Descrição

Objectives

Familiarize the students with the fundamental concepts of differential geometry of curves and surfaces in the Euclidean space, with emphasis on the computation in concrete examples.

Syllabus

Parametrized curves. Curves in the space: curvature and torsion; Frenet frame; existence and uniqueness of curves with prescribed curvature and torsion. Global properties of plane curves: the isoperimetric inequality and the four- vertex theorem. Surfaces in the 3-dimensional Euclidean space. Tangent plane and smooth maps between surfaces. The first fundamental form; area. The Gauss map. The second fundamental form. Principal curvatures; Gaussian and mean curvatures; curvature lines. The Gauss map in local coordinates. Computation in classical examples. The intrinsic geometry of surfaces. Isometries and conformal maps. The Gauss theorem and the equations of compatibility. Parallel transport; geodesics. Computation of geodesics in surfaces of revolution; Clairaut integral. The Gauss-Bonnet theorem and its applications. Additional/optional topics: differential forms and their application to the differential geometry of surfaces.

Prerequisites

Differential and Integral Calculus III.

Cross Competence Component

The UC allows the development of transversal competences on Critical Thinking, Creativity and Problem Solving Strategies, in class, in autonomous work and in the several evaluation components. The percentage of the final grade associated with these competences should be around 15%.

Ethical Principles