**Course Description** -- MTHT 442 (Section Numbers 37035 and 37036) -- Fall 2014

**Instructor:** Louis H. Kauffman

**Office: **533 SEO

**Phone: **(312) 996-3066

**E-mail: ** kauffman@uic.edu

**Web page: **http://www.math.uic.edu/~kauffman

**Office Hours**: 3PM to 4PM on Monday, Wednesday, Friday, or by appointment.

**Course Hours**: 1:00PM to 2:00PM on MWF in 300 TH.

This is a course on Differential Geometry and its applications.

The textbook is "Differential Geometry - Curves, Surfaces, Manifolds" by Wolfgang Kuhnel.

A second textbook is "Differential Forms with Applications to the Physical Sciences" by Harley Flanders (Dover Paperback Edition -- see Amazon)

Excerpts from the book "Calculus" by Apostol can be found here: Apostol Curves and Apostol Conics

An article and excercises about the combinatorics and topology of the Gauss Bonnet Theorem can be found here: Gauss Bonnet

A discussion of the Fary-Milnor Theorem can be found here: Fary-Milnor Theorem

Milnor's original paper on curvature of knotted curves can be found here: Milnor

Notes on Inifinitesimal Calculus and Differential Forms by LK are here: Zeroid

Notes on Diagrammatic Matrix Algebra can be found here: Diagrams

Course notes can be found here: Diff Geom Notes 1 Diff Geom Notes 2

**First Assignment:** Obtain a copy of the textbook. Read Chapters 1. and Chapter 2. Read the excerpt from Apostol (see above).
On page 524 of Apostol, do problems 1-8. On page 538 of Apostol, do problems 2,3,4,5,6.
This homework is due on Friday, September 5,2014.

**Second Assignment:** Apostol, page 544, problem 4; page 549, problems 10, 11, 22. Read
Chapter 2 of Kuhnel. Kunhel, page 49, problems 1,10,11.
This homework is due on Monday, September 15,2014.

**Third Assignment:** Read
Chapter 2 of Kuhnel. Kunhel, page 49, problem 1. Do this problem again and find a good solution!
(not to hand in). Kunhel page 49. Problems 9,19,23. And attend to the example on page 31. Show what is
claimed there that "one can solve this system of four equations for a,b alpha, beta."
Read the "Zeroid" Notes on our website.
This homework is due on Monday, September 29,2014.

**Fourth Assignment:** Read Kuhnel Chapter 3 and Chapter 4 through page 146.
Solve the tollowing two problems:
1. Let A and B be vectors in R^{n} with the usual inner product, denoted here by A*B. Show that
(A*A)(B*B) - (A*B)^{2} = Sum_{1<= i < j<= n} (A_{i}B_{j} - A_{j}B_{i}).
2. Using f(u,v) = (u,v, F(u,v)) where F is a diferentiable real-valued function of u and v,
work out the three fundamenatal forms,the principal and Gaussian curvatures for this surface element.
Choose some specific examples and derive their curvatures as special cases of your work.
This homework is due on Monday, November 3, 2014.

**Fifth Assignment:** Read Kuhnel Chapter 3 and Chapter 4 through page 166.
Solve the following problem: Using f(u_1,...,u_n) = (u_1,...,u_n, F(u)) where F is
a differentiable function of u = (u_1,...,u_n),work out the three fundamenatal forms,
the principal and Gaussian curvatures for this surface element.
Choose some specific examples and derive their curvatures as special cases of your work.
(i.e. same problem as before but for hypersurfaces in arbitrary dimensions.)
This homework is due on Wednesday, November 12 2014.

**Sixth Assignment:** Read Flanders Differential Forms -- Section 4.1 and Section 4.5.
Apply Flander's formalism to f(u_1,u_2) = (u_1,u_2, F(u)) where F is
a differentiable function of u = (u_1,u_2). Compare with your previous work using
the three fundamenatal forms. This is for discussion and not to be handed in.
However, please apply our recent formalism for Christoffel symbols and curvature to some
examples of your choosing. We will discuss specifics on Monday, December 1. Please hand
in some worked examples of your choosing on Wednesday, December 3.