(suitable for scientists and engineers)

Spring 2011 MATH 481, Section C13

WWW:
https://faculty.math.illinois.edu/~kapovich/481-10/481-10.html

The final exam has been graded and the results have been posted via ScoreReports. The high score was 48/48, the low score was 31.5/48, the median score was 44/48, the mean score was 42.6/48, with the standard deviation of 10.2%. I have also assigned the final course grades. If you are interested in your course grade, please ask me by e-mail and I will e-mail you your course grade. On Monday, May 9, I will hold office hours from 10am to 11:30am. You can come take a look at your final exam then and discuss your course grade as well.

The second midterm has been graded and the results have been posted via Score Reports. Here are the solutions to Midterm 2. Although formal letter grades are not assigned for the midterms, you may approximately interpret the results as follows: A: 24-28; B: 19-23.75; C: 14-18.75. The median score was 24.5/28, the mean score was 24.1/28, with the standard deviation of 14.6%.

The first midterm has been graded and the results have been posted via Score Reports. Solutions are posted below. I do not assign formal letter grades for midterms. However, you may approximately interpret the scores as follows: A: 26-29; B: 22-25.75; C: 17-21.75. For what it's worth, here is some statistical data: the median score was 25/29, the mean score was 25.25/29, with the standard deviation of 8.3%

Instructor: Ilya Kapovich

Telephone:

__e-mail:__ kapovich@math.uiuc.edu. (Preferred
method
of reaching me!)

__
Office location__: Altgeld Hall, room 365

Office hours (preliminary, subject to change):

Grader:
Jiajie Xiao, xiao2@illinois.edu

Text:
Required: The Geometry of Physics, An Introduction,
T. Frankel,
Cambridge U.P. 1997 (paperback)

Recommended: Tensor Analysis on Manifolds, R.
Bishop and S. Goldberg,
Dover (paperback)

Prerequisites: Multivariable calculus -required; some course in linear algebra - recommended

**Brief course description.
**

- Manifolds: configuration spaces, differentiable manifolds, tangent spaces, tangent bundles, orientability.
- Calculus on manifolds: Vector fields, flows, tensor fields.
- Differential forms and exterior calculus.
- Integration theory: Generalized Stokes theorem, de Rham cohomology.
- Riemannian geometry: Riemannian metrics, geodesics.

Some course materials:

- A handout on the definition and examples of manifolds
- A handout on checking the definition of a manifold and on orientability
- A handout on the implicit function theorem
- Solutions to h/wk 1
- A handout on tangent vectors
- Solutions to h/wk 2
- A handout on: diffeomorphisms and homeomorphisms; flow lines of a vector field; tangent map
- Solutions to h/wk 3

- A handout on dual space and differential of a function and on tensors
- Tensors on a vector space
- Solutions to h/wk 4
- Handout on Riemannian metrics
- Solutions to h/wk 5
- Solutions to Midterm 1
- Handout on differential forms, the pullback of a differential form and on singular r-chains
- Solutions to h/wk 6
- Solutions to h/wk 7
- Solutions to h/wk 8
- Handout on orientability and the volume form
- Solutions to h/wk 9
- Solutions to h/wk 10

- Solutions to Midterm 2
- Handout on the Stokes' Theorem for manifold and on its classic analogs
- Solutions to h/wk 11
- Handout on Riemannian connections and geodesics
- Solutions to h/wk 12
- Solutions to the h/wk 13
- Solutions to extra credit problems

NOTE: The correct due date for each h/work is the one shown in this web page, NOT the one shown in the pdf file of a particular h/wk (the reason is that I am reusing h/wks from previous years when the semester calendar was different).

Homeworks:

- Assignment 1, Due Friday, January 28
- Assignment 2,
Due Friday, February 4

- Assignment 3, Due Monday, February 14 (deadline extended since the assignment was posted pretty late)
- Assignment 4, Due Friday, February 18
- Assignment 5, Due Friday, February 25
- Assignment 6, Due Friday, March 4
- Assignment 7, Due Friday, March 11
- Assignment 8, Due Friday, March 18
- Assignment 9, due Friday, April 1
- Assignment 10, due Friday, April 8
- No homework is due on Friday, April 15
- Assignment 11, due Friday, April 22
- Assignment
12, due Friday, April 29

- Optional Extra Credit problems, due Monday, May 2.
- Assignment
13, for information purposes only (will not be collected and graded)

Grading system:

- Weekly worksheet assignments [1/5 of the grade], two one-hour tests [1/5 of the grade for each test], final exam [2/5 of the grade]. A small number of extra credit problems may be assigned upon request.
- The first mideterm is scheduled for Mo, Feb 28; the
second midterm is tentatively scheduled for Mo, April 11

- Final Exam: 8:00-1:00 am, Friday, May 6