MATH505 DIFFERENTIABLE MANIFOLDS
| Course Code: | 2360505 |
| METU Credit (Theoretical-Laboratory hours/week): | 3 (3.00 - 0.00) |
| ECTS Credit: | 8.0 |
| Department: | Mathematics |
| Language of Instruction: | English |
| Level of Study: | Graduate |
| Course Coordinator: | Prof.Dr. MEHMETCİK PAMUK |
| Offered Semester: | Fall Semesters. |
Course Objectives
At the end of the course the student will learn
- the basic notions about manifolds and differentiability to use in his/her research or in the other areas like differential geometry, topology, analysis, applied mathematics, algebraic geometry, physics etc.
- basic notions about vector fields, tensor fields, differential forms and the methods to do calculations related to these concepts.
- the concept of integration on manifolds and basic theorems about it.
Course Content
Differentiable manifolds, smooth mappings, tangent, cotangent bundles, differential of a map, submanifolds, immersions, imbeddings, vector fields, tensor fields, differential forms, orientation on manifolds, integration on manifolds, Stoke's theorem.
Course Learning Outcomes
After completing this course succesfully, the student will know
- the basic definitions about manifolds, differentiable mappings, tangent space at a point and differentials of smooth mappings,
- what immersion, imbedding and submanifold means,
- what a vector field is and the meaning of integrability of vector fields,
- basic definitions and computational tools about tensor fields, differential forms, exterior differentiation and some computational techniques concerning these concepts,
- orientation of manifolds, volume form,
- the definition of integration on manifolds,
- the concept of manifolds with boundary,
- Stokes' Theorem on manifolds with boundary and the relation to other important theorems of Calculus.
Program Outcomes Matrix
| Level of Contribution | |||||
| # | Program Outcomes | 0 | 1 | 2 | 3 |
| 1 | Acquires mathematical thinking skills (problem solving, generating ways of thinking, forming correspondence, generalizing etc.) and can use them in related fields. | ✔ | |||
| 2 | Gains academic maturity through self-study. | ✔ | |||
| 3 | Can design mathematics related problems, devise solution methods and apply them when appropriate. | ✔ | |||
| 4 | Carries out parts of a mathematical research program independently. | ✔ | |||
| 5 | Has a command of Turkish and English languages so that he/she can actively communicate (read, write, listen and speak). | ✔ | |||
| 6 | Contributes to solving global, environmental and social problems either individually or as being part of a social group. | ✔ | |||
| 7 | Respects ethical values and rules; applies them in professional and social issues. | ✔ | |||
| 8 | Can work cooperatively in a team and also individually. | ✔ | |||
| 9 | Gets exposed to academic culture through interaction with others. | ✔ | |||
| 10 | Comprehends necessity of knowledge, can define it and acquires it; uses knowledge effectively and shares it with others. | ✔ | |||
0: No Contribution 1: Little Contribution 2: Partial Contribution 3: Full Contribution