Math 485 Differential Geometry
with Applications to Special Relativity
  Spring 2014
Class time: MW 4:40-5:55 Roop 105
Problem Session: Th 5:15-7:00 Roop 105
Make up class time: F 4:40-5:55 in Roop 105
The syllabus is linked and a (tentative) calendar for Differential Geometry is available as well.  Much of that information is repeated bellow (but not all of it, so read the syllabus!) 


Instructor
Rebecca E. Field
fieldre@math.jmu.edu
(540) 568-4962
Office: Roop 114
Office Hours: Tu 5:00-6:00pm,
  W 6:00-7:00pm,
F 12:00-1:00pm
Also by appointment
(not available on Thursdays)



Textbook
Differential Geometry and Its Applications, by John Oprea (second edition)
available at the campus bookstore


Course summary:
The goal of this offering of Math 485 is to provide an introduction to one of the main things that research mathematicians mean when they say Geometry.  Differential geometry is the study of spaces that live inside R^n . The most general version of this that we
will be talking about is called a manifold which are subsets of R^n that are built out of pieces of R^m for m < n.
Building all of this up rigorously will take some time, so we will run into smaller examples in the form of curves
in R^3 (which are built out of pieces of R) sooner.  We will cover Curves, Surfaces, Curvature, Geodesics, Manifolds and General Relativity.

Here is my most important piece of advise about this course:     DO NOT FALL BEHIND!!  This includes things like DO NOT MISS CLASS!!  (If you must miss a class, get notes from one of your classmates and read them before the next class.)  It also includes things like DO YOUR HOMEWORK!!  It is not possible to actually learn this material without doing problems.  You might be able to convince yourself you understand, but if you can't do problems, you aren't at the level of understanding required to pass the class.  In fact, if the class seems too easy at any point, do extra problems!

Exams:
Big Quiz 1: Monday, February 17, self scheduled noon-7pm in Roop 115
Midterm: week of March 24, evening k
Big Quiz 2: Monday, April 14, during class time
Final: Monday, May 5, 3:30-5:30pm

Homework:
I will try to point out easy problems in the homework. These problems are more by way of checking to make sure you understand the definitions in the section and consequently are very fair game for your weekly quizes (along with definitions/theorems/notation we have covered).  Be absolutely sure you can do all of them blindfolded.  Medium problems can also show up on quizzes and exams.  There will sometimes be Extra Credit problems.

1.1 Introduction                                                               easy 2-6 medium 13 (figure is not to scale!)-16, 25-27 harder 17,22,23
1.2 Arclength Parametrization                                        easy 2 medium 5,6 harder 7
1.3 Frenet Formulas                                                        easy 2 medium 11-13,19 due Feb 7 4pm harder 22-24,28 due Feb 21
1.4 Non-unit Speed Curves                                             medium 4,6 due Feb 21
1.5 Some Implications of Curvature and Torsion           easy 2-4 medium 7,9,13 harder 17 due Feb 21

2.1 Introduction to Surfaces                                            easy 6-8 medium 11,13,19,20 harder 27 due Feb 28
2.2 The Geometry of Surfaces                                        easy 5,6 medium 8,9,11 harder 14,15 due Feb 28
2.3 The Linear Algebra of Surfaces                                medium 4,9,10 harder 12 due March 7
2.4 Normal Curvature                                                     medium 4-9 due March 21  

3.1 Introduction to Curvatures                                        easy 2-5 medium 6 harder 9-11 due March 28
3.2 Calculating Curvatures                                              easy 3 medium 4,5,7,10 harder 13,14,16,18-20 due Tuesday, April 1
3.3 Surfaces of Revolution                                              medium 1,3,4 due April 1
3.4 A Formula for Gauss Curvature                                medium 2,4,6
3.5 Some Effects of Curvatures(s)                                  medium 1 harder 8-10
3.6 Surfaces of Delaunay                                                easy 3 medium 4 harder 5
3.7 Elliptic Functions, Maple and Geometry                  medium 4

5.1 Introduction to Geodesics, Metrics and Isometries
5.2 The Geodesic Equations and the Clairaut Relation
5.4 Surfaces not in R^3

8.1 Introduction to A Glimpse at Higher Dimensions
8.2 Manifolds

Notes:
Spacetime and Special Relativity
Introduction to General Relativity
Geodesics and the Spacetime Manifold

Extra Help:
Please come by my office hours or make an appointment if you need extra help!

Topics we will cover are roughly as follows:

Chapter 1: The Geometry of Curves
Chapter 2: Surfaces
Chapter 3: Curvatures
Chapter 5: Geodesics
Chapter 8: Manifolds
Supplemental Notes: General Relativity