MATHEMATICS (TURKISH, PHD) | |||||
PhD | TR-NQF-HE: Level 8 | QF-EHEA: Third Cycle | EQF-LLL: Level 8 |
Course Code | Course Name | Semester | Theoretical | Practical | Credit | ECTS |
MAT3006 | Differential Geometry II | Fall | 2 | 2 | 3 | 6 |
The course opens with the approval of the Department at the beginning of each semester |
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): |
Prof. Dr. MURAT SARI |
Course Objectives: | The aim of the course is to explain differential geometry and related basic concepts to the students. Inform students about surface definition and surface theory theorems. By examining the Gauss transformation, normal curvature, prime curvature and other curvatures, which are characteristic properties of surfaces, improve students ability on differential geometry and related subjects. To ensure students to improve their analytical ability by mentioning about the surfaces of geometric bodies which are in daily life. Also to ensure students to understand the difference between the surface structure and translation relation by mentioning surface ranges as ruled surface and isometric surface. To ensure students to obtain ability of solving examples easily by making practice lessons. |
The students who have succeeded in this course; Be able to define surface on space Be able to define and calculate, operator and curvature of a surface Be able to calculate basic forms of surfaces. Be able to identify the kind of the curve on the surface. Be able to express Darboux Ribaucour frame and these frame’s derivative formulas by making geometric interpretation. Be able to define isometric, ruled and developable surfaces. |
Surface definition, Shape operator of surface, Gauss Transformation, Normal and prime curvatures , Mean and Gauss curvatures, Asymptotic, prime and geodesic curvatures, isometric spaces,Ruled spaces |
Week | Subject | Related Preparation | |
1) | Definitions and examples of surfaces on E^3. Critical point and its value. Connected surface. Theorem related with diffeomorphism. Parameter curves and tangent vectors (and its relation with derivative transformation). | ||
1) | |||
2) | Tangent space of a surface, Differentiable functions on a surface. Theorems about tangent space of a surface. Vertical vector field of a surface and gradient vector field grad f . | ||
3) | Directional derivative of a differentiable function f on a surface in the direction of the tangent of a curve on surface. Theorems and examples about w(f) . Surface P eleman M and algebra diferentiable functions. Functions F^* , F* on a manifold (surface) and their relations, theorems. Jacobian matrix J(f)(P). Vector field and vector field’s Lie Algebra on a surface. | ||
4) | Cotangent vectors, cotangent vector fields, differential of a function, total differential, Hessian form , covariant derivative and properties of it on a manifold, definition of shape operator. | ||
5) | Shape operator, Gauss transformation and finding shape operator of a surface. Shape operator matrix. Shape operator of plane and sphere. Relation between shape operator of a surface and Gauss transformation. 1st basic form of a surface, arc length of a curve on a surface, angle between two curve on a surface. | ||
6) | Differential equation of surface‘s vertical paths, vertical intersection condition of parameter curves, definition of 2nd and 3rd basic forms, theorems about curvature’s of surfaces. | ||
7) | Curvature of a surface in the direction of an element of the tangent space, normal section curvature of surface, results based on normal section curvature and Meusnier theorem, other expressions of normal section curvature. Prime vectors and curvatures. | ||
8) | Umbilic point, examples of umbilic points, prime curvature on umbilic point and Euler formula. Quadric approach of surface, Gauss curvature of a surface and definition of mean curvature. Conjugate vectors, asymptotic vectors, prime vectors and related theorems. | ||
9) | Linear correlation between 1st , 2nd and 3rd basic forms of surface. Olin de Rodrigues formula. Dupin indicator and its interpretation and benefits. | ||
10) | Expressing Mean and Gauss(Total) curvature in terms of the coefficients of the basic forms of the surface. Differential equations of prime curves of a surface. Definition of asymptotic curve of a surface M, type of the points on a surface and the position of asymptotic points on a surface. Other definitions of asymptotic curves. | ||
11) | Conjugate nets, definition and differential equation of conjugate direction, envelopes, translation surfaces, definition and differential equation of prime curves and related theorems. | ||
12) | Definition of geodesic curve of surface, finding geodesics of some structures like sphere and planes, differential equation of geodesic curves. Darboux Ribaucour frame on surface, this frame’s derivative formulas and geometric interpretation. | ||
13) | Spherical representation of surface, Beltrami formula of torsion of asymptotic lines, definition and examples of isometric surfaces. | ||
14) | Definition of comfort transformation, introduction to ruled surfaces and congruences. |
Course Notes: | 1) O’Neill, B., Elementary Differential Geometry, Academic Press, New York, 1966. |
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 | 0 | % 0 |
Laboratory | 0 | % 0 |
Application | 0 | % 0 |
Field Work | 0 | % 0 |
Special Course Internship (Work Placement) | 0 | % 0 |
Quizzes | 2 | % 5 |
Homework Assignments | 2 | % 5 |
Presentation | 0 | % 0 |
Project | 0 | % 0 |
Seminar | 2 | % 50 |
Midterms | 2 | % 50 |
Preliminary Jury | 1 | % 40 |
Final | 1 | % 40 |
Paper Submission | 0 | % 0 |
Jury | 0 | % 0 |
Bütünleme | % 0 | |
Total | % 190 | |
PERCENTAGE OF SEMESTER WORK | % 150 | |
PERCENTAGE OF FINAL WORK | % 40 | |
Total | % 190 |
Activities | Number of Activities | Duration (Hours) | Workload |
Course Hours | 14 | 3 | 42 |
Laboratory | 0 | 0 | 0 |
Application | 0 | 0 | 0 |
Special Course Internship (Work Placement) | 0 | 0 | 0 |
Field Work | 0 | 0 | 0 |
Study Hours Out of Class | 0 | 0 | 0 |
Presentations / Seminar | 0 | 0 | 0 |
Project | 2 | 15 | 30 |
Homework Assignments | 2 | 20 | 40 |
Quizzes | 2 | 5 | 10 |
Preliminary Jury | 0 | 0 | 0 |
Midterms | 2 | 5 | 10 |
Paper Submission | 0 | 30 | 0 |
Jury | 0 | 0 | 0 |
Final | 1 | 30 | 30 |
Total Workload | 162 |
No Effect | 1 Lowest | 2 Low | 3 Average | 4 High | 5 Highest |
Program Outcomes | Level of Contribution |