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Information
| Code | MT321 |
| Name | Differential Geometry |
| Semester | 5. Semester |
| Duration (T+A) | 4-0 (T-A) (17 Week) |
| ECTS | 8 ECTS |
| National Credit | 4 National Credit |
| Teaching Language | Türkçe |
| Level | Lisans Dersi |
| Type | Normal |
| Mode of study | Yüz Yüze Öğretim |
| Catalog Information Coordinator | Dr. Öğr. Üyesi NERGİZ POYRAZ |
Course Goal
To teach theories and applications about classical and generalized Stokes theorems, to give basic knowledge of curves and surfaces theories, to gain the ability of using analytical geometry, vector calculus and linear algebra knowledge, to teach understanding of abstract mathematical concepts and abstract thinking.
Course Content
Classical Stokes theorem and some applications, diferential forms and pull-back of diferential forms under diferentiable functions, Generalized Stokes theorem, curves and characterization of curves by curvature and torsion, Diferentiable surfaces and ruled surfaces.
Course Precondition
None
Resources
Diferansiyel Geometri, H. Hilmi hacısalihoğlu
Notes
1) Calculus and Analytic geometry, Authors : Shermann K. Stein, Anthony Barcellos. 2) A geometric Approach to differential forms, Authors : David Bachman. 3) Differential Geometry (Schaum's outline series), Authors : Martin M. Lipschutz.
Course Learning Outcomes
| Order | Course Learning Outcomes |
|---|---|
| LO01 | Is able to solve the problems about classical Stokes theorem. |
| LO02 | Is able to explain the concept of differential forms in space and cubical simplexes. |
| LO03 | Is able to explain the generalized Stokes theorem |
| LO04 | Knows the basic theorems about space curves. |
| LO05 | Knows the basic theorems about differentiable surfaces. |
Relation with Program Learning Outcome
| Order | Type | Program Learning Outcomes | Level |
|---|---|---|---|
| PLO01 | Bilgi - Kuramsal, Olgusal | Comprehend the ability to prove the mathematical knowledge gained in secondary education on the basis of theoretical basis. | 5 |
| PLO02 | Bilgi - Kuramsal, Olgusal | Understands importance of basic consepts of Algebra, Analaysis and Topology. | 4 |
| PLO03 | Yetkinlikler - Öğrenme Yetkinliği | Mathematical reasoning demonstrates the ability to develop and write mathematical proofs by gaining maturity. | |
| PLO04 | Bilgi - Kuramsal, Olgusal | Demonstrate the ability to express the basic theories of mathematics both correctly. | |
| PLO05 | Bilgi - Kuramsal, Olgusal | Understands the relationship between the different fields of mathematics and its relation to other disciplines. | 3 |
| PLO06 | Bilgi - Kuramsal, Olgusal | Comprehends the ability to understand the relationships between the objects in the most understandable way while creating a model for any problem. | 4 |
| PLO07 | Bilgi - Kuramsal, Olgusal | Comprehend and explain mathematical models such as formulas, graphs, tables and schema. | |
| PLO08 | Bilgi - Kuramsal, Olgusal | Demonstrate the ability to mathematically rearrange, analyze, and model the problems they encounter. | 5 |
| PLO09 | Bilgi - Kuramsal, Olgusal | Comprehends at least one of the computer programming languages. | |
| PLO10 | Bilgi - Kuramsal, Olgusal | Demonstrate the ability to use scientific methods and appropriate technologies effectively in problem solving. | 5 |
| PLO11 | Yetkinlikler - Bağımsız Çalışabilme ve Sorumluluk Alabilme Yetkinliği | Understands sufficient knowledge of foreign language to be able to understand Mathematical concepts and communicate with other mathematicians | |
| PLO12 | Yetkinlikler - Bağımsız Çalışabilme ve Sorumluluk Alabilme Yetkinliği | In addition to their professional development, they demonstrate their ability to continuously improve themselves by identifying their educational needs in scientific, cultural, artistic and social areas in line with their interests and abilities. | 5 |
