The first 4 weeks

Curves ( in R³ ): arc-length, curvature, torsion, Frenet-equations, Fundamental Theorem of the Local Theory of Curves, isoperimetric inequality. Surfaces ( in R³ ): parametrisation, orientability, area (functional), minimal surfaces, first- and second fundamental form, Gauss-map, normal- and principal curvatures, Weingarten-map, Theorema Egregium, Christoffel-symbols, Fundamental Theorem of the Local Theory of Surfaces, Mainardi-Codazzi-equations (for hyper-surfaces in R^N )