| Language of instruction: |
En |
| Type of course: |
Departmental Elective |
| Course Level: |
|
| 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
|
| 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. |
|
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: |
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 |