Description & Syllabus :
This course builds on MTH164 to describe calculus on curves, and surfaces. It provides useful tools for theoretical physics (in particular relativity theory) and the theory of hydrodynamics. It is also a useful basis for computer graphics. The foundation of differential geometry is the concept of curvature. The course will focus on understanding this and related concepts very clearly, both geometrically and computationally, for the case of surfaces in Euclidean space . For this, you'll need a solid background in multivariable calculus and linear algebra. We hope to give some idea of how curvature is understood in higher dimensions; this is the basis of Riemannian geometry and General Relativity.
Grading: Notes: Notes about Differential Forms and Vector Fields by Rob Hladky
The following is the outine of the course.References are to O'Neill. Supplementary material will only be covered if time permits.
The main portion of this course develops the mathematical machinary required to do calculus on surfaces and describe the geometric objects from both the extrinsic viewpoint (looking at the object from outside) and the intrinsic (the object is the universe you live within).
Chapter 1: Calculus in Euclidean space. The tangent bundle and vector fields. The cotangent bundle and differential forms. Recasting old M164 results in a robust mathematical language.
Chapter 2: Frames , curves and connections. Differentiating vector fields.
Chapter 4: Surfaces and calculus on surfaces. Stokes, Greens and Divergence theorems unified as one result. Some topology.
Chapter 5: The shape of space: introduction to curvature - the extrinsic viewpoint. Geodesics.
Chapter 6: Geometry of Surfaces in Euclidean Space : Intrinsic Viewpoint
Homework: As always in a mathematics class, doing the homework is an essential part of learning the course material.
It is often good to study together with other students. You can form groups and hand in just one homework for the group.
However, the number of students in a group can't be more than 2.
The homework assignments are listed below. They are due at the beginning of class on their due date.
Submissions consisting of multiple pages must be stapled together. If you cannot get to class, hand them in to my office (Hylan Building 820) before class.
Late homework will be penalized severely (by at least 25%), and will not be accepted if it is too much overdue.
The grader has the final say on all homework grades.
Week
Topics
Homework
Reading
08/31
1.1Euclidean Space
1.2Tangent Vectors
1.3Directional Derivatives
1.1 #4
1.2 #3(d)(e),#4,#5
1.3 #1,#2
Due 09/16
1.1, 1.2, 1.3
09/07
1.4 Curves in R^3
1.51-Forms
1.6Differential Forms
No Homework
1.4, 1.5, 1.6,
Notes about Differential Forms
09/14
1.6 Differential Forms
Notes Wedge Product
1.7 Mappings
1.4 #4,#6,#7
1.5 #6(b),#9,#10,#11
1.6 #7,#8
Due 09/25 1.6, 1.7
Notes about Differential Forms
09/21
Notes Pushforward and Pullback
2.1 Inner Product
2.2 Curves
No Homework
2.1, 2.2
Notes about Differential Forms
09/28
2.3 The Frenet Formula
2.4 Arbitrary-Speed Curves
1.7 #6,#8,#10
2.1 #8,#12
2.2 #5,#6
Due 10/07 2.3, 2.4, 2.5
10/05
2.5 Covariant Derivatives
2.6 Frame Fields
No Homework
2.5, 2.6
10/12
2.7 Connection Forms
2.8 The Structural Equations
2.3 #2,#7,#8
2.4 #1a,c,#9, #12,#17
2.5 #3,#4,#5
Due 10/21 2.7, 2.8
10/19
2.8 The Structural Equations
4.1 Surfaces in R^3
No Homework
2.8, 4.1
10/26
4.1 Surfaces in R^3 cont.
4.2 Patch Computations
Midterm 1
4.1, 4.2
11/02
4.3 Differentiable Functions and Tangent Vectors
4.4 Differential Forms on a Surface
No Homework
4.3, 4.4
11/09
4.5 Mappings of Surfaces
5.1 The Shape Operator of M in R^3
4.1 #4,#10,#11
4.1 #3,#5a-c
4.3 #4, #11a-b, #12
Due 11/18 4.5, 5.1
Midterm 30%
Final Exam 40%
Homework 30%