## MATH2051 Geometry of Curves and Surfaces

### 10 creditsClass Size: 200

Module manager: Dr K. Houston
Email: khouston@maths.leeds.ac.uk

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2008/09

### Pre-requisite qualifications

MATH1050 or MATH1932 or MATH1960 plus MATH1970.
Also MATH1015 or MATH1060 or MATH1331.

This module is approved as an Elective

### 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 higher dimensional 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

To introduce students to the geometry of curves, surfaces and hypersurfaces both in 3-dimensions and in n-dimensions. To develop students' geometrical intuition. To show how several variable calculus can be used to measure geometrical quantities. On completion of this module, students should be able to:
(a) recognise when a level set is a surface and describe it;
(b) describe and integrate a vector field;
(c) calculate and manipulate the arc length and curvature apparatus of a curve;
(d) calculate and manipulate the Weingarten map and associated quantities of a surface given either as a level surface or as a parameterised surface.

### Syllabus

This course will discuss the geometry of curves, surfaces and hyper-surfaces in both 3 and N dimensions. Surfaces and hypersurfaces will be exhibited initially as level sets of a regular function, this approach being a natural outgrowth of several variable calculus and allowing rapid appreciation of N-dimensional geometrical objects such as hyperspheres and N-dimensional tori. Such objects are much used in modern physics and dynamical systems in applied mathematics. The study of curvature of surfaces in 3 dimensions is also useful in computer-aided design. The topics covered, with chapter numbers as in Thorpe's book, (see booklist below) are:
1. Graphs and level sets in Rn (ch.1);
2. Vector fields on Rn , and integral curves (ch. 2);
3. The tangent space to a level set (ch.3);
4. Hypersurfaces of Rn defined as level sets of a regular functions (ch.4);
5. Curvature of curves in Rn and Frenet formulae for curves in R3 (ch.10 + extra material);
6. Geodesics on hypersurfaces (ch. 7);
7. Tangent and normal vector fields on surfaces (ch.5);
8. The Weingarten map of a hypersurface (ch.9);
9. Curvature of surfaces and hypersurfaces (normal and principal curvatures, first and second fundamental form, Gauss curvatures of a surface (ch.12); Parameterised surfaces (local charts) (ch.14); Minimal surfaces (ch.18).

### Teaching methods

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

 Delivery type Number Length hours Student hours Example Class 11 1.00 11.00 Lecture 22 1.00 22.00 Private study hours 67.00 Total Contact hours 33.00 Total hours (100hr per 10 credits) 100.00

### Methods of assessment

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) 2 hr 00 mins 100.00 Total percentage (Assessment Exams) 100.00

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