Undergraduate Course: Geometry (MATH10074)
|School||School of Mathematics
||College||College of Science and Engineering
|Credit level (Normal year taken)||SCQF Level 10 (Year 3 Undergraduate)
||Availability||Available to all students
|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.
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.
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).
Information for Visiting Students
|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.
|High Demand Course?
Course Delivery Information
|Academic year 2022/23, Available to all students (SV1)
|Learning and Teaching activities (Further Info)
Lecture Hours 22,
Seminar/Tutorial Hours 5,
Summative Assessment Hours 2,
Programme Level Learning and Teaching Hours 2,
Directed Learning and Independent Learning Hours
|Assessment (Further Info)
|Additional Information (Assessment)
||Coursework 20%, Examination 80%
||Hours & Minutes
|Main Exam Diet S1 (December)||2:00|
On completion of this course, the student will be able to:
- State definitions and theorems and present standard proofs accurately without access to notes/books.
- Compute the Frenet-Serret frame of space curves and determine their torsion and curvature in simple examples.
- Work accurately with differential forms and perform the basic operations of wedge product and exterior derivative.
- Compute the first and second fundamental forms and curvatures of a surface in simple examples.
- Apply theory developed in the course to solve simple problems.
|Recommended books for supplementary reading (not essential):|
Differential forms and Applications, Manfredo P. Do Carmo, Springer 1994
Differential Geometry of Curves and Surfaces, Manfredo P. Do Carmo, Prentice-Hall 1976
|Graduate Attributes and Skills
|Course organiser||Dr Johan Martens
Tel: (0131 6)51 7759
|Course secretary||Miss Greta Mazelyte