# CURVES AND SURFACES - 2024/5

Module code: MAT2047

## Module Overview

The module provides an introduction to the theory and properties of curves and surfaces in Euclidean space.

Basic concepts and tools from a branch of mathematics known as differential geometry are used to study curves and surfaces, and to understand their geometric properties.

This module builds on material from MAT1043 Multivariable Calculus and MAT1034 Linear Algebra.

The module is recommended for students intending to select the Year 3 modules MAT3009 Manifolds and Topology, and MAT3044 Riemannian Geometry, although it is not a pre-requisite for either module.

### 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: 91

Lecture Hours: 22

Guided Learning: 15

Captured Content: 22

## Module Availability

Semester 1

## Prerequisites / Co-requisites

None.

## Module content

Indicative content includes:

**Curves:**Parametrised curves. Length of a curves. Reparametrisations. Arc-length parametrisation. Frenet curves. Curvature and torsion. Fundamental theorem of the local theory of curves.**Surfaces:**Parametrised surface patches. Fundamental forms. Area of a surface patch. Gauss and Weingarten maps. Curves on surfaces and geodesics. Curvature. Minimal surfaces. Local normal forms. Gauss-Weingarten equations. Gauss equation and Codazzi-Meinardi equation. Riemann curvature tensor. Ricci tensor. Curvature scalar. Theorema Egregium. Fundamental theorem of the local theory of hypersurface patches.

## Assessment pattern

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

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

Examination | End-of-Semester Examination (2 hours) | 80 |

## Alternative Assessment

N/A

## Assessment Strategy

The __assessment strategy__ is designed to provide students with the opportunity to demonstrate:

- Understanding of subject knowledge, and recalled of key definitions, theorems and properties in the theory of elementary differential geometry of curves and surfaces in Euclidean space.
- The ability fo identify and use the appropriate techniques to solve geometric problems relating to curves and surfaces.

Thus, the

__summative assessment__for this module consists of:

- One in-semester test (50 minutes), worth 20% of the module mark, corresponding to Learning Outcomes 1, 2 and 5.
- A synoptic examination (2 hours), worth 80% of the module mark, corresponding to all Learning Outcomes 1 to 5.

Formative assessment

There are two formative unassessed courseworks over an 11 week period, designed to consolidate student learning.

Feedback

Students will receive individual written feedback on both the formative unassessed courseworks and the in-semester test. The feedback is timed such that feedback from the first coursework will assist students with preparation for the in-semester test. The feedback from both courseworks and the in-semester test will assist students with preparation for the synoptic examination. Students also receive verbal feedback in office hours.

## Module aims

- Provide students with an introduction to the geometry of curves and surfaces in Euclidean space.
- Develop students' understanding of concepts and basic tools of differential geometry in the context of curves and surfaces.
- Enable students to apply mathematical techniques to problems involving curves and surfaces, and calculate relevant quantities.

## Learning outcomes

Attributes Developed | ||

001 | Students will understand concepts, definitions and theorems relating to the geometric properties of curves. | KC |

002 | Students will be able to apply mathematical techniques to concrete examples of curves and calculate relevant quantities. | KT |

003 | Students will understand concepts, definitions and theorems relating to the geometric properties of surfaces. | KC |

004 | Students will be able to apply mathematical techniques to concrete examples of surfaces and calculate relevant quantities. | KC |

005 | Students will be able to analyse unseen geometric problems. | 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 students with an introduction to the theory and properties of curves and surfaces in Euclidean space.
- Provide students with experience of techniques from differential geometry used to understand and solve geometric problems relating to curves and surfaces, and calculate relevant quantities.

The learning and teaching methods include:

- Two one-hour lectures for eleven weeks. A flipped-learning approach will be taken in lectures, with module notes and an additional two hours of video recordings for eleven weeks provided to students in advance via a Virtual Learning Environment to complement lectures. These lectures provide a structured learning environment and opportunities for students to ask questions and to practice methods taught.
- Two unassessed courseworks to provide students with further opportunity to consolidate learning. Students receive individual written feedback on these courseworks as guidance on their progress and understanding.

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**

## Other information

The School of Mathematics and Physics is committed to developing graduates with strengths in Digital Capabilities, Employability, Global and Cultural Capabilities, Resourceness and Resilience, and Sustainability. This module is designed to allow students to develop knowledge, skills and capabilities in the following areas:

**Digital Capabilities**: The SurreyLearn page for MAT2047 features a dynamic discussion forum where students can pose questions and engage with others using e.g. LaTeX and MathML tools. This enhances their digital competencies while facilitating collaborative learning and information sharing.

**Employability**: The module MAT2047 equips students with skills which significantly enhance their employability. Students learn to visualise geometric problems and formulate these problems mathematically using tools from differential geometry. Differential geometry has numerous applications in fields such as Physics, Engineering and Computer Science. For instance, in Engineering, differential geometry is used to design optimal shapes for structures such as aircraft wings and, in Computer Science, differential geometry is used to study geometric algorithms for computer graphics and machine learning. Students will learn to evaluate complex geometric problems, break them into manageable components, and apply their knowledge and logical reasoning to arrive at solutions – these are highly sought after skills in many professions.

**Global and Cultural Capabilities:** Student enrolled in MAT2047 originate from a variety of countries and have a wide range of cultural backgrounds. Students are encouraged to work together during problem-solving teaching activities in seminars and lectures, which naturally facilitates the sharing of different cultures.

**Resourcefulness and Resilience**: MAT2047 is a module which demands the ability to visualise geometric problems, and formulate and solve these problems using the abstract mathematics of differential geometry. This rigorous branch of mathematics blends abstract mathematics with geometry, in a new approach which stretches their understanding of geometry and to which students will learn to adapt. Students will gain skills in analysing geometric problems using lateral thinking, and will complete assessments which challenge them and build resilience.

## Programmes this module appears in

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

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

Mathematics with Statistics MMath | 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 |

Mathematics 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 2024/5 academic year.