MATA6350 Differential geometry of curves and surfaces (5–7 cr)
Local and global properties of curves from the point of view of differential geometry. For instance: parametrization of curves, curvature and torsion of curves, the local canonical form, the Jordan curve theorem, isoperimetric inequality.Basic theory of surfaces, for instance: different ways to define and express surfaces, tangents and derivations, the first fundamental form, area and different notions of curvature, Gauss’ Theorema egregium. Possibly also geodesics and the Gauss-Bonnet theorem.
- can examine the length and parametrization of curves
- masters the definitions of curvature and torsion and can apply these
- knows the local canonical form of curves
- knows the contents and the significance of the Jordan curve theorem and the isoperimetric inequality.
- can examine the properties of surfaces using different expressions for surfaces
- knows the first fundamental form of surfaces
- can determine areas and various curvatures of surfaces
- knows the contents and the significance Gauss’ Theorema egregium
28h lectures, and exercises
Description of prerequisites
Vector Analysis 1 and 2
M. Abate, F. Tovena: Curves and Surfaces, Chapters 1 - 4 (at least)
- M. Abate, F. Tovena: Curves and Surfaces, Springer-Verlag Mailand, 2012; ISBN: 978-88-470-1940-9