MATH 457- Calculus on Manifolds

2012-2013 Fall Semester

 

Instructor       : Mustafa Korkmaz

Office              : M135

Lecture           : Tuesday 13:40 — 15:30, Thursday 11:40 —12:30 at M106

 

 

Textbook

“Calculus on Manifolds” by Michael Spivak.

 

References

“Differential Topology” by Victor Guillemin& Allan Pollack.

“A Geometric Aproach to Differential Forms” by David Bachman, available on internet at http://front.math.ucdavis.edu/math.GT/0306194

 

Tentative Outline of the Course

  1. Review of differentiation (The differential of a map between Euclidean spaces, the directional derivative, Jacobian matrix). Review of Inverse and Implicit Function Theorems (Formulations only). (Chapters 1 and 2) (2 weeks).
  2. Tangent vectors and tangent space at a point of Rn .  Vector fields and 1-forms on  Rn. The differential of a scalar valued function on Rn as a 1-form.  Pull- back of 1-forms. Differential of a mapping between Euclidean spaces as a linear mapping between tangent spaces. Tensor product, alternating forms. Wedge product. Orientation on vector spaces. Volume element. Differential k-forms on Rn. Pull-back of differential k-forms. Exterior differentiation of differential forms. Statement of the Poincaré Lemma (Chapter 4 up to "Geometric Preliminaries section") (4 weeks). 
  3.  Review of integration on Rn. Chains, integration on chains. Stokes' theorem (Chapter 3 and the part of Chapter 4 starting with "Geometric Preliminaries" section.) (3 weeks).
  4.  Differentiable Manifolds. Tangent vectors and tangent space at a point of a manifold. Vector fields and 1-forms on a manifold. Differential k-forms on a manifold. Pull-back of differential k-forms on manifolds. Exterior differentiation of differential forms on manifolds. Orientation of manifolds. Manifolds with boundary, induced orientation. (Chapter 5, up to the section about Stokes' Theorem) (3 weeks). 
  5. Integration on manifolds. Stokes' Theorem for differential forms on manifolds.  (Chapter 5, "Stokes' Theorem" section) (2 weeks)

 

Grading and attendance

There will be two midterm exams each of which will affect the total grade by 30%  and a final exam of  weight 40%. Students are required to attend at least 70% of lectures.

In order to be eligible to enter the final exam and a grade other than NA: 1) you must satisfy the attendance requirement, and 2) one of the grades of midterm exams must be above 10.

 

Exam Dates

Midterm 1:  8 November 2012, Thursday, 11:40 at M106

Midterm 2:  20 December 2012, Thursday, 11:40 at M106

Final Exam: To be determined later by the Registrar Office.