Rapor Tarihi: 13.04.2026 03:06
| Course Title | Code | Language | Type | Semester | L+U Hour | Credits | ECTS |
|---|---|---|---|---|---|---|---|
| Differential Geometry I | MAT303 | Turkish | Compulsory | 5. 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 students the fundamental concepts related to differentiable functions and vector fields in Euclidean and affine spaces. Within the scope of the course, advanced concepts such as differential operators like gradient, divergence, curl, and Laplace operators, tangent and cotangent vectors, Lie operators and algebras, as well as differential forms and exterior derivatives will be covered. The course aims to equip students with both theoretical and practical problem-solving skills. |
| Course Content | The course begins with the introduction of the fundamental structures and coverings of Euclidean and affine spaces. Differentiable functions and their examination thru tangent vectors are discussed. The analysis of vector fields is conducted thru gradient, divergence, and curl (rotational) operators, as well as Laplace operators. The directional derivative, the derivative of a function along a vector field, and the derivatives of vector fields along each other are taught. Derivative transformations and regular transformations with the Jacobian matrix are explained with examples. The Lie operator and Lie algebras are introduced, and the concepts of cotangent vector and cotangent vector field are covered. The course is reinforced by focusing on differential operators, differential forms, the wedge product, and the exterior derivative, along with general problem-solving and applied problem analysis. |
| # | Öğrenme Kazanımı |
| 1 | Explains the basic concepts of Euclidean and affine spaces and the properties of their structures. |
| 2 | Differentiable functions and tangent vectors interpret their functions. |
| 3 | Analyzes vector fields and applies gradient, divergence, curl, and Laplace operators. |
| 4 | Calculates and solves the derivatives of functions and vector fields along each other. |
| 5 | Performs regular transformations using derivative transformations and the Jacobian matrix. |
| 6 | Studies and analyzes structures related to Lie operators and Lie algebras. |
| 7 | Produces differential problem solutions using the cotangent vector and cotangent vector field. |
| 8 | Differential forms, the concept of the wedge product, and the exterior derivative are applied, and general vector analysis problems are solved. |
| Week | Topics/Applications | Method |
|---|---|---|
| 1. Week | Euclidean Space, Affine Space, Euclidean Affine Space | Preparation, After Class Study, Research, Other Activities, Interview |
| 2. Week | Differentiable Functions | Preparation, After Class Study, Research, Other Activities, Interview |
| 3. Week | Tangent Vectors | Preparation, After Class Study, Research, Other Activities, Interview |
| 4. Week | Vector Fields | Preparation, After Class Study, Research, Other Activities, Interview |
| 5. Week | Gradient, Divergence Operators | Preparation, After Class Study, Research, Other Activities, Interview |
| 6. Week | Curl (Rotational), Laplace Operators | Preparation, After Class Study, Research, Other Activities, Interview |
| 7. Week | Directional Derivative | Preparation, After Class Study, Research, Other Activities, Interview |
| 8. Week | general problem solving | |
| 9. Week | The Derivative of a Function Along a Vector Field, The Derivative of a Vector Field Along a Vector Field | Preparation, After Class Study, Research, Other Activities, Interview |
| 10. Week | Derivative Transformation, Finding Thru the Jacobian Matrix, Regular Transformation | Preparation, After Class Study, Research, Other Activities, Interview |
| 11. Week | Lie Operator and Lie Algebra | Preparation, After Class Study, Research, Other Activities, Interview |
| 12. Week | Cotangent Vector and Cotangent Vector Field | Preparation, After Class Study, Research, Other Activities, Interview |
| 13. Week | Differential Operator, Differential Forms | Preparation, After Class Study, Research, Other Activities, Interview |
| 14. Week | Cross Product, Exterior Derivative | 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 | DK8 |
|---|---|---|---|---|---|---|---|---|
| PY1 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 |
| PY2 | 2 | 2 | 3 | 2 | 3 | 3 | 2 | 3 |
| PY3 | 4 | 5 | 5 | 5 | 5 | 4 | 5 | 5 |
| PY4 | 4 | 5 | 5 | 5 | 5 | 5 | 5 | 5 |
| PY5 | 4 | 5 | 5 | 5 | 5 | 4 | 5 | 5 |
| PY6 | 1 | 2 | 3 | 3 | 3 | 2 | 3 | 3 |
| PY7 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| PY8 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| PY9 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| PY10 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| Ders Kitabı veya Notu | Ders Kitabı veya Ders Notu bulunmamaktadır. |
|---|---|
| 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 | 2 | 28 | |
| Final | 1 | 2 | 2 | |
| Practice | 14 | 1 | 14 | |
| Practice End-Of-Term | 14 | 1 | 14 | |
| Classroom Activities | 14 | 1 | 14 | |
| Total Workload | 158 | |||
| *AKTS = (Total Workload) / 25,5 | ECTS Credit of the Course | 5.0 | ||
