Applied Real Analysis (Master Degree) – Course Syllabus

1. Course Information

• Course Title: Applied Real Analysis

• Level: Master (Graduate)

• Credits: 3

• Prerequisite: Advanced Calculus / Real Analysis

• Course Description:
  This course studies the fundamental concepts of real analysis with emphasis on applications in applied mathematics, differential equations, optimization, probability, and numerical analysis.
  Module 1: Review of Real Numbers and Sequences

• Completeness property of ℝ

• Supremum and infimum

• Sequences and convergence

• Cauchy sequences

• Bolzano–Weierstrass theorem

• Applications in numerical approximation
  Module 2: Series of Real Numbers

• Infinite series

• Absolute and conditional convergence

• Convergence tests (comparison, ratio, root tests)

• Power series

• Applications in function approximation
  Module 3: Functions and Continuity

• Limits of functions

• Continuous functions

• Uniform continuity

• Intermediate value theorem

• Applications in modeling
  Module 4: Differentiation and Applications

• Differentiability

• Mean value theorem

• Taylor theorem and Taylor series

• Optimization problems

• Applications in economics and physics
  Module 5: Integration Theory

• Riemann integral

• Improper integrals

• Applications to probability and physics

• Numerical integration ideas
  Module 6: Sequences and Series of Functions

• Pointwise convergence

• Uniform convergence

• Weierstrass M-test

• Interchange of limit, integral, derivative
  Module 7: Metric Spaces

• Metric space definition

• Open and closed sets

• Compactness

• Connectedness

• Applications to fixed point problems
  Module 8: Fixed Point Theorems

• Banach fixed point theorem

• Contraction mappings

• Applications to differential equations

• Iterative numerical methods
  Module 9: Introduction to Functional Analysis

• Normed vector spaces

• Banach spaces

• Linear operators

• Applications to PDE and optimization
  Module 10: Applications

• Existence and uniqueness of differential equations

• Fourier series basics

• Optimization problems

• Applications in applied mathematics and engineering

Course Learning Outcomes (CLO)

After completing this course students will be able to:

1. Apply concepts of real analysis to applied mathematical problems.

2. Analyze convergence of sequences and functions in applications.

3. Use fixed point theorems to solve differential equations.

4. Apply analytical methods to optimization and modeling problems.

5. Use mathematical rigor in applied mathematical research.

Suggested Assessment

• Attendent – 10%

• Assignments / Problem Sets – 20%

• Midterm Examination – 20%

• Final Examination – 40%

• Project / Presentation – 10%

Textbooks 

    Real analysis and Applications

  A course in real analysis

     Functional analysis and Application

Recommended Textbooks

1. Applied Analysis – John K. Hunter & Bruno Nachtergaele

2. Understanding Analysis – Stephen Abbott

3. Real Analysis – H.L. Royden & P.M. Fitzpatrick

4. Principles of Mathematical Analysis – Walter Rudin

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• បើសំណល់ 0 គឺនៅក្រុមទី​5

• បើសំណល់ 1 គឺនៅក្រុមទី​1

• បើសំណល់ 2 គឺនៅក្រុមទី​2

• បើសំណល់ 3 គឺនៅក្រុមទី​3

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