## MATH2051 Geometry of Curves and Surfaces

### 10 creditsClass Size: 200

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2005/06

### Pre-requisites

MATH1941 or MATH1931 or equivalent.

This module is approved as an Elective

### 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

Lectures (22 hours) and examples classes (11 hours).

### Methods of assessment

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

Two hour written examination (100%)