Differential Geometry (MAT417) | GSÜ - ECTS Information Guide

Content

Objective

The aim of the course is to provide the student with basic knowledge and skills in elementary differential geometry of curves and surfaces in local parametric treatment.

Course Content

Curves in R3: Frenet formulas and Fundamental Theorem. Regular surfaces. Inverse image of regular values. Differentiable functions on surfaces. Tangent plane; the differential of a map, vector fields, the first fundamental form. Gauss map, second fundamental form, normal, principal curvatures. Manifolds, tangent spaces and Lie bracket

Course Learning Outcomes

At the end of the course the student should be able to
1. cope with modelling problems from diverse fields.
2. solve various problems in differential geometry and mechanics

Teaching and Learning Methods

Lectures, exercises

References

Millman, R.S. & Parker, G.D., Elements of Differential Geometry
Kühnel, W., Differential Geometry: Curves, Surfaces, Manifolds
Ethan D. Bloch; A first course in Geometric Topology and Differential Geometry
doCarmo, M. Differential Geometry of Curves and Surfaces
Montiel, S. & Ros, A. Curves and Surfaces

Weekly Contents

Theory Topics
Week	Weekly Contents
1	Recall on smooth functions, inverse function theorem
2	Curves in the Euclidean space and their reparametrization
3	Tangent, normal and binormal vectors
4	Curvature and torsion of space curves
5	Fundamental theorem of curves
6	Surfaces in space and coordinate patches
7	Smooth surfaces
8	Tangent and normal vectors to a surface, first fundamental form and arc lengths
9	Second fundamental form and Weingarten endomorphisms
10	Normal curvature, mean curvature and Gaussian curvature
11	Theorema Egregium of Gauss and isometries
12	Gauss – Bonnet formula and its consequences
13	Manifolds and tangent spaces
14	Tangent spaces and Lie bracket

Assessment System

Contribution to Overall Grade
Activities	Number	Contribution
Contribution of in-term studies to overall grade	2	50
Contribution of final exam to overall grade	1	50
Total	3	100

In-Term Studies
Activities	Number	Contribution
Assignments	7	0
Presentation	0	0
Midterm Examinations (including preparation)	2	50
Project	0	0
Laboratory	0	0
Other Applications	0	0
Quiz	0	0
Term Paper/ Project	0	0
Portfolio Study	0	0
Reports	0	0
Learning Diary	0	0
Thesis/ Project	0	0
Seminar	0	0
Other	0	0
Total	9	50

ECTS

Activities	Number	Period	Total Workload
Class Hours	14	4	56
Working Hours out of Class	14	5	70
Assignments	7	3	21
Midterm Examinations (including preparation)	2	6	12
Final Examinations (including preparation)	1	16	16
Total Workload			175
Total Workload / 25			7.00
Credits ECTS			7