| 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). |
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| 1) |
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| 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 . |
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| 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. |
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| 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. |
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| 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. |
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| 13) |
Spherical representation of surface, Beltrami formula of torsion of asymptotic lines, definition and examples of isometric surfaces.
|
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| 14) |
Definition of comfort transformation, introduction to ruled surfaces and congruences.
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| Course Notes / Textbooks: |
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 |
| |
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 |