| MATHEMATICS | |||||
| Bachelor | TR-NQF-HE: Level 6 | QF-EHEA: First Cycle | EQF-LLL: Level 6 | ||
| Course Code | Course Name | Semester | Theoretical | Practical | Credit | ECTS |
| MAT3005 | Differential Geometry I | Fall | 2 | 2 | 3 | 7 |
| Language of instruction: | English |
| Type of course: | Must Course |
| Course Level: | Bachelor’s Degree (First Cycle) |
| Mode of Delivery: | Face to face |
| Course Coordinator : | Assoc. Prof. MAKSAT ASHRAYYEV |
| Course Lecturer(s): |
RA DUYGU ÜÇÜNCÜ Prof. Dr. MURAT SARI Prof. Dr. NAFİZ ARICA |
| Recommended Optional Program Components: | There is no optional program component. |
| Course Objectives: | The purpose of this course is to provide students the basic concepts of the theory of curve definition and curves, and further expansions to provide information on space curves. A characteristic feature of the curves, tangent space, vector space, the transformation of derivatives, directional derivative, and the roof of Serret-Frenet curvatures, osculator, rektifiyan and normal planes, osculator (circle, sphere) and get the understanding of geometric properties of the curve by examining some special curves. Mentioning the differential properties of geometric shapes in everyday life, and to develop students' analytical thinking. Helix, involute-evolute, Bertrand pair, Monge curves, special curves, such as spherical curves is to establish relations with the students about why-why. Making practical applications as described in the discussion of topics to teach problem-solving ability. |
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The students who have succeeded in this course; 1 Be able to define affine space and frame, Euclidean space and frame, and be able to express the difference between them. 2 Be able to define topological space, topological frame and Hausdorff space. 3 Be able to take directional derivative by defining tangent space and tangent vector. 4 Be able to define and apply derivative transformation 5 Be able to express space curve definition. 6 Be able to generate Frenet-Serret frame by calculating normal, tangent and binormal vector field of curves. 7 Be able to define and apply Helix, involute-evolute, Bertrand and Monge curves 8 Be able to define and characterization of spherical curves. |
| Affine and Euclidean space and frame, Topological Space, Hausdorff Space, Topological Manifold, Tangent Space, directional derivative, derivative transformation, Curve definition and specialities of curvesFrenet-Serret derivative formulas, Osculating, Rectifying and Normal planes, Helix, involute-evolute, Bertrand pairMonge curve, Spherical curves, Curves on E^n and their characterization |
| Week | Subject | Related Preparation |
| 1) | ||
| 1) | Vector spaces, R^n standard real vector space, inner product spaces, orthonormal vector systems | |
| 2) | Linear transformations, Linear transformations and matrices | |
| 3) | Affine space, Euclidean space and their frames, coordinate functions and system. Reminding topological space, continuity and homeomorphism. Hausdorff space and metric space and its relation with the space E^n | |
| 4) | Definitions and examples of a topological manifold. Differentiable functions and components of a function (coordinate functions). Diffeomorphism and examples. | |
| 5) | The tangent vector and tangent vector spaces, algebra of differentiable functions, vector fields and spaces function theorem and its applications, Directional derivative definition. Tangent vector and vector space application. | |
| 6) | Directional derivative theorems and applications; vector field derivative that theorem and its applications. Derivative transformation and its applications. Definition of the curve. | |
| 7) | Tangent space of the curve, velocity vector, scalar speed; parameter transform, theorems, results and examples of the curve arc length, arc parameter and related theorems. | |
| 8) | Vector fields on the curve, its derivative, and related theorems. Covariant derivative and related theorems and examples. Vector field on a curved design | |
| 9) | Serret-Frenet formulas and derivatives roof of the unit-speed curves. Frenet vectors and planes at some point in the curve. | |
