Differential Geometry module (MA32007)

About the module

The basic ideas at the foundations of many physical theories, such as continuum mechanics, fluid dynamics, electromagnetism, thermodynamics, general relativity and gauge theories, are geometrical. This course develops some of the geometrical concepts and tools that are essential for understanding classical and modern physics and engineering. This module is mandatory for students taking the BSc or MMath in Mathematics and may optionally be taken in combination with other modules by students on any of the Mathematics combined degrees. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.

Prerequisites

Students taking this module must have achieved a pass mark in each of the modules MA21001 and MA22001, or equivalents, and have taken MA31007 or MA32002.

Indicative Content

• Manifolds

  Submanifolds in Rⁿ. Implicit Function theorem. Examples

• Vector fields on manifolds

  The tangent space. Vectors as differential operators. Vector fields and flows.

• Covectors and exterior forms

  Linear functionals and the dual space. Differential of a function. The pull-back of a covector.

• The exterior algebra

  The geometric meaning of forms in R^n. Exterior product. Inner product. Exterior differential. Relation to vector analysis.

• Integration of forms

  Line and surface integrals. Independence of parameterisation. Integrals and pull-backs. Stoke′s Theorem.

• Lie-derivative

  The Lie-derivative of forms. Relation to equations in physics.

Delivery and Assessment

The module is delivered in the form of lectures and workshops/tutorials and assessed via an exam (80%) and coursework (20%).

Credit Rating

This module is a Scottish Higher Education Level 3 or SCQF level 9 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.