Unit 2946: Advanced Real Analysis

1. Introduction to Real Analysis

• Definition and scope of real analysis
• Sets and functions: domain, range, types of functions
• Limits of sequences and functions
• Continuity of functions
• Importance of rigor and precision in mathematical proofs

2. Sequences and Series

• Definition of sequences and series
• Convergence and divergence
• Tests for convergence:
  • Ratio test
  • Comparison test
  • Limit comparison test
• Absolute and conditional convergence
• Power series and radius of convergence

3. Continuity and Differentiability

• Detailed study of continuity
• Intermediate value theorem
• Differentiability:
  • Definition and geometric intuition
  • Mean value theorem
• Applications to optimization problems
• Higher order derivatives and Taylor’s theorem

4. The Riemann Integral

• Motivation and definition of the Riemann integral
• Riemann sums and partitions
• Properties of the Riemann integral
• Fundamental theorem of calculus
• Techniques of integration

5. Sequences of Functions

• Pointwise convergence
• Uniform convergence:
  • Definition and examples
  • Difference between pointwise and uniform convergence
• Consequences of uniform convergence on continuity and integration
• Relation between convergence of function sequences and their integrals

6. Metric Spaces

• Definition of a metric space
• Open and closed sets in metric spaces
• Convergence and continuity in metric spaces
• Compactness:
  • Definition and examples
  • Heine-Borel theorem
• Completeness and Cauchy sequences
• Role of metrics in topology

7. Differentiation in Several Variables

• Functions of several variables: definition and examples
• Partial derivatives
• Gradient vector and its interpretation
• Directional derivatives
• Differentiability in higher dimensions
• Chain rule for multivariable functions

8. Integration in Several Variables

• Multiple integrals:
  • Double and triple integrals
  • Iterated integrals
• Change of variables theorem
• Jacobian determinant
• Applications:
  • Calculating volumes and surface areas
  • Physical applications involving mass and center of mass

Each section will include definitions, theorems with proofs, examples, and exercises to reinforce understanding.