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| MAT2023 | Analysis - III | 4+2+0 | ECTS:7 | | Year / Semester | Fall Semester | | Level of Course | First Cycle | | Status | Compulsory | | Department | DEPARTMENT of MATHEMATICS | | Prerequisites and co-requisites | None | | Mode of Delivery | Face to face | | Contact Hours | 14 weeks - 4 hours of lectures and 2 hours of practicals per week | | Lecturer | Prof. Dr. Mehmet AKBAŞ | | Co-Lecturer | ASSOC. PROF. DR. Mehmet KUNT, | | Language of instruction | Turkish | | Professional practise ( internship ) | None | | | | The aim of the course: | | The aim of the course to provide students with a general knowledge on n-dimensional analysis, and give them how to generalize analysis from one variable to several variables. |
| Learning Outcomes | CTPO | TOA | | Upon successful completion of the course, the students will be able to : | | | | LO - 1 : | apply the concepts to solve problems in high dimensional analysis. | 3,5,6,7 | 1 | | LO - 2 : | consolidate one dimensional Analysis concepts well. | 3,5,6,7 | 1 | | LO - 3 : | see differences between one dimensional and high dimensional derivatives | 3,5,6,7 | 1 | | LO - 4 : | use the theorems of inverse and implicit functions. | 3,5,6,7 | 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), LO : Learning Outcome | | |
| Functions of several real variables, Topology of Rn, Limit, Continuity, Compactness, Sequences of functions, Series of functions. Series in Rn, Linear operators and matrices. Derivative, Chain rule. Mean value theorems. Partial derivatives. Implicit and inverse function theorems. Maximum and minimum, Lagrange multiplier rul |
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| Course Syllabus | | Week | Subject | Related Notes / Files | | Week 1 | n- dimensional Euclid Space, topology of R^n | | | Week 2 | Sequences in R^n, Cauchy sequences, nested intervals theorem, Cantor's theorem, limit points of sets
| | | Week 3 | Countability, countability of rationals, uncountability of R^n, approximation by rationals | | | Week 4 | Compactness, Heine-Borel theorem, continuous functions, equivalent sequential property, combinations of functions, function limits | | | Week 5 | Continuity and compactness, preservation of compactness, uniform continuity, function sequences, uniform convergence, function spaces | | | Week 6 | Series in R^n, series of functions, Weierstrass M-test, Abel's partial summation formula, Dirichlet's test
| | | Week 7 | Topological aspects of continuity, inverse images, continuity on a set, connected sets, pathwise and poligonally connected sets | | | Week 8 | Mid-term exam | | | Week 9 | Linear operators and matricesi uniform continuity and boundedness, rank-nullity theorem, determinants
| | | Week 10 | Differantiable functionsi, directional and partial derivatives, matrix represantation, class C^1
| | | Week 11 | Chain rule, Gradient vector, direction of fastest increase and orthogonal to level sets | | | Week 12 | Mean value theorems, approximation of a function by the derivative, higher order partial derivatives, equality of mixed partial derivatives
| | | Week 13 | The implicit and inverse function theorems, existence theorems, contraction mappings, Banach's fixed point theorem, inverse function theorem, injectivity and surjectivity theorems, open mapping theorem | | | Week 14 | Implicit function theorem, block partial derivatives, differentiability of the implicitly defined function
| | | Week 15 | global homeomorphisms and solutions of problems | | | Week 16 | End-of-term exam | | | |
| 1 | Webb, J.R.L. 1991; Functions of Several Real Variables, Ellis Horwood Limited, England | | | |
| 1 | Fleming, W.H. 1977; Functions of Several Variables, Springer, 2nd Ed., New York | | | 2 | Spivak, M. 1967; Calculus, W. A. Benjamin Inc., ABD | | | |
| Method of Assessment | | Type of assessment | Week No | Date | Duration (hours) | Weight (%) | | Mid-term exam | 9 | 11/2024 | 2 | 50 | | End-of-term exam | 16 | 01/2025 | 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 | 4 | 14 | 56 | | Sınıf dışı çalışma | 6 | 14 | 84 | | Arasınav için hazırlık | 1 | 8 | 8 | | Arasınav | 1 | 2 | 2 | | Uygulama | 2 | 14 | 28 | | Dönem sonu sınavı için hazırlık | 3 | 10 | 30 | | Dönem sonu sınavı | 2 | 1 | 2 | | Total work load | | | 210 |
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