ECTS - Differential Geometry
Differential Geometry (MATH374) Course Detail
| Course Name | Course Code | Season | Lecture Hours | Application Hours | Lab Hours | Credit | ECTS |
|---|---|---|---|---|---|---|---|
| Differential Geometry | MATH374 | 6. Semester | 3 | 0 | 0 | 3 | 6 |
| Pre-requisite Course(s) |
|---|
| MATH251 |
| Course Language | English |
|---|---|
| Course Type | Compulsory Departmental Courses |
| Course Level | Bachelor’s Degree (First Cycle) |
| Mode of Delivery | Face To Face |
| Learning and Teaching Strategies | Lecture, Question and Answer, Team/Group. |
| Course Lecturer(s) |
|
| Course Objectives | This is classical differential geometry, i.e. differential geometry of curves and surfaces in space. The basic concern is the investigation of geometric properties of curves and surfaces using analysis and linear algebra. The content of the course serves as the intuitive motivation of theory of differentiable manifolds, Riemannian geometry and other modern geometries as well as the indispensable tool for mathematical modeling in classical physics and engineering applications. |
| Course Learning Outcomes |
The students who succeeded in this course;
|
| Course Content | Curves in the plane and space, curvature and torsion, global properties of plane curves, surfaces in space, the First Fundamental Form, curvatures of surfaces, Gaussian curvature and the Gauss Map, geodesics, minimal surfaces, Gauss`s Theorema Egregium, the Gauss-Bonnet Theorem. |
Weekly Subjects and Releated Preparation Studies
| Week | Subjects | Preparation |
|---|---|---|
| 1 | What is a Curve? Arc-length, Reparametrization | pp. 1-15 |
| 2 | Level Curves vs. Parametrized Curves, Curvature Plane Curves | pp. 16-34 |
| 3 | Space Curves, Simple Closed Curve | pp. 36-51 |
| 4 | The Isoperimetric Inequality, The Four Vertex Theorem, What is a Surface? | pp. 51-65 |
| 5 | Smooth Surfaces, Tangents, Normals and Orientability, Examples of Surfaces | pp. 66-82 |
| 6 | Quadric Surfaces, Triply Orthogonal Systems, Applications of the Inverse Function Theorem | pp. 84-95 |
| 7 | Lengths of Curves on Surfaces, Isometries of Surfaces, Conformal Mappings of Surfaces | pp. 97-111 |
| 8 | Surface Area, Equiareal Maps and a Theorem of Archimedes, The Second Fundamental Form | pp. 112-126 |
| 9 | The Curvature of Curves on a Surface, The Normal and Principal Curvatures, Geometric Interpretation of Principal Curvatures | pp.127-145 |
| 10 | Gaussian and Mean Curvatures, The Pseudosphere, Flat Surfaces | pp. 147-161 |
| 11 | Surfaces of Constant Mean Curvature, Gaussian Curvature of Compact Surfaces, The Gauss Map | pp. 161-169 |
| 12 | Definition and Basic Properties of Geodesics, Geodesic Equations, Geodesic on Surfaces of Revolution, Geodesics as Shortest Paths | pp. 171-196 |
| 13 | Plateau’s Problem, Examples of Minimal Surfaces, Gauss Map of a Minimal Surface | pp. 201-219 |
| 14 | Gauss’s Remarkable Theorem, The Gauss-Theorem | pp. 229-236, 247-267 |
| 15 | Review | |
| 16 | Final Exam |
Sources
| Course Book | 1. Elementary Differential Geometry, A. Pressley, Springer Verlag, 2000. |
|---|---|
| Other Sources | 2. Differential Geometry of Curves and Surfaces, M. Do Carmo, Prentice-Hall, 1976. |
| 3. Elements of Differential Geometry, R. S. Millman and G. D. Parker, Prentice-Hall, 1977. |
Evaluation System
| Requirements | Number | Percentage of Grade |
|---|---|---|
| Attendance/Participation | - | - |
| Laboratory | - | - |
| Application | - | - |
| Field Work | - | - |
| Special Course Internship | - | - |
| Quizzes/Studio Critics | - | - |
| Homework Assignments | 4 | 10 |
| Presentation | - | - |
| Project | - | - |
| Report | - | - |
| Seminar | - | - |
| Midterms Exams/Midterms Jury | 2 | 50 |
| Final Exam/Final Jury | 1 | 40 |
| Toplam | 7 | 100 |
| Percentage of Semester Work | 60 |
|---|---|
| Percentage of Final Work | 40 |
| Total | 100 |
Course Category
| Core Courses | X |
|---|---|
| Major Area Courses | |
| Supportive Courses | |
| Media and Managment Skills Courses | |
| Transferable Skill Courses |
The Relation Between Course Learning Competencies and Program Qualifications
| # | Program Qualifications / Competencies | Level of Contribution | ||||
|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | ||
| 1 | Acquires skills to use the advanced theoretical and applied knowledge obtained at the mathematics bachelors program to do further academic and scientific research in both mathematics-based graduate programs and public or private sectors. | X | ||||
| 2 | Transplants and applies the theoretical and applicable knowledge gained in their field to the secondary education by using suitable tools and devices. | X | ||||
| 3 | Acquires the skill of choosing, using and improving problem solving techniques which are needed for modeling and solving current problems in mathematics or related fields by using the obtained knowledge and skills. | X | ||||
| 4 | Acquires analytical thinking and uses time effectively in the process of deduction | X | ||||
| 5 | Acquires basic software knowledge necessary to work in the computer science related fields and together with the skills to use information technologies effectively. | X | ||||
| 6 | Obtains the ability to collect data, to analyze, interpret and use statistical methods necessary in decision making processes. | X | ||||
| 7 | Acquires the level of knowledge to be able to work in the mathematics and related fields and keeps professional knowledge and skills up-to-date with awareness in the importance of lifelong learning. | X | ||||
| 8 | Takes responsibility in mathematics related areas and has the ability to work affectively either individually or as a member of a team. | X | ||||
| 9 | Has proficiency in English language and has the ability to communicate with colleagues and to follow the innovations in mathematics and related fields. | X | ||||
| 10 | Has the ability to communicate ideas with peers supported by qualitative and quantitative data. | X | ||||
| 11 | Has professional and ethical consciousness and responsibility which takes into account the universal and social dimensions in the process of data collection, interpretation, implementation and declaration of results in mathematics and its applications. | X | ||||
ECTS/Workload Table
| Activities | Number | Duration (Hours) | Total Workload |
|---|---|---|---|
| Course Hours (Including Exam Week: 16 x Total Hours) | |||
| Laboratory | |||
| Application | |||
| Special Course Internship | |||
| Field Work | |||
| Study Hours Out of Class | 14 | 3 | 42 |
| Presentation/Seminar Prepration | |||
| Project | |||
| Report | |||
| Homework Assignments | 5 | 8 | 40 |
| Quizzes/Studio Critics | |||
| Prepration of Midterm Exams/Midterm Jury | 2 | 15 | 30 |
| Prepration of Final Exams/Final Jury | 1 | 20 | 20 |
| Total Workload | 132 | ||