Differential Geometry module (MA32007)

This module will provide you with an introduction to differential geometry. In differential geometry, you explore the mathematics of curves, surfaces, and their properties using calculus and linear algebra. You’ll delve into how geometrical objects behave in different spaces and dimensions. You'll see how more conventional geometry can be generalised. This is done by introducing new concepts like manifolds and the exterior algebra. This module develops some of the geometrical concepts and tools. These are essential for understanding both classical and modern physics and engineering.

Differential geometry is a branch of maths which is used in modern presentations of physical theories. These include continuum mechanics, fluid dynamics, electromagnetism, thermodynamics, and general relativity. Through this module, you'll develop a deep understanding of how geometry connects with physics and engineering. You’ll work on analytical techniques to solve geometrical problems. You will also gain insight into the elegant mathematics that governs the shapes and structures around us.

What you will learn

In this module, you will:

• learn about a key structure called a manifold and learn how a type of function called a vector field can be defined on a manifold
• find out how ideas from calculus can be generalised with the introduction of differential forms
• explore how vector calculus can be generalised by studying Lie derivatives

By the end of this module, you will be able to:

• give examples of new concepts of manifolds and submanifolds, covectors, and exterior forms
• determine the differential of a function and integrate differential forms
• compute line and surface integrals
• apply Stoke's Theorem in more abstract settings

Assignments / assessment

• coursework (20%)
• final exam (80%)

Teaching methods / timetable

• two one-hour lectures weekly
  • key points of the week's content will be discussed
  • lecture notes covering the full module content will be available before classes
  • in-class time will be prioritised for interactive discussion
• one hour of tutorials weekly
  • solve problems individually and in groups
  • support with difficulties will be provided by your lecturers and peers