Syllabus ( MATH 690 )
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Basic information
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| Course title: |
Exterior Form Analysis |
| Course code: |
MATH 690 |
| Lecturer: |
Prof. Dr. Oğul ESEN
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| ECTS credits: |
7.5 |
| GTU credits: |
3 (3+0+0) |
| Year, Semester: |
1/2, Fall and Spring |
| Level of course: |
Third Cycle (Doctoral) |
| Type of course: |
Area Elective
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| Language of instruction: |
English
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| Mode of delivery: |
Face to face
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| Pre- and co-requisites: |
none |
| Professional practice: |
No |
| Purpose of the course: |
The purpose of this course is to teach student computations of interior, exterior and Lie derivatives of the tensor fields on manifolds as well as the computations of the integrals of exterior forms. In order to establish the required geometrical framework, detail analysis of the tangent and the cotangent bundles of manifolds will be presented. |
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Learning outcomes
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Upon successful completion of this course, students will be able to:
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Identify differential manifolds, tangent and cotangent bundles.
Contribution to Program Outcomes
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Define and manipulate advanced concepts of Mathematics in a specialized way
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Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems
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Gain original, independent and critical thinking, and develop theoretical concepts and tools,
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Acquire scientific knowledge and work independently,
Method of assessment
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Written exam
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Homework assignment
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2. Compute the interior, exterior and Lie derivatives of tensor fields
Contribution to Program Outcomes
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Define and manipulate advanced concepts of Mathematics in a specialized way
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Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems
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Gain original, independent and critical thinking, and develop theoretical concepts and tools,
-
Acquire scientific knowledge and work independently,
Method of assessment
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Written exam
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Homework assignment
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Perform integration of differential forms on the manifolds
Contribution to Program Outcomes
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Define and manipulate advanced concepts of Mathematics in a specialized way
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Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems
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Gain original, independent and critical thinking, and develop theoretical concepts and tools,
-
Acquire scientific knowledge and work independently,
Method of assessment
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Written exam
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Homework assignment
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Contents
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| Week 1: |
Basic concepts and motivation |
| Week 2: |
Differentiable manifolds |
| Week 3: |
Differentiable mappings, submanifolds |
| Week 4: |
Tangent bundle, vector fields |
| Week 5: |
Jacobi-Lie bracket of vector fields |
| Week 6: |
Cotangent bundle |
| Week 7: |
Differential forms |
| Week 8: |
Interior and exterior derivative |
| Week 9: |
Lie derivative |
| Week 10: |
Homotopy operator. Midterm Exam. |
| Week 11: |
Poincaré Lemma |
| Week 12: |
Canonical coordinates, Darboux theorem |
| Week 13: |
Chains and simplices, integration fo differential forms |
| Week 14: |
Stokes’ theorem |
| Week 15*: |
- |
| Week 16*: |
Final Exam. |
| Textbooks and materials: |
• Abraham, R., Marsden, J. E., & Ratiu, T. (1993). Manifolds, Tensor Analysis, and Applications (Vol. 75). Springer Science & Business Media.
• Dubrovin, B. A., Fomenko, A. T., & Novikov, S. P. (2012). Modern geometry—methods and applications: Part II. Springer Science & Business Media.
• Spivak, M. (1999). A Comprehensive Introduction to Differential Geometry Vol. I. Publish or Perish.
• Şuhubi, E. (2013). Exterior Analysis. Elsevier. |
| Recommended readings: |
• Darling, R. W. R. (1994). Differential forms and connections. Cambridge University Press.
• Flanders, H. (1963). Differential Forms with Applications to the Physical Sciences. Elsevier.
• Nakahara, M. (2003). Geometry, topology and physics. CRC Press. |
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* Between 15th and 16th weeks is there a free week for students to prepare for final exam.
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Assessment
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| Method of assessment |
Week number |
Weight (%) |
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| Mid-terms: |
10 |
30 |
| Other in-term studies: |
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0 |
| Project: |
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0 |
| Homework: |
4,7,13 |
30 |
| Quiz: |
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0 |
| Final exam: |
16 |
40 |
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Total weight: |
(%) |
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Workload
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| Activity |
Duration (Hours per week) |
Total number of weeks |
Total hours in term |
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| Courses (Face-to-face teaching): |
3 |
14 |
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| Own studies outside class: |
3 |
14 |
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| Practice, Recitation: |
0 |
0 |
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| Homework: |
16 |
3 |
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| Term project: |
0 |
0 |
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| Term project presentation: |
0 |
0 |
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| Quiz: |
0 |
0 |
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| Own study for mid-term exam: |
20 |
1 |
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| Mid-term: |
3 |
1 |
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| Personal studies for final exam: |
30 |
1 |
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| Final exam: |
3 |
1 |
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Total workload: |
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Total ECTS credits: |
* |
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* ECTS credit is calculated by dividing total workload by 25.
(1 ECTS = 25 work hours)
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