Module manager: Ben Sharp
Email: b.g.sharp@leeds.ac.uk
Taught: Semester 2 (Jan to Jun) View Timetable
Year running 2026/27
MATH2150 Calculus, Curves and Complex Analysis
| MATH2150 | Calculus, Curves and Complex Analysis |
MATH2051 Geometry of Curves and Surfaces MATH3113 Differential Geometry
This module is not approved as an Elective
Differential geometry is the application of calculus to describe, analyse and discover facts about geometric objects. It provides the language in which almost all modern physics is understood, and is a beautiful subject in its own right. This module develops the geometry of curves and surfaces embedded in Euclidean space. A recurring fundamental theme is curvature (in its many guises) and its interplay with topology.
Students will be introduced to the study of geometry using functions and maps of several variables. Key geometric-analytical quantities will be rigorously defined and intuitively un-packed e.g. the first and second fundamental forms, abstract integration along curves and surfaces, various notions of curvature. These formal objects will then be used to exhibit both deep and fundamental laws which underpin our geometric understanding of the world around us: The Gauss equations (a law governing the relationship between the intrinsic curvature of a surface with its extrinsic curvatures) and the Gauss-Bonnet formula (an integral identity which directly relates intrinsic curvatures to the global shape of a surface).
On successful completion of the module students will be able to: 1. Identify and exhibit examples of curves and surfaces in Euclidean space. 2. Compute geometric quantities for explicit examples. 3. Utilise the fundamental definitions appropriate to curves and surfaces in Euclidean space to verify mathematical statements. 4. Demonstrate an understanding of the difference between intrinsic and extrinsic geometric quantities and discriminate between these in practice. 5. Translate between symbolic and visual representations of geometric objects.
Regular parametrised curves in Euclidean space: length, curvature and Frenet frames. 2. Local theory of regularly parametrized surfaces: tangent planes, the first fundamental form, area, orientation, the second fundamental form, principal, normal, mean and Gauss curvatures. 3. Global theory of regular surfaces: submanifolds of Euclidean space, surfaces as level sets, the Regular Value Theorem, integration on surfaces. 4. Isometries of surfaces. Extrinsic versus intrinsic quantities. Gauss's Theorema Egregium. Geodesics and geodesic curvature. 5. Triangulations and the Euler characteristic of a surface. The Gauss-Bonnet Theorem. Additional topics that build on these may be covered as time allows. Such topics may be drawn from the following, or similar: 6. Regular closed plane curves, rotation index and the Whitney-Graustein Theorem. 7. Variational problems in differential geometry, for example geodesics as critical points of energy, minimal surfaces, constant mean curvature surfaces. 8. The Gauss-Codazzi equations.
| Delivery type | Number | Length hours | Student hours |
|---|---|---|---|
| Lecture | 44 | 1 | 44 |
| Private study hours | 156 | ||
| Total Contact hours | 44 | ||
| Total hours (100hr per 10 credits) | 200 | ||
There will be regular small pieces of homework which students can hand in for formative feedback
Check the module area in Minerva for your reading list
Last updated: 12/05/2026
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