MAT257 Analysis II, University of Toronto 2011-12
 

MATH 257 Analysis II 2011-12

Where to find information: All course information, documents, etc (announcements, handouts, ...) will be posted on Blackboard. Some fraction of this material (such as this syllabus and the course schedule) will also be posted on the instructor's web pages.

instructor: Robert Jerrard
office: 1001B, 215 Huron (on 10th floor - take elevator to 9th floor then walk up one flight of stairs).
web: www.math.utoronto.ca/rjerrard/ .
phone: (416) 946-5441 .
e-mail: rjerrard at math dot toronto dot edu. I try to reply to student email within 1 business day. I prefer to discuss complicated questions in person.
department mailing address: Toronto, Ontario M5S 2E4, CANADA.
office hours: T F 4-5 pm and by appointment. (I will normally be able to continue office hours to 5:30 if people are there, but 215 Huron is locked after 5pm, so you need to show up before then.)

Course schedule
Teaching assistants:
Tutorial: R 4-6pm, SS 1074.

Text:   Calculus on Manifolds, by M. Spivak. Westview Press, ISBN 0-8053-9021-9, plus selected additional material.
A good supplemental reference is Analysis on Manifolds by J. Munkres. This covers mostly the same material as Spivak, but with much more detail and motivation.

Course content:
  1. Topology of R^n: Metrics and norms, compactness, continuous functions, extreme value theorem.
  2. Differentiation: inverse and implicit function theorems, Taylor expansion, maxima and minima, Lagrange multipliers.
  3. Riemann Integration in R^n: Integrable functions, Fubini's theorem, partitions of unity, change of variables.
  4. Differential forms: Poincaré lemma, Surface integrals. Vector fields: Gradient, divergence, and curl.
  5. Manifolds in R^n: integration on manifolds; Stokes' theorem for differential forms and its classical versions in R^2 and R^3.
Prerequisites: MAT157Y1, MAT240H1
Co-requisite: MAT247H1
Evaluation:
  1% : Class participation
15% : 10 hand-in homework sets (drop two)
39% : 3 mid-term tests
45% : Final examination (3 hours) Homework assignments will be collected at the end of the lecture on the dates on which they are due.
Remarks.
  • Our goal is to understand the entire book, fill in the details, and master the exercises by the end of the year. We will at times take a slightly more general point of view than the book (which focuses exclusively on R^n), and also expand on examples and applications.

  • The weekly tutorial offers an opportunity to share insight, ask additional questions, look at more examples, and work on problems. You are encouraged to discuss lectures, readings, and homework problems with each other and with us, and you may consult other sources. However, you should write up your assignments yourself, in your own words, and be ready to defend them!

  • Exams will be closed-book, closed notes.
  • Attendance in lectures and tutorials is expected. However, the Class participation mark will be evaluated rather generously; in order to receive a poor score, it is necessary to exhibit a consistent pattern of apparent lack of interest and involvement in the course.

  • The teaching staff will use Blackboard to send out announcements by email. Since these messages will go to your U of T email account, you must check that account regularly.

  • Suitable accomodations will be made for students who miss a test or fail to turn in an assignment for a valid reason, as long as the reason is documented by a Uof T Student Medical Certificate, a Student Health or Disability Related Certificate, a College Registrar's Letter, or an Accessibility Services Letter. This documentation should be presented to the instructor within one week of your return to class.