ECTS - Differential Geometry
Differential Geometry (MATH374) Course Detail
| Course Name | Course Code | Season | Lecture Hours | Application Hours | Lab Hours | Credit | ECTS |
|---|---|---|---|---|---|---|---|
| Differential Geometry | MATH374 | 3 | 0 | 0 | 3 | 6 |
| Pre-requisite Course(s) |
|---|
| MATH 251 Advanced Calculus I |
| Course Language | English |
|---|---|
| Course Type | N/A |
| 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;
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| 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 | Has the ability to apply scientific knowledge gained in the undergraduate education and to expand and extend knowledge in the same or in a different area | X | ||||
| 2 | Can apply gained knowledge and problem solving abilities in inter-disciplinary research | X | ||||
| 3 | Has the ability to work independently within research area, to state the problem, to develop solution techniques, to solve the problem, to evaluate the obtained results and to apply them when necessary | X | ||||
| 4 | Takes responsibility individually and as a team member to improve systematic approaches to produce solutions in unexpected complicated situations related to the area of study | X | ||||
| 5 | Can develop strategies, implement plans and principles on the area of study and can evaluate obtained results within the framework | X | ||||
| 6 | Can develop and extend the knowledge in the area and to use them with scientific, social and ethical responsibility | X | ||||
| 7 | Has the ability to follow recent developments within the area of research, to support research with scientific arguments and data, to communicate the information on the area of expertise in a systematically by means of written report and oral/visual presentation | X | ||||
| 8 | To have an oral and written communication ability in at least one of the common foreign languages ("European Language Portfolio Global Scale", Level B2) | X | ||||
| 9 | Has software and hardware knowledge in the area of expertise, and has proficient information and communication technology knowledge | X | ||||
| 10 | Follows scientific, cultural, and ethical criteria in collecting, interpreting and announcing data in the research area and has the ability to teach. | X | ||||
| 11 | Has professional 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 | ||