# Module Information

Module Identifier
MA30510
Module Title
Differential Geometry of Curves and Surfaces
2021/2022
Co-ordinator
Semester
Semester 1
Pre-Requisite
Other Staff

#### Assessment

Due to Covid-19 students should refer to the module Blackboard pages for assessment details

Assessment Type Assessment length / details Proportion
Semester Exam 2 Hours   (Written Examination)  100%
Supplementary Exam 2 Hours   (Written Examination)  100%

### Learning Outcomes

On successful completion of this module students should be able to:

1. Calculate the curvature and torsion of a space curve, demonstrate and use them to determine the shape of the curve.

2. Give definitions of the various different types of curvature associated to a surface and show how to compute them.

3. Demonstrate the first and second fundamental forms of a surface, how to compute them, and how they suffice to determine the local shape of the surface.

4. Appreciate the distinction between intrinsic and extrinsic aspects of surface geometry.

### Brief description

Differential geometry is a branch of mathematics that uses calculus to study the geometric properties of curves and surfaces. Differential geometry provides the mathematical framework of many advanced theories in physics ranging from general relativity to soft matter (e.g. polymers, colloids and liquid crystals).

This module provides an introduction to differential geometry of curves and surfaces from both its local and global aspects. The presentation of the material will emphasize basic geometrical facts and requires a background in vector calculus.

### Content

Review of basic concepts (vector calculus, linear transformations, Green’s theorem).

Local theory of curves – (Arc Length, The Frenet-Serret Apparatus).

Global Theory of Curves.

Local theory of surfaces (area element, minimal surfaces, mean and Gaussian curvature, the first and second fundamental forms).

Global theory of surfaces (The Gauss-Bonnet theorem)

Problem classes (4 practical sessions) to help students answer questions in the problem sheets.

### Module Skills

Skills Type Skills details
Application of Number Inherent in module.
Communication The written exams will require students to clearly explain their working.
Improving own Learning and Performance Problem sheets will allow students to assess their progress.
Information Technology Use of computer software to help solve problem sheets.
Personal Development and Career planning Familiarity with numerical methods for solving differential equations is a valuable career skill, particularly for those going on to work in computer graphics and in industry.
Problem solving Inherent in module.
Research skills Inherent in module.
Subject Specific Skills Inherent in module.
Team work Students will be encouraged to work in groups for the practical sessions.

### Notes

This module is at CQFW Level 6