Prof.Dr.Esen İYİGÜN e-posta: esen@uludag.edu.tr telefon: 0.224.2941766 adres: Uludağ Üniversitesi, Fen-Edebiyat Fakültesi, Matematik Bölümü, 16059, Görükle Kampüsü, Bursa
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Website:
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Objective of the Course:
In classical analysis we are traditionally concerned with real-valued functions in the number space Rn.In order to be able to define continuous function between more general sets it is necessary to give these sets a topological structure.They then become topological spaces.The idea can be taken a stage further.To define a differentiable function between two general sets we give these sets what is called a differentiable structure.They then become differentiable manifolds.This generalization of a differentiable function is the elementary starting point for some far-reaching extensions of classical mathematics, both in analysis and geometry, and it has many applications.
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Contribution of the Course to Professional Development
Knows the differentiable structures and gains knowledge about their basic properties.
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Learning Outcomes:
1
Learns set, function, continuous functions, topological space concepts and some special topolojical space.;
2
Learns differentiable manifold, differentiable function and differential varieties.;
3
To obtain information on Grassman manifolds.;
4
Learns manifolds structure on a topolojical space and their properties.;
5
Understands partitions of unity, partial differentiation, tangent vector and derived linear function concept.;
6
Learns the inverse function theorem and their application and also Leibniz’s formula.;
7
Learns immersions, submanifolds and submersions concepts.;
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Course Content:
Week
Theoretical
Practical
1
Sets and functions, Continuous functions
2
Topological spaces, Some special topolojical spaces
3
Differentiable manifolds
4
Differentiable functions
5
The induced topology on a manifold, Differentiable varieties
6
Grassmann manifolds
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Manifold structure on a topological space
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Midterm Exam + Repeating courses
9
Properties of the induced topology
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Partitions of unity, Partial differentiation
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Tangent vectors, Derived linear functions
12
The inverse function theorem, Leibniz’s Formula
13
Immersions, General properties of immersions
14
Submanifolds, Submersions.
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Textbooks, References and/or Other Materials:
Differentiable Manifolds An Introduction, F.Brickell and R.S.Clark, Van Nostrand Reinhold Company Ltd, 1970.
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Assesment
TERM LEARNING ACTIVITIES
NUMBER
PERCENT
Midterm Exam
1
40
Quiz
0
0
Homeworks, Performances
0
0
Final Exam
1
60
Total
2
100
Contribution of Term (Year) Learning Activities to Success Grade
40
Contribution of Final Exam to Success Grade
60
Total
100
Measurement and Evaluation Techniques Used in the Course
The system of relative evaluation is applied.
Information
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ECTS / WORK LOAD TABLE
Activites
NUMBER
TIME [Hour]
Total WorkLoad [Hour]
Theoretical
14
3
42
Practicals/Labs
0
0
0
Self Study and Preparation
14
4
56
Homeworks, Performances
0
0
0
Projects
0
0
0
Field Studies
0
0
0
Midtermexams
1
30
30
Others
0
0
0
Final Exams
1
52
52
Total WorkLoad
180
Total workload/ 30 hr
6
ECTS Credit of the Course
6
26
CONTRIBUTION OF LEARNING OUTCOMES TO PROGRAMME QUALIFICATIONS
