Course Code : PMAT 41783
Title : Differential Geometry
Pre-requisites : PMAT 22583
Learning Outcomes:
At the end of this course the student should be able to
- demonstrate the fundamental knowledge of curves and surfaces in space
- identify the importance of the two factors curvature and torsion, and their intrinsic properties.
Course Contents:
Theory of Curves: Concept of a curve, Arc length, Curvature and torsion, Frenet-serret formulae, General helix, intrinsic equations, Fundamental existence and uniqueness theorems for space curves, Canonical representation of a curve. Involutes and Evolutes, Theory of contact.
Theory of Surfaces: Concept of a surface, Topological properties of a surface, Surface of revolution, Ruled surfaces, Length of arc on a surface, Vector element of an area, First and second fundamental forms, Curves on a surface, Direction coefficients, Direction ratios, Family of curves on a surface, Double family of curves. Umbilical point, Intrinsic properties of a surface, Geodesies.
Curvature: Principle curvature and directions, Gaussian and mean curvatures, Lines of curvature Rodrigues formula.
Introduction to Riemannian geometry
Method of Teaching and Learning: A combination of lectures, tutorials and presentations.
Assessment: Based on tutorials, tests, presentations and end of course examination.
Recommended Reading:
- Lipschutz, L., (1969) Differential Geometry, McGraw-
- Willmore, T.J., (2013) An Introduction to Differential Geometry, Oxford University Press.
- Do Carmo, M.P., (2016) Differential Geometry of Curves and Surfaces, Prentice-Hall, New Jersey.