MAT 342E - Differantial Geometry

Course Objectives

1. To give students essentials of curves and surfaces in three dimensional Euclidean space.
2. To reinforce students’ multivariable calculus and linear algebra knowledge and to give them the ability to apply this knowledge on geometry problems.

Course Description

Differential Geometry of Curves: Parametrizations of regular curves in R^3 space, curvature, torsion, Frenet trihedral, Frenet-Serret equations. Spherical curves. Special curve families; helices, involutes and evolutes, Bertrand’s curves. Fundamental theorems of the theory of curves. Some applications of symbolic computation.


Differential Geometry of Surfaces: Regular surfaces. The tanget plane. First fundamental form of a surface, Local geometry of Gauss map. Second fundamental form and shape operator. Normal curvature, principal curvatures and principal directions, Gausssian curvature, mean curvature. The lines of curvature, asymptotic curves, Dupin indicatrix, Meunier’s theorem. Special surface families. Some applications of symbolic computation. Theorema Egregium of Gauss.