Geometry & Calculus of Variations (U01612)

? Credit Points : 10 ? SCQF Level : 9 ? Acronym : MAT-3-GCV

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.

Entry Requirements

? Pre-requisites : Prior attendance at MAT-2-SVC

? Prohibited combinations : Similar courses from Mathematics 3 (Hons) prior to 2004-05

Subject Areas

Home subject area

Specialist Mathematics & Statistics (Honours), (School of Mathematics, Schedule P)

Delivery Information

? Normal year taken : 3rd year

? Delivery Period : Semester 2 (Blocks 3-4)

? Contact Teaching Time : 2 hour(s) 30 minutes per week for 11 weeks

All of the following classes

Type	Day	Start	End	Area
Lecture	Tuesday	10:00	10:50	KB
Lecture	Friday	10:00	10:50	KB

? Additional Class Information : Tutorials: at times to be arranged.

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.