Mechanical Engineering

Knowledge and understanding: Knowledge and understanding of basic concepts of differential geometry: differentiable manifold, differentiable map, tangent bundle, vector field, flow of a field, tensor field, differential form, Lie group. Local theory of curves: Frenet formulas; local theory of surfaces: metric properties, gaussian curvature, Gauss theorem. Skills and attributes: Be able to calculate the Frenet apparatus of a curve defined by parametric or cartesian equations. Be able to calculate the gaussian curvature, principal curvatures and normal sections of a surface defined by parametric or cartesian equations. Be able to determine local coordinates on a manifold and write down transition maps. Be able to study a differential map, using local coordinates. Find the tangent space to a manifold. Calculate the flow of a vector field. Be able to operate with tensors. Calculate line integrals and recognize exact forms.