Undergraduate Course: Geometry (MATH10074)
Course Outline
| 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 |
| SCQF Credits | 10 |
ECTS Credits | 5 |
| Summary | This course is an introduction to differential geometry, which is the study of geometry using methods of calculus and linear algebra. 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 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. The culmination of the course will be a sketch proof of the Gauss-Bonnet theorem, a remarkable connection between the curvature of surfaces and their topology. |
| Course description |
Curves in Euclidean space, regularity, velocity, arc-length, Frenet-Serret frame, curvature and torsion, equivalence problem.
Tangent vectors, vector fields, differential forms, Poincare¿s Lemma, moving frames, connection forms, structure equations.
Surfaces, first and second fundamental forms, curvature, isometry, Theorema Egregium, geodesics on surfaces, integration of forms, statement of general Stokes' theorem, Euler characteristic, Gauss-Bonnet theorem (sketch proof).
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Information for Visiting Students
| Pre-requisites | None |
Course Delivery Information
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| Academic year 2014/15, Available to all students (SV1)
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Quota: None |
| Course Start |
Semester 1 |
Timetable |
Timetable |
| Learning and Teaching activities (Further Info) |
Total Hours:
100
(
Lecture Hours 22,
Seminar/Tutorial Hours 5,
Summative Assessment Hours 2,
Programme Level Learning and Teaching Hours 2,
Directed Learning and Independent Learning Hours
69 )
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| Assessment (Further Info) |
Written Exam
95 %,
Coursework
5 %,
Practical Exam
0 %
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| Additional Information (Assessment) |
Coursework 5%, Examination 95% |
| Feedback |
Not entered |
| Exam Information |
| Exam Diet |
Paper Name |
Hours & Minutes |
|
| Main Exam Diet S1 (December) | MATH10074 Geometry | 2:00 | |
Learning Outcomes
1. An ability to compute the Frenet-Serret frame of space curves and determine their torsion and curvature.
2. An ability to perform simple manipulations with differential forms.
3. An understanding of the first and second fundamental forms of a surface and its principal curvatures. An ability to compute simple examples.
4. An ability to calculate geodesics on surfaces in simple examples.
5. An ability to apply Stokes' Theorem to simple examples.
6. An ability to apply the Gauss-Bonnet theorem to simple examples.
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Additional Information
| Graduate Attributes and Skills |
Not entered |
| Study Abroad |
Not Applicable. |
| Keywords | Geom |
Contacts
| Course organiser | Dr James Lucietti
Tel: (0131 6)51 7179
Email: J.Lucietti@ed.ac.uk |
Course secretary | Mrs Kathryn Mcphail
Tel: (0131 6)50 4885
Email: k.mcphail@ed.ac.uk |
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