Syllabus ( MATH 595 )

Basic information


Course title: 
Tensor Analysis and General Relativity 
Course code: 
MATH 595 
Lecturer: 
Prof. Dr. Oğul ESEN

ECTS credits: 
7.5 
GTU credits: 
3 (3+0+0) 
Year, Semester: 
20162017 20172018, Fall and Spring 
Level of course: 
Second Cycle (Master's) 
Type of course: 
Area Elective

Language of instruction: 
English

Mode of delivery: 
Face to face

Pre and corequisites: 
none 
Professional practice: 
No 
Purpose of the course: 
The target of this lecture is to give the basic knowledge of Tensors to the graduate students who wish to make research on the topics of Applied Mathematics, Theoretical Physics and Differential ;Geometry. In the meantime we would like to introduce the topic of General Relativity to the graduate students . 



Learning outcomes


Upon successful completion of this course, students will be able to:

The mathematical structure of Newton equation
Contribution to Program Outcomes

Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems
Method of assessment

Written exam

Homework assignment

The determination of particle orbits via Newton equation.
Contribution to Program Outcomes

Define and manipulate advanced concepts of Mathematics

Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems

Work effectively in multidisciplinary research teams

Develop mathematical, communicative, problemsolving, brainstorming skills.

Effectively express his/her research ideas and findings both orally and in writing

Demonstrating professional and ethical responsibility.
Method of assessment

Written exam

Homework assignment

Lagrange function and the application of EulerLagrange equations in Classical Mechanics.
Contribution to Program Outcomes

Define and manipulate advanced concepts of Mathematics

Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems

Acquire scientific knowledge and work independently

Develop mathematical, communicative, problemsolving, brainstorming skills.

Effectively express his/her research ideas and findings both orally and in writing

Demonstrating professional and ethical responsibility.
Method of assessment

Written exam

Homework assignment

The Determination of the orbits of planets under the influence of Gravitational force.
Contribution to Program Outcomes

Define and manipulate advanced concepts of Mathematics

Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems

Acquire scientific knowledge and work independently

Work effectively in multidisciplinary research teams

Develop mathematical, communicative, problemsolving, brainstorming skills.

Effectively express his/her research ideas and findings both orally and in writing

Demonstrating professional and ethical responsibility.
Method of assessment

Written exam

Homework assignment

The learning of the basic concepts of Special and General Relativity theory.
Contribution to Program Outcomes

Define and manipulate advanced concepts of Mathematics

Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems

Acquire scientific knowledge and work independently

Work effectively in multidisciplinary research teams

Develop mathematical, communicative, problemsolving, brainstorming skills.

Effectively express his/her research ideas and findings both orally and in writing

Demonstrating professional and ethical responsibility.
Method of assessment

Written exam

Homework assignment

Learning the Invariance concept and its combination with Tensors.
Contribution to Program Outcomes

Define and manipulate advanced concepts of Mathematics

Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems

Acquire scientific knowledge and work independently

Work effectively in multidisciplinary research teams

Develop mathematical, communicative, problemsolving, brainstorming skills.

Effectively express his/her research ideas and findings both orally and in writing

Demonstrating professional and ethical responsibility.
Method of assessment

Written exam

Homework assignment

Observing the natural fitting of Electromagnetic Theory with Relativity Theory.
Contribution to Program Outcomes

Define and manipulate advanced concepts of Mathematics

Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems

Review the literature critically pertaining to his/her research projects, and connect the earlier literature to his/her own results

Acquire scientific knowledge and work independently

Work effectively in multidisciplinary research teams

Continuously develop their knowledge and skills in order to adapt to a rapidly developing technological environment

Develop mathematical, communicative, problemsolving, brainstorming skills.

Effectively express his/her research ideas and findings both orally and in writing

Demonstrating professional and ethical responsibility.
Method of assessment

Written exam

Homework assignment

The awareness of the shortcoming of Special Relativity in the analysis of Graviatation.
Contribution to Program Outcomes

Define and manipulate advanced concepts of Mathematics

Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems

Acquire scientific knowledge and work independently

Work effectively in multidisciplinary research teams

Develop mathematical, communicative, problemsolving, brainstorming skills.

Effectively express his/her research ideas and findings both orally and in writing
Method of assessment

Written exam

Homework assignment

The generalization of Special Relativity to General Relativity.
Contribution to Program Outcomes

Define and manipulate advanced concepts of Mathematics

Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems

Acquire scientific knowledge and work independently

Work effectively in multidisciplinary research teams

Continuously develop their knowledge and skills in order to adapt to a rapidly developing technological environment

Effectively express his/her research ideas and findings both orally and in writing
Method of assessment

Written exam

Homework assignment

The basic calculational methods of Tensor Analysis
Contribution to Program Outcomes

Define and manipulate advanced concepts of Mathematics

Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems

Acquire scientific knowledge and work independently

Work effectively in multidisciplinary research teams

Develop mathematical, communicative, problemsolving, brainstorming skills.
Method of assessment

Written exam

Homework assignment

Learning the application of Tensors in Riemannian Geometry.
Contribution to Program Outcomes

Define and manipulate advanced concepts of Mathematics

Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems

Review the literature critically pertaining to his/her research projects, and connect the earlier literature to his/her own results

Acquire scientific knowledge and work independently

Work effectively in multidisciplinary research teams

Continuously develop their knowledge and skills in order to adapt to a rapidly developing technological environment

Develop mathematical, communicative, problemsolving, brainstorming skills.