| PLO13 | Yetkinlikler - Öğrenme Yetkinliği | Understands the programming techniques and shows the ability to do programming. | 3 |
| PLO14 | Yetkinlikler - Öğrenme Yetkinliği | Demonstrates the ability to study mathematics both independently and as a group. | |
| PLO15 | Bilgi - Kuramsal, Olgusal | Demonstrate an awareness of the universal and social impacts and legal consequences of mathematical applications in the field of study. | 3 |
| PLO16 | Bilgi - Kuramsal, Olgusal | Demonstrate the ability to select, use and develop effectively for contemporary mathematical applications. | 5 |
| PLO17 | Bilgi - Kuramsal, Olgusal | It has ability of lifelong learning awareness, access to information, monitoring developments in science and technology and self-renewal ability. | |
| PLO18 | Bilgi - Kuramsal, Olgusal | Gains the ability to use information technologies effectively for contemporary mathematical applications. | 4 |
| PLO19 | Bilgi - Kuramsal, Olgusal | Gains the ability to design, conduct experiments, field work, data collection, analysis, archiving, text solving and / or interpretation according to mathematics fields. | 5 |
| PLO20 | Bilgi - Kuramsal, Olgusal | Gains the consciousness of prefesional ethics and responsibility. | 5 |
Week Plan
| Week | Topic | Preparation | Methods |
|---|---|---|---|
| 1 | A brief introduction to Green s theorem, divergence theorem and surface integral | Review of the relevant pages from sources | Öğretim Yöntemleri: Anlatım |
| 2 | Classical Stokes theorem | Review of the relevant pages from sources | Öğretim Yöntemleri: Anlatım |
| 3 | Differential forms and exterior derivative of differential forms | Review of the relevant pages from sources | Öğretim Yöntemleri: Anlatım |
| 4 | Pull back of diferential forms under differentiable functions | Review of the relevant pages from sources | Öğretim Yöntemleri: Anlatım |
| 5 | Generalised Stokes theorem | Review of the relevant pages from sources | Öğretim Yöntemleri: Anlatım |
| 6 | The theory of curves and reparametrization by arc length | Review of the relevant pages from sources | Öğretim Yöntemleri: Anlatım |
| 7 | Curvature, torsion and Frenet-Serre equations | Review of the relevant pages from sources | Öğretim Yöntemleri: Anlatım |
| 8 | Mid-Term Exam | Review of the topics discussed in the lecture notes and sources | Ölçme Yöntemleri: Yazılı Sınav |
| 9 | Central curves, helices and involutes | Review of the relevant pages from sources | Öğretim Yöntemleri: Anlatım |
| 10 | Isometries and isometry group of space | Review of the relevant pages from sources | Öğretim Yöntemleri: Anlatım |
| 11 | Characterization of a curve by curvature and torsion | Review of the relevant pages from sources | Öğretim Yöntemleri: Anlatım |
| 12 | Characterization of a plane curve by curvature | Review of the relevant pages from sources | Öğretim Yöntemleri: Anlatım |
| 13 | Differentiable surfaces and implict function theorem | Review of the relevant pages from sources | Öğretim Yöntemleri: Anlatım |
| 14 | Ruled surfaces | Review of the relevant pages from sources | Öğretim Yöntemleri: Anlatım |
| 15 | Solving problems | Review of the relevant pages from sources | Öğretim Yöntemleri: Problem Çözme |
| 16 | Term Exams | Review of the topics discussed in the lecture notes and sources | Ölçme Yöntemleri: Yazılı Sınav |
| 17 | Term Exams | Review of the topics discussed in the lecture notes and sources | Ölçme Yöntemleri: Yazılı Sınav |
Student Workload - ECTS
| Works | Number | Time (Hour) | Workload (Hour) |
|---|---|---|---|
| Course Related Works | |||
| Class Time (Exam weeks are excluded) | 14 | 4 | 56 |
| Out of Class Study (Preliminary Work, Practice) | 14 | 8 | 112 |
| Assesment Related Works | |||
| Homeworks, Projects, Others | 2 | 4 | 8 |
| Mid-term Exams (Written, Oral, etc.) | 1 | 12 | 12 |
| Final Exam | 1 | 24 | 24 |
| Total Workload (Hour) | 212 | ||
| Total Workload / 25 (h) | 8,48 | ||
| ECTS | 8 ECTS | ||