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# Basic and Real Analysis

Enrolled: 1 student
Lectures: 67
Level: Intermediate

Real Analysis is a rigorous and foundational course in advanced mathematics that delves into the study of real numbers, functions, and their properties. Real Analysis provides the tools and techniques necessary for a deeper understanding of calculus, as well as for the exploration of more advanced mathematical subjects. This course aims to equip students with a solid theoretical foundation and problem-solving skills essential for further studies in mathematics, physics, engineering, and related disciplines.

Course Objectives:

1. Develop a rigorous understanding of the real number system, including properties of real numbers, order relations, and completeness.
2. Study the concepts of limits, continuity, and differentiability of functions, and explore their applications in calculus.
3. Investigate the convergence and divergence of sequences and series, and analyze their properties.
4. Introduce the theory of integration, including Riemann integration and its fundamental properties.
5. Explore the fundamental concepts of metric spaces and topological spaces, and their relevance in analysis.
6. Examine the properties of continuous functions and their implications on connectedness and compactness.
7. Introduce the concept of differentiation in higher dimensions, including partial derivatives and the chain rule.
8. Study the theory of infinite series and power series expansions, and analyze their convergence and differentiability.
9. Investigate the fundamental theorems of calculus, such as the mean value theorem and the fundamental theorem of calculus.
10. Develop proficiency in constructing rigorous mathematical proofs and logical reasoning.