Rapor Tarihi: 13.04.2026 03:08
| Course Title | Code | Language | Type | Semester | L+U Hour | Credits | ECTS |
|---|---|---|---|---|---|---|---|
| Differential Geometry II | MAT304 | Turkish | Compulsory | 6. Semester | 2 + 2 | 3.0 | 5.0 |
| Prerequisite Courses | |
| Course Level | Undergraduate |
| Mode of delivery | Lecturing |
| Course Coordinator | Doç. Dr. GÜLHAN AYAR |
| Instructor(s) | |
| Goals | The aim of this course is to teach the differential geometry of space curves both theoretically and practically; to develop students' abilities to mathematically establish, calculate, and geometrically interpret the concepts of derivative along a curve, arc parametrization, Frenet frame, curvature, and torsion. Additionally, it is aimed to strengthen the students' competencies in analyzing abstract mathematical structures and in proof-based thinking. |
| Course Content | Introduction to Curves and Basic Concepts, Derivative Along a Curve and Covariant Derivative, Arc Length and Arc Parameterization, Frenet Vector Fields and Frenet Formulas, Unit Speed and Non-Unit Speed Curves, Geometric Structure of the Frenet Roof, Planes Associated with a Curve: Osculating Plane, Normal Plane, Rectifying Plane, Geometric Interpretation of Curvature and Torsion |
| # | Öğrenme Kazanımı |
| 1 | Defines the fundamental concepts of space curves and expresses them mathematically. |
| 2 | He calculates the derivative and covariant derivative along the curve and explains their geometric meaning. |
| 3 | It calculates the arc length and arc parameter, reparameterizes a curve to be unit speed. |
| 4 | Obtains and applies Frenet vector fields (T, N, B) and Frenet formulas. |
| 5 | It calculates curvature and torsion for unit-speed and non-unit-speed curves. |
| 6 | The Frenet frame determines the Darboux vector and the associated planes (osculating, normal, rectifying). |
| 7 | It analyzes the effect of curvature and torsion on the geometric behavior of the curve in space. |
| Week | Topics/Applications | Method |
|---|---|---|
| 1. Week | Basic information about curves | Preparation, After Class Study, Research, Other Activities, Interview |
| 2. Week | The relationship between the curve and the direction | Preparation, After Class Study, Research, Other Activities, Interview |
| 3. Week | The derivative of a vector field along a curve (covariant derivative) | Preparation, After Class Study, Research, Other Activities, Interview |
| 4. Week | Spring parameter, spring length | Preparation, After Class Study, Research, Other Activities, Interview |
| 5. Week | Frame areas, Frenet vector fields | Preparation, After Class Study, Research, Other Activities, Interview |
| 6. Week | Frenet vector fields and Frenet formulas | Preparation, After Class Study, Research, Other Activities, Interview |
| 7. Week | Frenet frame for unit speed curves | Preparation, After Class Study, Research, Other Activities, Interview |
| 8. Week | general problem solving | Preparation, After Class Study, Research, Other Activities, Interview |
| 9. Week | Frenet frame for curves that are not unit speed | Preparation, After Class Study, Research, Other Activities, Interview |
| 10. Week | Frenet frame for curves that are not unit speed | Preparation, After Class Study, Research, Other Activities, Interview |
| 11. Week | Darboux vector of the Frenet frame,Darboux indicator | Preparation, After Class Study, Research, Other Activities, Interview |
| 12. Week | Finding the Frenet frame using the Gram-Schmidt method | Preparation, After Class Study, Research, Other Activities, Interview |
| 13. Week | Oscillator, Normal, Rectifier planes | Preparation, After Class Study, Research, Other Activities, Interview |
| 14. Week | Geometric interpretations of curvature and torsion | Preparation, After Class Study, Research, Other Activities, Interview |
| No | Program Requirements | Level of Contribution | |||||
|---|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | |||
| 1 | Possesses theoretical and applied knowledge of the fundamental areas of mathematics (Analysis and Function Theory, Algebra and Number Theory, Geometry, Applied Mathematics, Topology and Foundations of Mathematics, and Mathematical Logic). | ✔ | |||||
| 2 | Explains the historical development of mathematical concepts, their relationship with other branches of science, and their application areas. | ✔ | |||||
| 3 | Defines mathematical problems, selects appropriate methods, and solves them using analytical/numerical techniques. | ✔ | |||||
| 4 | Constructs mathematical expressions with logical integrity and reaches conclusions using proof methods. | ✔ | |||||
| 5 | Formulates and interprets real-life problems by performing mathematical modeling. | ✔ | |||||
| 6 | Effectively uses information technologies and mathematical software in data analysis and computation processes. | ✔ | |||||
| 7 | Takes responsibility in individual or team work; plans and executes projects. | ✔ | |||||
| 8 | Follows current developments in their field; improves themselves with a lifelong learning awareness. | ✔ | |||||
| 9 | Expresses mathematical ideas verbally and in writing clearly and in accordance with academic rules. | ✔ | |||||
| 10 | Acts in accordance with professional and academic ethical values; acts with a sense of social responsibility. | ✔ | |||||
| Program Requirements | DK1 | DK2 | DK3 | DK4 | DK5 | DK6 | DK7 |
|---|---|---|---|---|---|---|---|
| PY1 | 5 | 5 | 5 | 5 | 5 | 4 | 5 |
| PY2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| PY3 | 3 | 5 | 5 | 5 | 5 | 4 | 4 |
| PY4 | 3 | 4 | 4 | 5 | 5 | 4 | 5 |
| PY5 | 2 | 3 | 3 | 4 | 4 | 3 | 5 |
| PY6 | 1 | 2 | 2 | 2 | 2 | 1 | 2 |
| PY7 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| PY8 | 2 | 3 | 3 | 3 | 3 | 2 | 4 |
| PY9 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| PY10 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
| Ders Kitabı veya Notu |
|
|---|---|
| Diğer Kaynaklar |
|
| ECTS credits and course workload | Quantity | Duration (Hour) | Total Workload (Hour) | |
|---|---|---|---|---|
|
Ders İçi |
Class Hours | 14 | 4 | 56 |
|
Sınavlar |
Midterm | 1 | 2 | 2 |
| Homework | 14 | 2 | 28 | |
| Homework Preparation | 14 | 1 | 14 | |
| Final | 1 | 2 | 2 | |
| Practice | 14 | 1 | 14 | |
| Practice End-Of-Term | 14 | 2 | 28 | |
| Classroom Activities | 14 | 1 | 14 | |
| Total Workload | 158 | |||
| *AKTS = (Total Workload) / 25,5 | ECTS Credit of the Course | 5.0 | ||
