Differential Geometry

I developed an upper-level differential geometry course for Bard College which I taught on two occasions. This page contains information about this course.

Summary

The course required only multivariable calculus as a prerequisite, and most of the students who took the course were junior or senior math or physics majors. I focused the course heavily on vector geometry and parametrizations of curves, surfaces, and manifolds, while also covering classical differential geometry topics such as curvature of curves and surfaces.

The textbook for the course was Differential Geometry of Curves and Surfaces, by Banchoff and Lovett.

Homework Assignments

Here are the weekly homework assignments for the course, which were worth 40% of the course grade. Students were encouraged to work together on the homework but were required to write up their own solutions individually.

• Homework 1 (Roulettes, Parallel Curves)
• Homework 2 (Integration on Curves, Arc Length, Tractrix)
• Homework 3 (Parametrizations and Curvature in Two Dimensions)
• Homework 4 (More Curvature, Line Integrals, Envelopes, Pedal Curves)
• Homework 5 (Parametric Curves in Three Dimensions)
• Homework 6 (Torsion, Frenet–Serret Formulas)
• Homework 7 (Parametric Surfaces)
• Homework 8 (Manifolds and Higher Dimensions)
• Homework 9 (Tangent Vectors, Critical Points)
• Homework 10 (First Fundamental Form, Hyperbolic Plane)
• Homework 11 (Maps Between Surfaces, Gauss Map, Gaussian Curvature)
• Homework 12 (Quadratic Forms, Principle Curvatures, Darboux Frame)

Exams

The course included three in-class exams, which were worth 60% of the course grade. Here are the exams:

• Exam 1 Practice Problems (and Solutions)
• Exam 1 (and Solutions)
• Exam 2 Practice Problems (and Solutions)
• Exam 2 (and Solutions)
• Final Exam Practice Problems (and Solutions)
• Final Exam (and Solutions)

Outlines

Though the course used a textbook, I also wrote outlines of some of the material we covered:

• Outline – Parametric Curves
• Outline – Curvature
• Outline – Finding Parametric Equations
• Outline – Line Integrals
• Outline – First Fundamental Form
• Outline – Maps Between Surfaces
• Outline – The Gauss Map
• Outline – Quadratic Forms
• Outline – Curvature of Surfaces
• Outline – Curves on Surfaces

Syllabus

Here is the syllabus from the last time I taught the course.

Textbook Information

The textbook is Differential Geometry of Curves and Surfaces, by Banchoff and Lovett. You will need a copy of the textbook for reading and homework problems, though you do not need to bring it to class. The book is currently available for $54 from Amazon.com, or you can rent it for the semester for $16. Amazon also has a Kindle version of the book, which has excellent quality. The Kindle version ought to work on Kindles, PC's, Macs, iPads, and Androids.

Course Description

This course will use methods from vector calculus to study the geometry of curves and surfaces in three dimensions. Topics covered will include curvature and torsion of curves, geometry of surfaces, geodesics, spherical and hyperbolic geometry, minimal surfaces, Gaussian curvature, and the Gauss-Bonnet theorem. Time permitting, we may also discuss applications to subjects such as cartography and navigation, shapes of soap bubbles, computer graphics, image processing, and general relativity.

The prerequisite is Math 241 (Vector Calculus). Though we will occasionally see proofs in class, Proofs & Fundamentals (Math 261) is not required.

Homework, Exams, and Grading

Homework

There will be a weekly homework assignment due on Friday evening. You are encouraged to work together on the homework, but you should write up your own solutions individually, and you must acknowledge any collaborators.

Exams and Grading

The grade will be based on the weekly homework and three exams:

The exams are two hours long, and are tentatively scheduled for the following dates:

• Exam 1: Friday, October 3
• Exam 2: Friday, November 7
• Final Exam: Thursday, December 18