Course Language:
English
Course Objectives:
This course constitutes the pillar of many topics in mathematics such as complex analysis, differential equations, differential and integral calculus, and differential geometry. It is impossible to assimilate these areas of mathematics without having this basic knowledge of analysis. The aim of the course is to equip students with this basic knowledge.
Course Content:
Elements of point set theory.Functions and the Real Number System. Open and closed sets in Rp. Accumulation points. Bolzano-Weierstrass theorem in Rp. Compactness and connectedness.Heine Borel Theorem. Convergence of sequences in Rp. Sequences of Functions, Uniform Convergence Continuity and Uniform continuity.Sequences of Continuous Functions,Limits of Functions.
Course Methodology:
1: Lecture, 2: Problem Solving
Course Evaluation Methods:
A: Written examination, B: Homework
Vertical Tabs
Course Learning Outcomes
| Learning Outcomes | Program Learning Outcomes | Teaching Methods | Assessment Methods |
| 1) Grasp the structure of the real numbers as a complete ordered field | 2,3 | 1,2 | A,B |
| 2) Learn how to handle the convergence of sequences, series | 1,2,4,7 | 1,2 | A,B |
| 3) Master the concept of the limit of functions and the concept of continuity; | 2,3,4,7 | 1,2 | A,B |
| 4) Acquire the knowledge of differentiability of functions | 1,2,4,7 | 1,2 | A,B |
| 5) Learn integration and classes of Riemann integrable functions | 1,2,4,7 | 1,2 | A,B |
Course Flow
| COURSE CONTENT | ||
| Week | Topics | Study Materials |
| 1 | Limits of Functions, Limit Theorems, Cauchy Convergence criterion | Textbook |
| 2 | Continuous Functions, Combinations of Continuous Functions, Continuous Functions on Intervals | Textbook |
| 3 | Uniform Continuity. Monotone functions. Inverse Function Theorem | Textbook |
| 4 | Continuous and Monotone Functions, Sequences of Functions, Pointwise and Uniform Convergence | Textbook |
| 5 | Cauchy Criterion for Uniform Convergence | Textbook |
| 6 | The Derivative, The Mean Value Theorem, L'Hospital Rules, Taylor's Theorem | Textbook |
| 7 | Partitions and Tagged Partitions, Riemann sum, Riemann integrability | Textbook |
| 8 | Some Properties of the Integral, Boundedness Theorem | Textbook |
| 9 | Riemann Integrable Functions, Cauchy Criterion, | Textbook |
| 10 | Squeeze Theorem, Classes of Riemann Integrable Functions | Textbook |
| 11 | Additivity Theorem, The Fundamental Theorem of Calculus, | Textbook |
| 12 | Substitution Theorem, Lebesgue’s lntegrability Criterion, | Textbook |
| 13 | Composition Theorem, The Product Theorem, | Textbook |
| 14 | Integration by Parts, Taylor’s Theorem with the Remainder | Textbook |
Recommended Sources
| RECOMMENDED SOURCES | |
| Textbook |
Robert G. Bartle, Donald R. Sherbert, Introduction to Real Anlaysis, Fourth Edition, John Wiley & Sons, Inc.(2011),ISBN-13: 978-0471433316ISBN-10: 9780471433316.
https://sciencemathematicseducation.files.wordpress.com/2014/01/0471433314realanalysis4.pdf |
| Additional Resources | Stephen Abbott, Understanding Analysis, Springer, 2. Edition (2015) |
Material Sharing
| Documents | |
| Assignments | |
| Exams |
Assessment
| IN-TERM STUDIES | NUMBER | PERCENTAGE |
| Mid-terms | 2 | 70 |
| Quizzes | - | |
| Assignments | 3 | 30 |
| Total | 100 | |
| CONTRIBUTION OF FINAL EXAMINATION TO OVERALL GRADE | 40 | |
| CONTRIBUTION OF IN-TERM STUDIES TO OVERALL GRADE | 60 | |
| Total | 100 |
| COURSE CATEGORY | Expertise/Field Courses |
Course’s Contribution to Program
| COURSE'S CONTRIBUTION TO PROGRAM | |||||||
| No | Program Learning Outcomes | Contribution | |||||
| 1 | 2 | 3 | 4 | 5 | |||
| 1 | The ability to make computation on the basic topics of mathematics such as limit, derivative, integral, logic, linear algebra and discrete mathematics which provide a basis for the fundamental research fields in mathematics (i.e., analysis, algebra, differential equations and geometry) | x | |||||
| 2 | Acquiring fundamental knowledge on fundamental research fields in mathematics | x | |||||
| 3 | Ability form and interpret the relations between research topics in mathematics | x | |||||
| 4 | Ability to define, formulate and solve mathematical problems | x | |||||
| 5 | Consciousness of professional ethics and responsibility | x | |||||
| 6 | Ability to communicate actively | x | |||||
| 7 | Ability of self-development in fields of interest | x | |||||
| 8 | Ability to learn, choose and use necessary information technologies | x | |||||
| 9 | Lifelong education | x | |||||
ECTS
| ECTS ALLOCATED BASED ON STUDENT WORKLOAD BY THE COURSE DESCRIPTION | |||
| Activities | Quantity |
Duration (Hour) |
Total Workload (Hour) |
| Course Duration (14x Total course hours) | 14 | 5 | 70 |
| Hours for off-the-classroom study (Pre-study, practice) | 14 | 4 | 56 |
| Mid-terms (Including self-study) | 2 | 10 | 20 |
| Quizzes | - | - | - |
| Assignments | 3 | 5 | 15 |
| Final examination (Including self-study) | 1 | 15 | 15 |
| Total Work Load | 176 | ||
| Total Work Load / 25 (h) | 7 | ||
| ECTS Credit of the Course | 7 | ||