MTH4110 · Differential geometry

Overview

This unit will explore the metric structure of curves and surfaces, primarily in 3-dimensional Euclidean space. The major focus is on the various concepts of curvature and related notions, and the relationships between them. Curvature and torsion of a curve. First and second fundamental forms of a surface. Geodesic and normal curvatures of a curve on a surface. Gaussian, mean and principal curvatures of a surface. Important theorems relating these concepts. Links will be drawn with many other areas of mathematics, including real and complex analysis, linear algebra, differential equations, and general relativity

Areas of study: Applied mathematics Mathematical statistics Mathematics Pure mathematics

Offerings

Assessment

The Handbook lists an examination for this unit.

Assessment details may change. Please refer to the assessment information in Moodle closer to the start of the teaching period.

Requisites

The Handbook lists no prerequisite, corequisite or prohibition for this unit.

Learning outcomes

1. Critically analyse the significance and implications of intrinsic measures of curvature, for curves and surfaces in 3-dimensional space
2. Perform calculations to determine curvature and related quantities for curves and surfaces in 3-dimensional space.
3. Synthesise and apply advanced concepts and theorems regarding the geometry of curves and surfaces in 3-dimensional space, demonstrating a deep understanding.
4. Formulate and prove theorems, and solve complex problems about curves and surfaces in 3-dimensional space using differential geometry techniques.
5. Evaluate and explore the intricate connections between differential geometry and other advanced areas of mathematics and physics, such as real and complex analysis, abstract linear algebra, differential equations, and general relativity.
6. Articulate complex mathematical ideas and findings related to differential geometry with clarity, precision, and rigour, suitable for academic and professional audiences

Workload

Three 1-hour seminars; One 2-hour applied class (in weeks 2-12) and 7 hours of independent study per week.

Student experience, not official rules. Nothing here changes what the Handbook says.