 Prerequisites

18.100B (Analysis I) and either 18.700 (Linear Algebra) or 18.701 (Algebra I) are the official prerequisites for this course. 18.901 (Introduction to Topology) is helpful but not required.

Textbooks

The required texts are Analysis on Manifolds by James Munkres and Calculus on Manifolds by Michael Spivak.

Exams

There will be three exams in this course: two midterm exams, taken in class; and a final exam administered during finals week.

The first midterm will be in class on October 3

The second midterm will be in class on November 14

The final will be in 56-154 from 1:30 to 4:30 on Monday December 17

Assignments

There will be exercises assigned in class every day and posted to this website; these will not be graded, but the material will appear on the exams. There will also be biweekly problem sets which will be graded.

Grades

Graded assignments will consist of 50% of your final grade; exams will be the other 50%.

Exercises

  • Sep 5: Munkres section 1: 2,3, 4b.  section 3: #2,3,4,7,9. Read Munkres section 1 and 3, and Spivak pp1-13
  • Sep 7: Munkres section 5: #2,3,4,6   section 6: #1.
  • Sep 7 — Homework: Problem Set 1 (due September 21 at 11 am in room 2-108)
  • Sep 10: Munkres section 6: #10  
  • Sep 12: Munkres section 7: #1,2,3  
  • Sep 14: Munkres section 8: #1,2,3,4  
  • Sep 17: Munkres section 9: #1,2  
  • Sep 19: exercises from handout Manifold lecture notes 1
  • Sep 21: exercises from handout Manifold lecture notes 2
  • Sep 21 — Homework: Problem Set 2 (due October 5 at 11 am in room 2-108)
  • Sep 28: exercises from handout Manifold lecture notes 3
  • October 5 Homework: Problem Set 3
  • October 5: First 5 exercises from handout `vector fields and one forms'
  • October 10: All other exercises from `vector fields and one forms' (The exercises on integrating one forms are optional, but you should find them interesting.)
  • October 12: Exercises from Birkhoff Rota A, 2a,2b, 6a, 6b, 14
  • October 15: B.R. D: 1,3,6,7,11
  • October 17: B.R. A: 1, B.R. C: 1,2,3,5
  • October 19 Homework: Problem Set 4 Due Friday November 2 at 11 am.
  • October 22: Read the proof of theorem 3 in Birkoff Rota. We shall be covering Theorem 11 next lecture.
  • October 24: Problems 7-17 in flows of vector fields on manifolds handout
  • October 29: All other problems from the flows of vector fields handout.
  • November 2: Munkres chapter 10 problems 1,2,3
  • November 2 Homework: Problem Set 5 Due Wednesday November 21 at 11 am. Note the different due date! As this problem set is a bit harder than usual, you should start it early.
  • November 5 Munkres chapter 11 problems 1,2,4,5,6
  • November 7 Munkres chapter 11 problems 8,9. Spivak problem 3-14
  • November 9 Munkres chapter 14 problems 1,2,3,4
  • November 16 Munkres chapter 12 problem 4, Spivak 3-31, 32
  • November 19 Prove the properties of the extended integral (Theorem 15.3 in Munkres) without the assumption that the functions involved are continuous. Munkres chapter 15 problems 1,2,4.
  • November 30 Homework: Problem Set 6 Due Friday December 7.
  • November 30 Munkres chapter 17, 1,2,3,4,5,6.
  • December 3. Exercises 1,2,3,4,7 on Densities handout.
  • December 5. Exercise 5 from Densities handout, Exercises 1,2,3,4 from Integrating Densities handout.
  • December 7. Exercises 5,6,7 from Integrating densities handout
  • December 10 Exercises from the handout on Lie derivatives of densities.