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| MATL7911 | @ | 3+0+0 | ECTS:7.5 | | Year / Semester | Spring Semester | | Level of Course | Third Cycle | | Status | Elective | | Department | DEPARTMENT of MATHEMATICS | | Prerequisites and co-requisites | None | | Mode of Delivery | | | Contact Hours | 14 weeks - 3 hours of lectures per week | | Lecturer | Doç. Dr. Filiz OCAK | | Co-Lecturer | | | Language of instruction | Turkish | | Professional practise ( internship ) | None | | | | The aim of the course: | | This course aims to introduce the concept of "tensor", which is frequently used in fields such as theoretical mathematics and physics, and to give algebraic operations on this subject.
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| Programme Outcomes | CTPO | TOA | | Upon successful completion of the course, the students will be able to : | | | | PO - 1 : | Learn the concepts of multilinear functions and dual spaces | 1,2,8 | 1, | | PO - 2 : | Knows algebraic operations on tensors | 1,2,8 | 1, | | PO - 3 : | Recognize mixed type tensors | 1,2,8 | 1, | | PO - 4 : | Knows the concepts of Contraction, Symmetry and Alternating Operator. | 1,2,8 | 1, | | CTPO : Contribution to programme outcomes, TOA :Type of assessment (1: written exam, 2: Oral exam, 3: Homework assignment, 4: Laboratory exercise/exam, 5: Seminar / presentation, 6: Term paper), PO : Learning Outcome | | |
| Linear functional, Dual vector space, Dual Transforms, Tensors and multilinear functions, Tensorial multiplication of vector spaces, Covariant tensors, Components of covariant tensor, Contravariant tensors, Components of contravariant tensor, Mixed Tensors, Components of mixed tensors, Base change, Contraction function (Contraction Operator), Symmetric and Alternating tensors, External multiplication, Parallel vectors in the sense of Levi-Civita parallelism, Riemannian structure, Covariant derivative, Concept of Connection, Riemann Curvature Tensor, Christoffel symbols, First and Second field Bianchi identities, Arbitrary tensor covariant derivative, Lie derivative
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| Course Syllabus | | Week | Subject | Related Notes / Files | | Week 1 | Linear Functional and Dual Vector (Covector) Space | | | Week 2 | Tensors, Tensorial Products of Vector Spaces | | | Week 3 | Covariant and Contravariant Tensors | | | Week 4 | Mixed Tensors and Components | | | Week 5 | Contraction Function (Contraction Operator) | | | Week 6 | Symmetric Tensors and Symmetrizing Operator, Symmetric Multiplication | | | Week 7 | Alternating Tensors and Alternating Operators | | | Week 8 | Outer Product Space | | | Week 9 | Mid term exam | | | Week 10 | Parallel Vector Fields, Parallelism in the Levi-Civita Sense | | | Week 11 | Riemannian Structure, Covariant Derivative | | | Week 12 | Concept of Connection, Riemann Curvature Tensor | | | Week 13 | Christoffel Symbols, First and Second Bianchi identities | | | Week 14 | Covariant Derivative of Arbitrary Tensor Fields | | | Week 15 | Lie differentation | | | Week 16 | end of term exam | | | |
| 1 | Hacısalihoglu, H.H., Ekmekci, F.N., Tensör Geometri, Ankara Üniversitesi Fen Fakültesi, 2003 | | | |
| 1 | Bishop, R.L., Goldberg, S.I., Tensor Analysis on Manifolds, The Macmillan Company, New Yor | | | |
| Method of Assessment | | Type of assessment | Week No | Date | Duration (hours) | Weight (%) | | Mid-term exam | 9 | 16/04/2025 | 1 | 30 | | In-term studies (second mid-term exam) | 11 | 30.04.2025 | 1 | 20 | | End-of-term exam | 16 | 13.06.2025 | 2 | 50 | | |
| Student Work Load and its Distribution | | Type of work | Duration (hours pw) | No of weeks / Number of activity | Hours in total per term | | Yüz yüze eğitim | 3 | 14 | 42 | | Sınıf dışı çalışma | 10 | 14 | 140 | | Arasınav | 1 | 1 | 1 | | Dönem sonu sınavı için hazırlık | 8 | 5 | 40 | | Dönem sonu sınavı | 2 | 1 | 2 | | Total work load | | | 225 |
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