Undergraduate Course: Geometry & Calculus of Variations (MATH09003)
Course Outline
| School | School of Mathematics |
College | College of Science and Engineering |
| Course type | Standard |
Availability | Available to all students |
| Credit level (Normal year taken) | SCQF Level 9 (Year 3 Undergraduate) |
Credits | 10 |
| Home subject area | Mathematics |
Other subject area | Specialist Mathematics & Statistics (Honours) |
| Course website |
https://info.maths.ed.ac.uk/teaching.html |
Taught in Gaelic? | No |
| Course description | Optional course for Honours Degrees involving Mathematics and/or Statistics. Plane curves, regularity, curvature(moving frame analysis). Space curves, biregularity, curvature and torsion. Families of plane curves, functionals and their variation, Euler-Lagrange equations. Motion in a potential, energy. Surfaces, regularity, shape operator, mean and Gauss curvature. Geodesics as a variational problem. |
Information for Visiting Students
| Pre-requisites | None |
| Displayed in Visiting Students Prospectus? | Yes |
Course Delivery Information
| Not being delivered |
Summary of Intended Learning Outcomes
1. Isometry
2. How to define planar curves, check their regularity, and determine arc-length.
3. How to determine tangent, normal and curvature of a planar curve.
4. Definition of families of planar curves and construction of their envelopes.
5. The Equivalence Problem for planar curves.
6. Definition of a functional and its first variation.
7. Derivation of the Euler-Lagrange equation of a functional.
8. Integration of the Euler-Lagrange equation in the case of ignorable coordinates and other examples.
9. Definition of Space Curves and Biregularity.
10. Determination of Tangent, Normal, Binormal, Curvature and Torsion
11. The Equivalence Problem for space curves.
12. Definition of a surface and regularity. Calculation of Tangent Space and Normal.
13. Definition of a curve within a surface, its arc-length and calculation of the first fundamental form.
14. Conditions for stationary arc-length and definition of Geodesics.
15. Examples of Geodesics.
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Assessment Information
Coursework: 15%; Degree Examination: 85%.
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Special Arrangements
| None |
Additional Information
| Academic description |
Not entered |
| Syllabus |
Not entered |
| Transferable skills |
Not entered |
| Reading list |
http://www.readinglists.co.uk |
| Study Abroad |
Not entered |
| Study Pattern |
Not entered |
| Keywords | GCV |
Contacts
| Course organiser | Dr Aram Karakhanyan
Tel: (0131 6)50 5056
Email: aram.karakhanyan@ed.ac.uk |
Course secretary | Dr Jenna Mann
Tel: (0131 6)50 4885
Email: Jenna.Mann@ed.ac.uk |
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