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Semester 1

Geometry (MATH10074)

Subject

Mathematics

College

SCE

Credits

10

Normal Year Taken

3

Delivery Session Year

2022/2023

Pre-requisites

Required knowledge may be deduced from the course descriptions and syllabuses of the pre-requisite University of Edinburgh courses listed above. Visiting students are advised to check that they have studied the material covered in the syllabus of each prerequisite course before enrolling.

Course Summary

Differential geometry is the study of geometry using methods of calculus and linear algebra. It has numerous applications in science and mathematics.This course is an introduction to this classical subject in the context of curves and surfaces in euclidean space. There are two lectures a week and a workshop every two weeks. There are biweekly assignments and a closed-book exam in December.

Course Description

The course begins with curves in euclidean space; these have no intrinsic geometry and are fully determined by the way they bend and twist (curvature and torsion). The rest of the course will then develop the classic theory of surfaces. This will be done in the modern language of differential forms. Surfaces possess a notion of intrinsic geometry and many of the more advanced aspects of differential geometry can be demonstrated in this simpler context. One of the main aims will be to quantify the notions of curvature and shape of surfaces. The culmination of the course will be a sketch proof of the Gauss-Bonnet theorem, a profound result which relates the curvature of surfaces to their topology.Syllabus:Curves in Euclidean space: regularity, velocity, arc-length, Frenet-Serret frame, curvature and torsion, equivalence problem.Calculus in R^n: tangent vectors, vector fields, differential forms, moving frames, connection forms, structure equations.Surfaces in Euclidean space: regularity, first and second fundamental forms, curvatures (principal, mean, Gauss), isometry, Gauss's Theorema Egregium, geodesics on surfaces, integration of forms, statement of Stokes' theorem, Euler characteristic, Gauss-Bonnet theorem (sketch proof).

Assessment Information

Written Exam 80%, Coursework 20%, Practical Exam 0%

Additional Assessment Information

Coursework 20%, Examination 80%

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