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:
- 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.
Sequences
2
Sequences and limits
3
Bounded sequences and unique limits
4
Theorem on limits
5
Sandwich theorem
6
Supremum and Infimum
7
Cauchy sequences and Completeness
8
Example Calculation
9
Subsequences and accumulation values
10
Bolzano-Weierstrass theorem
11
Limit superior and limit inferior
12
Examples for Limit superior and limit inferior
13
Open, Closed and Compact Sets
14
Heine-Borel theorem
Series
1
Series - Introduction
2
Geometric Series and Harmonic Series
3
Cauchy Criterion
4
Leibniz Criterion
5
Comparison Test
6
Ratio and Root Test
7
Reordering for Series
8
Cauchy Product
Functions
1
Sequence of Functions
2
Pointwise Convergence
3
Uniform Convergence
4
Limits for Functions
5
Continuity and Examples
6
Epsilon-Delta Definition
7
Combination of Continuous Functions
8
Continuous Images of Compact Sets are Compact
9
Uniform Limits of Continuous Functions are Continuous
10
Intermediate Value Theorem
11
Some Continuous Functions
Differentiation
1
Differentiability
2
Properties for Derivatives
3
Chain Rule
4
Uniform Convergence for Differentiable Functions
5
Examples of Derivatives and Power Series
6
Derivatives of Inverse Functions
7
Local Extrema and Rolle's Theorem
8
Mean Value Theorem
9
L'Hôpital's Rule
10
Other L'Hôpital's Rules
11
Higher Derivatives
12
Taylor's Theorem
13
Application for Taylor's Theorem
14
Proof of Taylor's Theorem
Integration
1
Riemann Integral - Partitions
2
Riemann Integral for Step Functions
3
Properties of the Riemann Integral for Step Functions
4
Riemann Integral - Definition
5
Riemann Integral - Examples
6
Riemann Integral - Properties
7
First Fundamental Theorem of Calculus
8
Second Fundamental Theorem of Calculus
9
Proof of the Fundamental Theorem of Calculus
10
Integration by Substitution
11
Integration by Parts
12
Integration by Partial Fraction Decomposition
13
Integrals on Unbounded Domains
14
Comparison Test for Integrals
15
Integral Test for Series
16
Improper Riemann-Integrals for Unbounded Functions
17
Cauchy Principal Value
Examination Questions
1
Problem Solving - Series and Convergent Criteria
2
Problem Solving - Continuous Functions
3
Problem Solving - Differentiable Functions
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