MATH 134
CALCULUS ON MANIFOLDS
- Class times and location: 11 MWF location 310 Science Center
- E-mail: denne [at] math [dot] harvard [dot] edu (usual substitutions)
- Office Hours: Monday 2-3pm, Tuesday 3-4pm, Friday 1-2pm, also by
appointment.
- Course Webpage:
http://www.math.harvard.edu/~denne/teach/math134.html
- Class Assistant: Sam Lichtenstein
- email: sflicht@fas.harvard.edu
- Sam's Office Hour: Tuesdays 9pm in Adams Dining Hall.
- Section time and location: Sundays 7.30-8.30pm SC 411.
- Final Exam:
CLICK HERE for the .pdf file.
This exam is due 4.30pm January 12th, 2006. Either handed to me in my
office or put in an envelope and put in my (snail) mailbox near SC 325.
Electronic versions may also be emailed to me. Please only send .pdf files.
I will email an acknowledgement.
- Office hours: I'll be available for consultation during reading
week. Times to be announced.
- Midterm exam 2:
- When: 4-6pm, Thursday 13th December.
- Where: Science Center 411.
- Material covered: Everything up to and including the Poincare Lemma.
- What is the exam like? There will be 5-7 questions to do. Some will
have parts to them. The style and level of the questions will be similar to
the homework questions (certainly no more difficult). Make sure you know
the definitions and main theorems from class and how to use them. I plan to
give you an exam you can do in 1 hour, so that time pressure should not be
a concern.
- Midterm exam 1:
- When: 6-8pm, Thursday 26th October.
- Where: Science Center 411.
- Material covered: Class material through Monday 23rd October. This is
roughly Boothby Ch 2 sect 1-7, Ch 3 sect 1-5, Ch 4 sect 1-4, plus any
additional material given as class notes.
- What is the exam like? There will be 5-7 questions to do. Some will
have parts to them. The style and level of the questions will be similar to
the homework questions (certainly no more difficult). Make sure you know
the definitions and main theorems from class and how to use them. I plan to
give you an exam you can do in 1 hour, so that time pressure should not be
a concern.
- Study guide: click here for
.pdf file (last updated 10/23/06).
- Class handouts:
- Extra Problems: Click here for
.pdf file (last updated 10/17/06).
These are extra practice problems for you to do in addition to the homework problems. They are also useful for studying for the exams.
- Extra credit (optional) Assignment: Due Monday 19th December, 2006. Click here for
.pdf file.
This assignment leads you step-by-step through the proof of the existence and uniqueness theorem (of solutions to first order differential equations). Solutions can be found here.
- Homework Assignments:
- Summary:
Generalization of multivariable calculus to the setting of manifolds
in real n-space, as used in the study of global analysis and
geometry. Differentiable mappings of linear spaces, the inverse and
implicit function theorems, differential forms, integration on
manifolds, the general version of Stokes's theorem, integral geometry,
applications.
- Prerequisite: Mathematics 21a,b and familiarity with proofs as in
Mathematics 101, 112, 121, or the equivalent.
- Texts: An Introduction to Differentiable Manifolds and
Riemannian Geometry by W.M. Boothby. Either the revised second edition or the second edition can be used. Warning, this text is advanced!
Other excellent sources include:
- Calculus on Manifolds by M. Spivak;
- Differential Topology by V. Guillemin and A. Pollack:
- Geometrical methods of mathematical physics by B.F. Schutz;
- The geometry of physics: an introduction by T. Frankel;
- Riemannian Geometry by M.P. Do Carmo, we'll use this more at the end of
semester.
All these texts are found on reserve in the Cabot library. Some of them are also in the Birkoff library on the third floor of the Mathematics Department.
- Examination policy: 2 midterm exams and 1 take-home final exam.
- Grading policy: worksheets 1/5; midterms 2/5; final
exam 2/5.
- Homework policy: weekly worksheets.
- The assignments and the
dates they are due will be posted on the course website.
- Homework should be turned in the course mailbox (outside 325 Science
Center) by noon on the day that the assignment is due.
- Late homework will be accepted only in special circumstances and
only with prior approval.
- It is OK to discuss the problems amongst yourselves. However each
student must hand in their own solutions that they have written
themselves. (Copying someone else's homework is unacceptable.)
To make the job of grading easier, could
you please follow the following guidelines for homework:
- Write your name on your HW.
- Neat, legible handwriting. We will not grade anything we cannot read!!!
- Write on one side of the paper only.
- The problems should be in the order assigned.
- Staple (or paper-clip) all pages together.
- Attendance: Attendance will not be taken at each
class.
However, it is much harder to learn the material on your
own, so you are strongly encouraged to attend each class.
You must attend each of the 2 midterm exams and final
exam. Make-up exams will only be given in exceptional circumstances.
- Drop Date: The drop date for the course is Monday October 16.
- Final exam date: Take-home final.
- Syllabus (last modified 11/03/06):
Start with your notes from class and use the references for further study.
(B refers to Boothby, G&P to Guillemin & Pollack.)
- Examples of manifolds, definition of a differentiable and
topological manifold. Class notes + B Ch 3 sect 1.
- Maps between manifolds: diffeomorphisms, immmersions, submersions B Ch 3
sect 2, 3, 4 and B Ch 2 sect 1,2.
- Inverse Function Theorem, Rank Theorem. Spivak Ch 2, B Ch 2 sect 6,7.
- Regular neighborhoods, solutions of constraint equations and Implicit Function Theorems. Spivak Ch 2, B Ch 3 sect 4,5 also G&P Ch 1 sect 1-4.
- Derivations, tangent vectors, tangent spaces. B Ch 2 sect 4 and
Ch 4 sect 1.
- Maps on Tangent spaces. B Ch 4 sect 1.
- Vector fields. B Ch 4 sect 2.
- Flows, integral curves and the existence and uniqueness theorem for
ODEs. Class notes + B Ch 4 sections 2, 3, 4.
- Midterm 1 During week of 23-27 October.
- Review of dual vector spaces. Class notes.
- Covectors, cotangent spaces. B Ch 5 sect 1.
- Tensors B Ch 5 sect 5 p193-196, sect 6 199-201. Spivak Ch 4.
- Transformation law for tensors. Class notes.
- Alternating tensions, wedge product, exterior algebra. B Ch5 sect 5 p.196-198, sect 6 p. 202-206. G&P Ch 4 Section 2. Spivak Ch 4.
- Orientation and volume forms. B Ch 5 sect 7 p. 207-211. G&P Ch 4 sect 3.
- Partitions of unity and Riemannian metrics. B Ch 5 sect 4 p. 186-189.
- Riemannian metric and relationship between the tangent and
cotangent bundles. Class notes.
- Note that Spivak Ch 4 and 5 is an excellent reference for much of
the material in all of these sections.
- Exterior Derivative. B Ch5 sect 8 p. 212-216. G&P Ch 4 sect 5.
- Integration of forms on manifolds. B Ch 6 sect 1,2 p.223-227, p.229-235. G&P Ch 4 sect 4.
- Manifolds with boundary. B Ch 6 sect 4 p.244-249.
- Stokes' Theorem. B Ch 6 sect 5. G&P Ch 6 sect 5 p. 251-257.
- Poincare Lemma. Spivak Ch 5 is best and class notes. (B Ch 6 sect 7.)
- Midterm 2 During week of December 4-8.
- Extra topics including:
- Lie derivatives and Lie algebras. Hand outs and B Ch 4
sect 7.
- Cartan's formula. B Ch 4 sect 8, B Ch 5 sect 8.
|