## MATH2051 Geometry of Curves and Surfaces

### 10 creditsClass Size: 95

Module manager: Dr Derek Harland
Email: D.G.Harland@leeds.ac.uk

Taught: Semester 1 View Timetable

Year running 2019/20

### Pre-requisite qualifications

[(MATH1010 or MATH1050) and (MATH1012 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 Standard exam (closed essays, MCQs etc) 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