# 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 problems in geometry.
- 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.