MAT4053 Differentiable ManifoldsBahçeşehir UniversityDegree Programs SPEECH AND LANGUAGE THERAPYGeneral Information For StudentsDiploma SupplementErasmus Policy StatementNational QualificationsBologna Commission
SPEECH AND LANGUAGE THERAPY
Bachelor TR-NQF-HE: Level 6 QF-EHEA: First Cycle EQF-LLL: Level 6

Course Introduction and Application Information

Course Code Course Name Semester Theoretical Practical Credit ECTS
MAT4053 Differentiable Manifolds Fall 3 0 3 6
This catalog is for information purposes. Course status is determined by the relevant department at the beginning of semester.

Basic information

Language of instruction: English
Type of course: Non-Departmental Elective
Course Level: Bachelor’s Degree (First Cycle)
Mode of Delivery: Face to face
Course Coordinator :
Recommended Optional Program Components: None
Course Objectives: The differentiable manifolds course aims to give the fundamental knowledge for the studies of graduate students who intends to study at geometry.

Learning Outcomes

The students who have succeeded in this course;
upon succeeding this course
1)be able to test a differentiable structure given on a set
2)be able to give examples of Differentiable structures on a set
3) be able to check differentiablity of a function
4) be able to solve problems involving the derived map of a transformation between two manifolds
5) be able to use the properties of induced topology on a manifold,
6) be able to coordinatize Grassmann manifolds and can evaluate their dimensions,
7) be able to understand the existence problems by using the unity of partition
8)be able to explain the derived function of a function by using the Leibniz rule,
9) be able to explain submanifolds as images under Immersions
10) be able to coordinatize quotient manifolds and calculate their dimensions,
11) be able to construct Klein bottle and Mobius strip as an example of a quotient manifold

Course Content

Differentiable (diff.able) functions, Atlas, diff.able structures on a set, Examples of diff.able structures, diff.able manifolds, diff.able functions, The induced topology on a manifold, diff.able varieties, Grassmann manifolds, Manifold structure on a topological space, properties of the induced topology, Topological restrictions on a manifold, Partitions of unity, Partial differentiation, tangent vectors, The invers function Theorem, Leibniz's rule. İmmersions, submanifolds, regular submanifolds, some topological properties of submanifolds. Submersions, The fibres of submersions, Quotient manifolds, Transformation groups, Examples of quotient manifolds.

Weekly Detailed Course Contents

Week Subject Related Preparation
1) Preliminaires
2) Some classical theory of differentiable functions
3) Atlas, differentiable structures on a set
4) Examples of differentiable structures on a set
5) Differentiable manifolds
6) Differentiable functions
7) The induced topology on a manifold
8) Differentiable varieties, Grassmann manifolds
9) Topological restrictions on a manifold, Partitions of unity
10) Manifold structure on a topological space, properties of the induced topology
11) Partial differentiation, tangent vectors, derived linear functions, The invers function Theorem, Leibniz's rule.
12) İmmersions, submanifolds, regular submanifolds, some topological properties of submanifolds.
13) Submersions, The fibres of submersions, Quotient manifolds
14) Transformation groups, Examples of quotient manifolds.

Sources

Course Notes / Textbooks: Differentiable Manifolds an Introduction ,F Brickell, R. S. Clark.
References: .

Evaluation System

Semester Requirements Number of Activities Level of Contribution
Midterms 2 % 45
Final 1 % 55
Total % 100
PERCENTAGE OF SEMESTER WORK % 45
PERCENTAGE OF FINAL WORK % 55
Total % 100

ECTS / Workload Table

Activities Number of Activities Duration (Hours) Workload
Course Hours 14 3 42
Study Hours Out of Class 7 2 14
Midterms 2 20 40
Final 1 30 30
Total Workload 126

Contribution of Learning Outcomes to Programme Outcomes

No Effect 1 Lowest 2 Low 3 Average 4 High 5 Highest
           
Program Outcomes Level of Contribution
1) To use theoretic and methodological approach, evidence-based principles and scientific literature in Speech and Language Therapy field systematically for practice.
2) To have theoretic and practical knowledge for individual's, family's and the community's health promotion and protection.
3) To use information and health technologies in practice and research in the field of Speech and Language Therapy.
4) To communicate effectively with advisee, colleagues for effective professional relationships.
5) To be able to monitor occupational information using at least one foreign language, to collaborate and communicate with colleagues at international level.
6) To use life-long learning, problem-solving and critical thinking skills.
7) To act in accordance with ethical principles and values in professional practice.
8) To take part in research, projects and activities within sense of social responsibility and interdisciplinary approach.
9) To be able to search for literature in health sciences databases and information sources to access to information and use the information effectively.
10) To take responsibility and participate in the processes actively for training of other therapist, education of health professionals and individuals about speech and languege therapy.
11) To carry out speech and languge therapy practices considering cultural differences and different health needs of different groups in the community.