| 10) | Curvature and torsion and related theorems for geometric interpretation. Definition of contact. Osculator circle of the curve. | |
| 11) | Definition osculator sphere, and finding the center and the radius of the osculator sphere. Finding the Frenet frame and its curvature calculation. | |
| 12) | Special curves helix (trend line), definitions and theorems of the helix. Special curves circular cylinder, evolute-involute curve equations and their properties. | |
| 13) | Definition and equation of Bertrand curve pair. Finding elements of Bertrand curve pair of a curve. Definitions, theorems and results of Monge curve and spherical curves. | |
| 14) | Manifolds. Characterization for lines of curvature in n-dimensional Euclid space, harmonic curvature and related theorems. |
| Course Notes / Textbooks: | 1) O’Neill, B., Elementary Differential Geometry, Academic Press, New York, 1966. 2) Hacısalihoğlu, H. H. , Diferensiyel Geometri, MEB Yayınları, 1983. 3) Hacısalihoğlu, H. H. Yüksek Boyutlu Uzaylarda Dönüşümler ve Geometriler, İnönü Üniversitesi Yayınları, Mat.NO: 1 ,1980 |
| References: | 1) Hacısalihoğlu, H. H. , Diferensiyel Geometri, MEB Yayınları, 1983. 2) Hacısalihoğlu, H. H. Yüksek Boyutlu Uzaylarda Dönüşümler ve Geometriler, İnönü Üniversitesi Yayınları, Mat.NO: 1 ,1980 |
| Semester Requirements | Number of Activities | Level of Contribution |
| Attendance | 14 | % 0 |
| Quizzes | 2 | % 5 |
| Homework Assignments | 2 | % 5 |
| Midterms | 2 | % 50 |
| Final | 1 | % 40 |
| Total | % 100 | |
| PERCENTAGE OF SEMESTER WORK | % 60 | |
| PERCENTAGE OF FINAL WORK | % 40 | |
| Total | % 100 | |
| Activities | Number of Activities | Duration (Hours) | Workload |
| Course Hours | 14 | 3 | 42 |
| Study Hours Out of Class | 14 | 4 | 56 |
| Homework Assignments | 2 | 5 | 10 |
| Quizzes | 2 | 5 | 10 |
| Midterms | 2 | 2 | 4 |
| Final | 2 | 21 | 42 |
| Total Workload | 164 | ||
| No Effect | 1 Lowest | 2 Low | 3 Average | 4 High | 5 Highest |
| Program Outcomes | Level of Contribution | |
| 1) | To have a grasp of basic mathematics, applied mathematics and theories and applications in Mathematics | 5 |
| 2) | To be able to understand and assess mathematical proofs and construct appropriate proofs of their own and also define and analyze problems and to find solutions based on scientific methods, | 5 |
| 3) | To be able to apply mathematics in real life with interdisciplinary approach and to discover their potentials, | 4 |
| 4) | To be able to acquire necessary information and to make modeling in any field that mathematics is used and to improve herself/himself, | 4 |
| 5) | To be able to tell theoretical and technical information easily to both experts in detail and non-experts in basic and comprehensible way, | 4 |
| 6) | To be familiar with computer programs used in the fields of mathematics and to be able to use at least one of them effectively at the European Computer Driving Licence Advanced Level, | 3 |
| 7) | To be able to behave in accordance with social, scientific and ethical values in each step of the projects involved and to be able to introduce and apply projects in terms of civic engagement, | 3 |
| 8) | To be able to evaluate all processes effectively and to have enough awareness about quality management by being conscious and having intellectual background in the universal sense, | 3 |
| 9) | By having a way of abstract thinking, to be able to connect concrete events and to transfer solutions, to be able to design experiments, collect data, and analyze results by scientific methods and to interfere, | 5 |
| 10) | To be able to continue lifelong learning by renewing the knowledge, the abilities and the competencies which have been developed during the program, and being conscious about lifelong learning, | 4 |
| 11) | To be able to adapt and transfer the knowledge gained in the areas of mathematics ; such as algebra, analysis, number theory, mathematical logic, geometry and topology to the level of secondary school, | 5 |
| 12) | To be able to conduct a research either as an individual or as a team member, and to be effective in each related step of the project, to take role in the decision process, to plan and manage the project by using time effectively. | 4 |