# MA 430 Differential Geometry

### Syllabus

## Office Hours

Monday 3-4, Thursday 4-5, Friday 2-4 (no appointment necessary). Feel free to make an appointment if you cannot come to my regular office hours.

## Textbook

**No textbook is required.** Handouts with new material and practice problems will be distributed for each teaching unit. The textbooks used for the class preparation include the following:

**Richard Milman, George Parker**, Elements of Differential Geometry

**Richard L. Faber**, Differential Geometry and Relativity Theory

## Topics Covered

1. Curves: parametrization, tangent vector, arc length, acceleration vector, curvature, normal and binormal vector, torsion, Frenet-Serret apparatus2. Surfaces: tangent plane, curvature, Theorema Egregium.

3. Surfaces.: coordinate patches, the First Fundamental Form.

4. Surfaces: the Second Fundamental Form, the Gauss curvature, geodesics, curvature tensor, manifolds.

## Grading

Exam 1 | 26% |

Exam 2 | 26% |

Exam 3 | 26% |

Project | 22% |

TOTAL | 100% |

A+ | A | A- | B+ | B | B- | C+ | C | C- | D+ | D | D- | F | |

grade | 97-100 | 93-96 | 90-92 | 87-89 | 83-86 | 80-82 | 77-79 | 73-76 | 70-72 | 67-69 | 63-66 | 60-62 | 0-59 |

## Prerequisites

Calculus 3 or permission of instructor.

## Attendance

It is imperative that students attend all classes. Students are responsible for all material covered in class, even if attendance is not checked or assignments collected.## Exams and Projects

There will be**three exams**and

**one project**during the semester. Project or exams turned in after their due date will receive an automatic reduction in grade. No exam grade will be dropped.

## More on the course topics

- The course will start with a review of Multivariable Calculus. The course can be considered a continuation of Calculus 3 course and the next step in deepening the students understanding of calculus and building students' problem solving skills.
- Then, the study of multivariable calculus will morph into the study of differential geometry - a mathematical discipline that uses methods of multivariable calculus to study geometrical features, such as shape and curvature, of objects. The curvature measures the extend of bending of a curve, a surface, a space or their generalizations to any dimension, the manifolds. Studying ways of describing such an extent of bending is one of the central ideas of the course and enables one to understand concepts like the expansion rate of the universe.
- Differential Geometry is used in natural sciences, especially in physics and computational chemistry.
- The course provides the students interested in continuing their education at a graduate level with mathematical techniques that certain graduate programs use.

## Course Objectives

- Identify situations that require the use of vector calculus and differential geometry.
- Solve certain classes of problems related to vector calculus, differential geometry or topology.
- Understand and write mathematical proofs using formal mathematical reasoning.
- Present solutions on computer or in a written form.

## Learning outcomes

Students will:- acquire knowledge of various mathematical concepts required for successful application of mathematics in other disciplines,
- be able to identify and solve problems that require the use of vector calculus and differential geometry,
- know how to use formal mathematical reasoning and write mathematical proofs when necessary,
- demonstrate ability to research and cover a topic independently and to present their findings in a form of a written report.