# Gwybodaeth Modiwlau

Module Identifier
MA37110
Module Title
DIFFERENTIAL GEOMETRY
2010/2011
Co-ordinator
Semester
Semester 1
Mutually Exclusive
Pre-Requisite
Pre-Requisite
Pre-Requisite
Other Staff

#### Course Delivery

Delivery Type Delivery length / details
Lecture 18 X 1 HOUR
Seminars / Tutorials 4 X 1 HOUR
Workload Breakdown (Every 10 credits carries a notional student workload of 100 hours.) Lectures and tutorials 22 hours Worksheets (4 x 5 hours) 20 hours Private study 56 hours Examination 2 hours

#### Assessment

Assessment Type Assessment length / details Proportion
Semester Assessment 2 Hours   conventional examination  100%
Supplementary Assessment 2 Hours   conventional examination  100%

### Learning Outcomes

On successful completion of this module students should be able to:

1. illustrate the basic theory of differential calculus in R^n;

2. compare the generalization of these concepts to the setting of differential manifolds;

3. define vector and tensor fields, and perform algebraic and analytic computions with them;

4. analyse and synthesise mathematical idenities in an intrinsic setting;

### Aims

IMAPS wishes to introduce new level 3 modules reflecting research interests and expertise of new staff, thereby rectifying the previous problem of very limited range of modules for final year students. This module introduces the important area of differential geometry, MAM has some common lectures but extends the applications. It is intended to offer this module in alternate years. Differential geometry is currently unrepresented in the curriculum and we would like to introduce it as an option.

### Brief description

Differential Geometry is an important subject in modern mathematics, essential in the description of several areas of mathematical physics, and engineering.

### Content

Differential Calculus on R^n
Functions, vector fields, tensors. Curvilinear coordinates and hypersurfaces. Stoke's and Gauss' theorems.
Geometry of Special Relativity
Minkowski space-time, indefinite metrics, tensor formulation.
Differentiable Manifolds,
Introduction of the notion of manifolds and generalization of the above concepts to this situation. Lie brackets of vector fields.
Connections
Affine connections, Koszul connections, torsion. Riemannian manifolds and the Levi-Civita connection. Geodesics.

### Module Skills

Skills Type Skills details
Application of Number Throughout the module.
Communication Students will be expected to submit written worksheet solutions
Improving own Learning and Performance Feedback via tutorials
Information Technology Extensive use of spreadsheets.
Personal Development and Career planning Students will be exposed to an area of application that they have not previously encountered.
Problem solving All situations considered are problem-based to a greater or lesser degree.
Research skills Students will be encouraged to consult various books and journals for examples of application.
Subject Specific Skills Using differential geometric techniques in modeling.
Team work n/a

### Notes

This module is at CQFW Level 6