# CURVES AND SURFACES - 2023/4

Module code: MAT2047

## Module Overview

The module has three parts. The first part is the study of plane curves in 2D and space curves in 3D and their properties. The second part develops the definition of surfaces in 3D and their properties. The third part is the study of curves such as geodesics within surfaces in 3D.

### Module provider

Mathematics & Physics

### Module Leader

WOLF Martin (Maths & Phys)

### Number of Credits: 15

### ECTS Credits: 7.5

### Framework: FHEQ Level 5

### Module cap (Maximum number of students): N/A

## Overall student workload

Independent Learning Hours: 106

Lecture Hours: 22

Captured Content: 22

## Module Availability

Semester 1

## Prerequisites / Co-requisites

MAT1005 Vector Calculus and MAT1034 Linear Algebra

## Module content

The module introduces the concepts of elementary differential geometry in the context of curves and surfaces in Euclidean space. In particular, the module consists of two main parts:

- Curves: parametrised curves and arc-length parametrisation; length of a curve; Frenet curves; curvature and torsion; fundamental theorem of curves
- Surfaces: parametrised surfaces; first and second fundamental forms; area of a surface; Gauss map; Weingarten map; curves on surfaces and geodesics; curvature; minimal surfaces; Gauss equation; Codazzi-Meinardi equation; Theorema Egregium; fundamental theorem of surfaces.

Examples from various applied areas of mathematics and other sciences are used for illustration.

## Assessment pattern

Assessment type | Unit of assessment | Weighting |
---|---|---|

School-timetabled exam/test | In-semester test (50 minutes) | 20 |

Examination | Exam (2 hours) | 80 |

## Alternative Assessment

N/A

## Assessment Strategy

The assessment strategy is designed to provide students with the opportunity to demonstrate:

- Understanding of and ability to interpret and manipulate mathematical statements;
- Subject knowledge through the recall of key postulates, theorems, and their proofs;
- Ability to apply subject knowledge to solve problems similar to the ones covered in the lectures.

The summative assessment for this module consists of:

- One one-hour in-semester test worth 20% of the module mark;
- One two-hour end-of-semester examination worth 80% of the module mark.

Formative assessment and feedback:

Students receive written feedback via a number of marked formative coursework assignments over the entire duration of the module. Students are also encouraged to arrange meetings with the module convenor for questions and verbal feedback on the weekly comprehension of the material.

## Module aims

- This module aims to:

1. Introduce the students to the geometry of curves and surfaces;

2. Enable students to understand the foundations and basic tools of differential geometry;

3. Illustrate standard applications in both pure and applied mathematics.

## Learning outcomes

Attributes Developed | ||

001 | Have a firm understanding of the concepts, theorems, and techniques of the geometric properties of curves and surfaces; | KC |

002 | Have a clear understanding as how to apply the mathematical techniques to concrete examples; | KT |

003 | Be able to analyse geometric problems and compute relevant quantities explicitly. | KCT |

Attributes Developed

**C** - Cognitive/analytical

**K** - Subject knowledge

**T** - Transferable skills

**P** - Professional/Practical skills

## Methods of Teaching / Learning

The learning and teaching strategy is designed to provide:

- A detailed introduction to the relevant theory and its tenets, and to the appropriate mathematical tools for their implementation;
- Experience (through demonstration) of the methods used to interpret, understand, and solve concrete problems.

The learning and teaching methods include:

- Teaching consists of captured content, tutorials, and lectures.
- A number of marked formative coursework assignments over 11 weeks with feedback to the students.
- Additional tutorial sessions and/or office hours may be arranged as and when needed.

A complete set of self-contained notes will be provided in advance to any topics to be treated.

Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.

## Reading list

https://readinglists.surrey.ac.uk

Upon accessing the reading list, please search for the module using the module code: **MAT2047**

## Programmes this module appears in

Programme | Semester | Classification | Qualifying conditions |
---|---|---|---|

Mathematics with Statistics MMath | 1 | Optional | A weighted aggregate mark of 40% is required to pass the module |

Mathematics with Statistics BSc (Hons) | 1 | Optional | A weighted aggregate mark of 40% is required to pass the module |

Mathematics BSc (Hons) | 1 | Optional | A weighted aggregate mark of 40% is required to pass the module |

Mathematics with Music BSc (Hons) | 1 | Optional | A weighted aggregate mark of 40% is required to pass the module |

Financial Mathematics BSc (Hons) | 1 | Optional | A weighted aggregate mark of 40% is required to pass the module |

Mathematics MMath | 1 | Optional | A weighted aggregate mark of 40% is required to pass the module |

Please note that the information detailed within this record is accurate at the time of publishing and may be subject to change. This record contains information for the most up to date version of the programme / module for the 2023/4 academic year.