Differential Geometry(MAT417)
| Course Code | Course Name | Semester | Theory | Practice | Lab | Credit | ECTS |
|---|---|---|---|---|---|---|---|
| MAT417 | Differential Geometry | 8 | 4 | 0 | 0 | 4 | 8 |
| Prerequisites | MAT116, MAT202 |
| Admission Requirements | MAT116, MAT202 |
| Language of Instruction | French |
| Course Type | Compulsory |
| Course Level | Bachelor Degree |
| Course Instructor(s) | SUSUMU TANABE tanabesusumu@hotmail.com (Email) |
| Assistant | |
| Objective | This course aims to give an introduction to the basic concepts of Differential Geometry, Differential Topology. |
| Content | Smooth manifolds, Smooth maps, The tangent space, The cotangent bundle, Submanifolds, Embedding and Approximation theorems, Lie group actions, Tensors, Differential forms, Integration on manifolds, De Rham cohomology, Stokes theorem. |
| Course Learning Outcomes |
1. will be able to deal with various examples of differentiable manifolds and smooth maps 2. will be able to have familiarity with tangent vectors, tensors and differential forms 3. will be able to work practically with vector fields and differential forms 4. will be able to appreciate the basic ideas of de Rham cohomology and its examples 5. will be able to apply the ideas of differentiable manifolds to other areas 6. will be able to apply the basic techniques, results and concepts of the course to concrete examples |
| Teaching and Learning Methods | Lecture notes, Problem solving |
| References |
Principe d'analyse mathématique", W. Rudin “Géométrie et calcul différentiel sur les variétés : Cours, études et exercices pour la maîtrise de mathématiques», F.Pham "Géométrie et Topologie des Surfaces” D. Lehmann, C.Sacre |
Theory Topics
| Week | Weekly Contents |
|---|---|
| 1 | experience 1 Jacobians |
| 2 | Experience 2 Local inversion |
| 3 | Experience 3 Implicit function theorem |
| 4 | experience 4 Implicit function theorem |
| 5 | Subvariety |
| 6 | Smooth subvariety |
| 7 | Midterm |
| 8 | Tangent space |
| 9 | Directional derivative |
| 10 | Integration on manifolds |
| 11 | Differential forms |
| 12 | Polar coordinates |
| 13 | Stokes theorem |
| 14 | Closed Exact forms |
Practice Topics
| Week | Weekly Contents |
|---|
Contribution to Overall Grade
| Number | Contribution | |
|---|---|---|
| Contribution of in-term studies to overall grade | 2 | 30 |
| Contribution of final exam to overall grade | 1 | 2 |
| Toplam | 3 | 32 |
In-Term Studies
| Number | Contribution | |
|---|---|---|
| Assignments | 0 | 0 |
| Presentation | 0 | 0 |
| Midterm Examinations (including preparation) | 0 | 0 |
| 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 | 14 | 10 |
| Toplam | 14 | 10 |
| 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 | ||