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: There will be one midterm exam and a final.
Both exams will be take home-exams of one week duration. There will be weekly written homeworks and daily reading assignments.
The grade breakdown for M255 is as follows: Notes: Notes about Differential Forms and Vector Fields by Rob Hladky
This is the outine of the main portion of both M255 and M256. The approximate split between the two is indicated below, but may shift if timing differs from expected. 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.
Split between M255 and M256.
Chapter 6: Curvature revisited - the instrinic viewpoint. Isometries. Some Global theorems. Orientability.
Chapter 7: Riemannian geometries. Metrics and covariant derivatives. The Gauss-Bonnett theorem.
Chapter 8: Lengths on surfaces - geodesics revisited. Covering spaces. Topology and curvature.
Supplementary material: Higher dimensions and applications - Possibly others.
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
09/01
1.1Euclidean Space
1.2Tangent Vectors
1.3Directional Derivatives
1.1 #4
1.2 #3(d)(e),#4,#5
1.3 #1,#2
Due 09/12
1.1, 1.2, 1.3
09/08
1.4 Curves in R^3
1.51-Forms
1.6Differential Forms
1.4 #4,#6,#7
1.5 #6(b),#9,#10,#11
Due 09/22 1.4, 1.5, 1.6,
Notes about Differential Forms
09/15
1.6 Differential Forms
Notes Wedge Product
1.6 #7,#8
Due 10/03 1.6, 1.7
Notes about Differential Forms
09/22
1.7 Mappings
2.1 Inner Product
1.7 #3,#4,#6,#10
Due 10/03 2.1, 2.2
Notes about Differential Forms
Midterm 25%
Final Exam 35%
Homework 40%