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| MAT4024 | Manifolds and Hypersurfaces | 4+0+0 | ECTS:6 | | Year / Semester | Spring Semester | | Level of Course | First Cycle | | Status | Elective | | Department | DEPARTMENT of MATHEMATICS | | Prerequisites and co-requisites | None | | Mode of Delivery | | | Contact Hours | 14 weeks - 4 hours of lectures per week | | Lecturer | Prof. Dr. Yasemin SAĞIROĞLU | | Co-Lecturer | Prof.Dr. Yasemin SAĞIROĞLU | | Language of instruction | Turkish | | Professional practise ( internship ) | None | | | | The aim of the course: | | To introduce the concept of manifold, which is an important structure in differential geometry. To make some differential calculations on the hypersurface by explaining the concept of hypersurface which is a special state of manifold. |
| Learning Outcomes | CTPO | TOA | | Upon successful completion of the course, the students will be able to : | | | | LO - 1 : | Learn the concept of manifold | 1,2,3,4,5,6,7,8 | 1 | | LO - 2 : | Learn Riemann Manifold | 1,2,3,4,5,6,7,8 | 1 | | LO - 3 : | Recognize hypersurfaces | 1,2,3,4,5,6,7,8 | 1 | | LO - 4 : | Calculate the normal vector field and know orientation | 1,2,3,4,5,6,7,8 | 1 | | LO - 5 : | Calculate fundamental forms | 1,2,3,4,5,6,7,8 | 1 | | LO - 6 : | Make the differential calculation on hypersurfaces | 1,2,3,4,5,6,7,8 | 1 | | CTPO : Contribution to programme outcomes, TOA :Type of assessment (1: written exam, 2: Oral exam, 3: Homework assignment, 4: Laboratory exercise/exam, 5: Seminar / presentation, 6: Term paper), LO : Learning Outcome | | |
| Affine space, Euclidean space, topological manifolds, differentiable manifolds, curves on manifolds, tangent vectors and tangent space, Riemannian manifolds and covariant derivative, hypersurfaces, normal vector field in hypersurfaces, orientation, geodesics and parallelism, shape operator, Gauss transformation, calculation of the matrix of Weingarten transformation, fundamental forms and algebraic invariants of the shape operator. |
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| Course Syllabus | | Week | Subject | Related Notes / Files | | Week 1 | Affine space, Euclidean space | | | Week 2 | Topological Manifols | | | Week 3 | Differentiable Manifols | | | Week 4 | Curves on differentiable manifols, tangent vectors and tangent space | | | Week 5 | Riemannian Manifols | | | Week 6 | Kovaryant Türev | | | Week 7 | Hypersurfaces | | | Week 8 | Normal vector field on hypersurfaces | | | Week 9 | Orientation in hypersurfaces | | | Week 10 | Mid-term | | | Week 11 | Geodesics on hypersurfaces, parallelism | | | Week 12 | Weingarten Transformation | | | Week 13 | Gauss Transformation, fundamental forms | | | Week 14 | Examples of hypersurfaces and calculations | | | Week 15 | Examples of hypersurfaces and calculations | | | Week 16 | Final exam | | | |
| 1 | Diferensiyel Geometri I-II, H. Hilmi HACISALİHOĞLU, A.Ü. Fen Fakültesi Yayınları, Türkiye, 1994. | | | |
| 1 | Notes on Differential Geometry, Noel J. HICKS, Van Nostrand Reinhold Company, London, 1971. | | | |
| Method of Assessment | | Type of assessment | Week No | Date | Duration (hours) | Weight (%) | | Mid-term exam | 9 | 18/04/2022 | 1,5 | 50 | | End-of-term exam | 16 | 06/06/2022 | 1,5 | 50 | | |
| Student Work Load and its Distribution | | Type of work | Duration (hours pw) | No of weeks / Number of activity | Hours in total per term | | Yüz yüze eğitim | 4 | 14 | 56 | | Sınıf dışı çalışma | 3 | 14 | 42 | | Arasınav için hazırlık | 10 | 1 | 10 | | Arasınav | 1.5 | 1 | 1.5 | | Dönem sonu sınavı için hazırlık | 10 | 1 | 10 | | Dönem sonu sınavı | 1.5 | 1 | 1.5 | | Total work load | | | 121 |
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