MATH 134 CALCULUS ON MANIFOLDS
- Class times and location: 12 MWF location 310 Science Center
- Office Hours: Mondays 1-2pm, Tuesdays 4-5pm, also by
appointment.
- Course Webpage:
http://www.math.harvard.edu/~denne/teach/math134.html
- Class Assistant: Joshua Reyes
- email: jreyes@fas.harvard.edu
- Joshua's Office Hour: Tues 4pm at 4th floor math lounge
- Section time and location: Wednesday 6pm 111 Science Center
- FINAL EXAM: Click here for
.pdf file. Due 5pm Friday 14th January, 2005.
Last modified/corrected December 23rd 2004.
- Extra Problems: Click here for
.pdf file. - last updated 12/01/04.
- 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: Calculus on Manifolds by M. Spivak; Differential
Topology by V. Guillemin and A. Pollack. Also good, but more advanced
An Introduction to Differentiable Manifolds and Riemannian Geometry by
W.M. Boothby. Another excellent text is Riemannian Geometry by M.P. Do
Carmo, we'll use this more at the end of semester.
- Examination policy: 2 midterm exams and 1 take-home final exam.
- Homework policy: weekly worksheets.
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. I will not grade anything I 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.
- LATE HW: We will not grade late HW, unless you have a good reason (ex illness).
(If you are sick, let me know and we can meet and sort out HW extensions and some extra help so you can catch up on the work.)
If you can't make it to the class when the HW is due, you have several options:
1) Give your HW to a friend to hand in.
2) Put it in Josh's mailbox (before class) in the Math Department
(near 325 Science Center).
3) Put it in my mailbox (before class) in the Math Department (near 325 Science Center).
- Grading policy: worksheets 1/5; midterms 2/5; final
exam 2/5.
- 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 special circumstances.
- Drop Date: The drop date for the course is Monday October 18.
- Final exam date: Take-home final.
- Syllabus (subject to final revision):
Use class notes as the start point to study. I've also included
other references.
- Review of pointset topology. Spivak Ch 1
- Examples of manifolds, definition of a differentiable and
topological manifold. Class notes + Do Carmo Ch 0 section sections 2 and 4.
- Orientation and Jacobian matrix. Class notes + Do Carmo Ch 0 section 4.
- Solutions of constraint equations and the Implicit and Inverse Function
Theorems. Spivak ch 2 + class notes.
- Maps between manifolds: diffeomorphisms, immersions, submersions,
embeddings. Boothby Ch 3 sections 4 and 5.
- Rank Theorem, regular values and Preimage Theorem. Boothby Ch 2
section 7, also G&P Ch 1 sections 3 and 4.
- Tangent vectors, tangent spaces. Boothby Ch 2 section 4 and Ch 4
section 1.
- Maps on Tangent spaces. Boothby Ch 4 Section 1
- Dual space and differential of a function. Class notes and
Boothby Ch 5 Section 1.
- Vector fields and flows. Class notes + Boothby Ch
5 sections 2, 3, 4.
- Tangent Bundle as a vector bundle. Class hand out. (Not examinable.)
- Midterm 1 Wed October 27
- Tensors and exterior algebra. G&P Ch 4 Section 2. Spivak Ch
4. Boothby Ch5 section 5 and 6.
- Differential forms. G&P Ch4 Section 3 Boothby Ch5 section 5 and 6.
- Transformation law for tensors. Class notes.
- Riemannian metric and relationship between the tangent and
cotangent bundles. Class notes.
- Orientation again and volume forms. G&P Ch4 section 3, Boothby Ch5
section 7 (includes partitions of unity Boothby Ch 5 section 4.)
- Note that Spivak Ch 4 and 5 is an excellent reference for much of
the material in all of these sections.
- Integration of forms on manifolds. G&P Ch4 section 4. Boothby Ch6
sections 1 and 2.
- Exterior Derivative. G&P Ch4 section 5, Boothby Ch5 section 8.
- Manifolds with boundary. Boothby Ch 6 section 4.
- Stokes Theorem. G&P Ch4 section 7. Boothby Ch 6 section 5.
- Poincare Lemma. Spivak Ch 5. Class notes. Boothby Ch6 sections 7
and 6.
- Classical Theorems - exercises in HW, G&P. Class notes and Boothby Ch 6 section 5.
- Midterm 2 Friday December 10
- Extra topics including:
- Lie derivatives and Lie algebras. Hand outs and Boothby Ch 4
section 7.
- Frobenius Theorem (briefly) Boothby Ch 4 section 8.
- Covariant Derivatives and Connections and Geodesics. Hand
outs. Boothby Ch 7 sections 1,2,3. Also Do Carmo's Riemannian Geometry.
- Applications to mechanical systems. Handouts.
|