Differential Geometry(MAT417)
| Course Code | Course Name | Semester | Theory | Practice | Lab | Credit | ECTS |
|---|---|---|---|---|---|---|---|
| MAT417 | Differential Geometry | 8 | 4 | 0 | 0 | 4 | 8 |
| Prerequisites | |
| Admission Requirements |
| Language of Instruction | French |
| Course Type | Compulsory |
| Course Level | Bachelor Degree |
| Course Instructor(s) | SUSUMU TANABE tanabesusumu@hotmail.com (Email) |
| Assistant | |
| 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. |
| 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 |
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 |
Practice Topics
| Week | Weekly Contents |
|---|
Contribution to Overall Grade
| Number | Contribution | |
|---|---|---|
| Contribution of in-term studies to overall grade | 2 | 50 |
| Contribution of final exam to overall grade | 1 | 50 |
| Toplam | 3 | 100 |
In-Term Studies
| 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 |
| Toplam | 9 | 50 |
| No | Program Learning Outcomes | Contribution | ||||
|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | ||
| 1 | understands principles of deductive reasoning; has experience to verify well-foundedness and exactness of mathematical statements in systematic ways; | X | ||||
| 2 | can properly state and use concepts and results of major mathematical interest; | X | ||||
| 3 | masters current computational techniques and algorithms; has a good ability in their use; can identify relevant tools, among those one has learned, suitable to solve a problem and is able to judge whether or not one is in possession of these tools; | X | ||||
| 4 | is able to express one’s mathematical ideas in an organised way both in written and oral forms; | X | ||||
| 5 | understands relations connecting substantial concepts and results; can switch from one viewpoint to another on mathematical objects (pictures, formulae, precise statements, heuristic trials, list of examples,...); | X | ||||
| 6 | has followed individually a guided learning strategy; has pursued steps toward the resolution of unfamiliar problems; | X | ||||
| 7 | has a theoretical and practical knowledge in computer science well adapted for learning a programming language; | X | ||||
| 8 | has investigated the relevance of modeling and using mathematical tools in natural sciences and in the professional life; is conscious about historical development of mathematical notions; | X | ||||
| 9 | has followed introduction to some mathematical or non-mathematical disciplines after one’s proper choice; had experience to learn selected subjects according to one’s proper arrangement; | X | ||||
| 10 | masters French language as well as other foreign languages, to a level sufficient to study or work abroad. | X | ||||
| 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 | ||