Differential Geometry
TuTh 11:00am-12:15pm in MSB303
Syllabus
Ralph
M. Kaufmann
Office: MSB M312 Phone: (860)
486-3850 e-mail: kaufmann@math.uconn.edu
Office hours: TBA and by
appointment
Course
description
The basic idea and of differential
geometry is to say something about the geometry of an object by moving
a little bit on this object - for instance moving along a curve or on a
surface. Turning this approach into a questions it reads: what kind of
information can I get about my curve or my surface if I move a little
bit on them? It turns out that there is indeed a lot one can learn.
For a curve, one gets tangent directions, curvature and other geometric
information in this way. For a surface, there are 2-d generalizations
of these concepts.
One
striking fact is that knowing this information everywhere allows you
for instance to discover that the earth is not flat. Furthermore it
allows to explain why there cannot be any maps of the earth which give
the right distances and angles at the same time. These types of
considerations are also the basis for the theory of general relativity.
In this course, we will treat curves and surfaces from the above
perspectives which lead us to the results discussed above. We will
provide a classical treatment, but the results and concepts have
applications in discretized versions for computer imaging and methods
of finite elements.
I will try to encorporate computer animations if possible. If all goes
well we should also have the time to discuss
special and a little bit of general relativity at the end of the course.
News: Sample final is
up: Look at it, we will discuss it on Th.
Sample midterm
Sample final
Homework
|
Chapter/Section
|
Numbers
|
Due
|
HW0 (optional)
|
1-4
Re-read e.g. Steward Multivariable calculus 5e Chapter 13 and do some
of the exercises
1-2
|
1b
1,4,5
|
|
HW1
|
1-3
1-5
|
4,6
1, 10 a,b)
12*
|
Sep 19
|
HW2
|
2-2 |
1,2,4,10,16
|
Oct 17
|
HW3
|
2-3
|
3, 4, 6, 11
|
Nov 7
|
HW4
|
2-4
|
3, 9, 10, 11,
13 (optional)
|
Nov 14
|
HW5
|
2-5
2-6
|
3, 9, 10, 11
2, 5 (harder)
|
Nov 30
|
Assignment 1
|
Prepare a ca 5-10min
presentation on "Your Favorite Plane Curve"** |
|
Sep 28
|
Assignment 2
|
Write an overview of the maping
techniques/functions used for the earth. *** |
|
Oct 26
|
*means optional
**The basic idea is that you write down the equations for a curve, give
its geometric properties, relate them to the equation and maybe give
history and/or applications.
Check the famous curve index for a starting point. You can also pick an
example from the book or come up with something on your own.
Other references can be found in the library or at Mathworld and
Wikipedia.
If you would like, you can work in groups of up to 2 people (you do not
have to).
***You can for instance check the first pages of any good atlas for a
start.
Supplemental material: Linear Algebra, Differentiable Maps,
Jacobians and the Chain Rule
Background
Information on Topology (Advanced
material which gives an introduction to topology)
