2024/25 Undergraduate Module Catalogue

MATH2051 Geometry of Curves and Surfaces

10 creditsClass Size: 130

Module manager: Martin Palmer-Anghel
Email: M.D.Palmer-Anghel@leeds.ac.uk

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2024/25

Pre-requisite qualifications

[(MATH1005 or MATH1010 or MATH1050) and (MATH1005 or MATH1060 or MATH1331)] or (PHYS1290 and PHYS1300 and MATH1060).

This module is not approved as a discovery module

Module summary

Differential geometry has played a central and influential role in the development of 20th century pure mathematics and is fundamental to our understanding of the natural world. It is a key element of modern theories of particle physics and cosmology, and a crucial ingredient of all advanced approaches to mechanics and dynamical systems theory.This course offers an introduction to the subject by examining the geometry of curves and surfaces embedded in Euclidean space. The approach is to use familiar ideas from multivariable calculus and linear algebra to construct and study geometric objects, with elegant abstract definitions being illustrated by many concrete examples.

Objectives

On completion of this module, students should be able to:
a) recognise a regularly parametrized curve and compute its arc length and curvature;
b) construct and manipulate the Frenet frame of a curve in R^3;
c) construct the tangent and normal spaces of a parametrized surface;
d) compute the shape operator of an oriented surface, and manipulate it to find the associated curvatures of the surface;
e) construct simple minimal surfaces, and surfaces of prescribed Gaussian curvature.

Syllabus

- Parametrized curves in Euclidean space, their arc length and curvature, evolutes and involutes, the Frenet formulas.
- Parametrized surfaces, their tangent and normal spaces.
- The shape operator on an oriented surface and associated curvatures.
- Minimal surfaces, surfaces of prescribed Gaussian curvature and surfaces of revolution in R^3.

The topics covered are:
1. Parametrized curves in R^n, reparametrization, arc length.
2. Curvature of curves in R^n, signed curvature of curves in R^2, evolutes and involutes of planar curves, the Frenet formulae for curves in R^3 .
3. Parametrized surfaces in R^3, and surfaces of revolution.
4. The tangent and normal spaces of a parametrized surface. Oriented surfaces. Directional derivatives.
5. The shape operator of an oriented surface: principal, Gauss and mean curvatures.
6. Minimal surfaces and surfaces of prescribed Gaussian curvature.

Teaching methods

Delivery type	Number	Length hours	Student hours
Workshop	10	1.00	10.00
Lecture	22	1.00	22.00
Private study hours			68.00
Total Contact hours			32.00
Total hours (100hr per 10 credits)			100.00

Private study

Studying and revising of course material.
Completing of assignments and assessments.

Opportunities for Formative Feedback

Regular problem solving assignments

Methods of assessment

Coursework

Assessment type	Notes	% of formal assessment
In-course Assessment	.	15.00
Total percentage (Assessment Coursework)		15.00

There is no resit available for the coursework component of this module. If the module is failed, the coursework mark will be carried forward and added to the resit exam mark with the same weighting as listed above.

Exams

Exam type	Exam duration	% of formal assessment
Open Book exam	2 hr 00 mins	85.00
Total percentage (Assessment Exams)		85.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated

Reading list

The reading list is available from the Library website

Last updated: 16/10/2024

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