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