Effectively express his/her research ideas and findings both orally and in writing

Demonstrating professional and ethical responsibility.
Method of assessment

Written exam

Homework assignment

The learning of the application of Riemannian Geometry in General Relativity.
Contribution to Program Outcomes

Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems

Acquire scientific knowledge and work independently

Work effectively in multidisciplinary research teams

Continuously develop their knowledge and skills in order to adapt to a rapidly developing technological environment

Develop mathematical, communicative, problemsolving, brainstorming skills.

Effectively express his/her research ideas and findings both orally and in writing

Demonstrating professional and ethical responsibility.
Method of assessment

Written exam

Homework assignment

Learning the analysis and solution techniques of Einstein Field Equations.
Contribution to Program Outcomes

Define and manipulate advanced concepts of Mathematics

Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems

Review the literature critically pertaining to his/her research projects, and connect the earlier literature to his/her own results

Acquire scientific knowledge and work independently

Work effectively in multidisciplinary research teams

Continuously develop their knowledge and skills in order to adapt to a rapidly developing technological environment

Develop mathematical, communicative, problemsolving, brainstorming skills.

Effectively express his/her research ideas and findings both orally and in writing

Demonstrating professional and ethical responsibility.
Method of assessment

Written exam

Homework assignment

Learning the frequently coincided analytıcal solutions of Einstein Field Equations and their physical interpretations.
Contribution to Program Outcomes

Define and manipulate advanced concepts of Mathematics

Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems

Acquire scientific knowledge and work independently

Work effectively in multidisciplinary research teams

Continuously develop their knowledge and skills in order to adapt to a rapidly developing technological environment

Develop mathematical, communicative, problemsolving, brainstorming skills.

Effectively express his/her research ideas and findings both orally and in writing

Demonstrating professional and ethical responsibility.
Method of assessment

Written exam

Homework assignment

The learning of Metric Tensors which are solutions of Einstein Field Equations and their phsical applications via Lagrange Equations.
Contribution to Program Outcomes

Define and manipulate advanced concepts of Mathematics

Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems

Review the literature critically pertaining to his/her research projects, and connect the earlier literature to his/her own results

Acquire scientific knowledge and work independently

Work effectively in multidisciplinary research teams

Continuously develop their knowledge and skills in order to adapt to a rapidly developing technological environment

Develop mathematical, communicative, problemsolving, brainstorming skills.

Effectively express his/her research ideas and findings both orally and in writing

Demonstrating professional and ethical responsibility.
Method of assessment

Written exam

Homework assignment


Contents


Week 1: 
Introduction to Classical Mechanics 
Week 2: 
Lagrange Function, EulerLagrange Equations and their application in Classical Mechanics 
Week 3: 
Introduction to Special Relativity 
Week 4: 
Lorentz Transformations and the Concept of Invariance 
Week 5: 
Special Relativity and Electromagnetic Theory 
Week 6: 
The Shortcomings of Special Relativity and need for a General Theory 
Week 7: 
Invariance and Tensors 
Week 8: 
Basic Calculations with Tensors 
Week 9: 
Midterm 
Week 10: 
The use of Tensors in Riemannian Geometry 
Week 11: 
Riemann Curvature, Ricci and Scalar Ricci Tensors 
Week 12: 
Einstein Field Equations 
Week 13: 
The Analytical solutions of Einstein Field Equations 
Week 14: 
The applications of the Analytical solutions of Einstein Field Equations to phsical problems via Lagrange function ve EulerLagrange equations 
Week 15*: 
The Fitting of General Relativity with experimental and observational physical data and introduction to General Relativistic Cosmology and Black Holes 
Week 16*: 
Final Exam 
Textbooks and materials: 
• Ray D’Inverno, Introducing Einstein’s Relativity, Oxford University Press, 1992 • David C. Kay, Tensor Calculus, Schaum’s Outline Series, 1988 • S. Chandrasekkhar, The Mathematical Theory of Black Holes, Oxford University Press,1992 • •

Recommended readings: 
J. Foster J. D. Nightingale, A Short Course in General Relativity, Second edition, SpringerVerlag, 1995 • Articles and Lecture Notes from Arxiv.org preprint server 

* Between 15th and 16th weeks is there a free week for students to prepare for final exam.




Assessment



Method of assessment 
Week number 
Weight (%) 

Midterms: 
9 
30 
Other interm studies: 

0 
Project: 

0 
Homework: 
5, 7, 12 
30 
Quiz: 

0 
Final exam: 
16 
40 

Total weight: 
(%) 



Workload



Activity 
Duration (Hours per week) 
Total number of weeks 
Total hours in term 

Courses (Facetoface teaching): 
3 
14 

Own studies outside class: 
3 
14 

Practice, Recitation: 
0 
0 

Homework: 
15 
3 

Term project: 
0 
0 

Term project presentation: 
0 
0 

Quiz: 
0 
0 

Own study for midterm exam: 
22.5 
1 

Midterm: 
3 
1 

Personal studies for final exam: 
30 
1 

Final exam: 
3 
1 



Total workload: 



Total ECTS credits: 
* 

* ECTS credit is calculated by dividing total workload by 25. (1 ECTS = 25 work hours)



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