DESCRIPTION AND OBJECTIVES

A continuation of Real Analysis-I, this course will continue to cover the fundamentals of real analysis, concentrating on the Riemann-Stieltjes integrals, Functions of Bounded Variation, Improper Integrals, and convergence of series. Emphasis would be on proofs of main results.

Pre-Requisite

Real Analysis-I

INTENDED LEARNING OUTCOMES

After the completion of the course, Students will be able to

1. Describe fundamental properties of the real numbers that lead to the formal development of real analysis.
2. Comprehend rigorous arguments developing the theory underpinning real analysis.
3. Demonstrate an understanding of limits and how they are used in sequences, series, differentiation  and integration.
4. Construct rigorous mathematical proofs of basic results in real analysis.
5. Appreciate how abstract ideas and rigorous methods in mathematical analysis can be applied to important practical problems.

COURSE CONTENTS

1. The Riemann-Stieltjes Integrals
2. Definition and existence of integrals, properties of integrals
3. Real Valued Functions of Several Variables
4. Continuous real valued functions
5. Partial derivatives and differentials
6. Geometric interpretation of differentiability
7. Chain rule, Taylor’s theorem. Maxima and Minima
8. Vector Valued Functions of Several Variables Linear transformations and matrices
9. Continuous and differentiable transformations
10. Chain rule for transformations
11. Inverse function theorem
12. Implicit function theorem, Jacobians
13. Method of Lagrange multipliers
14. Functions of Bounded Variation
15. Definition and examples, properties of functions of bounded variation
16. Improper Integrals: Types of improper integrals
17. Tests for convergence of improper integrals
18. Absolute and conditional convergence of improper integrals
19. Sequences and Series of Functions
20. Power series, definition of point-wise and uniform convergence
21. Uniform convergence and continuity
22. Uniform convergence and differentiation
23. Examples of uniform convergence.

Recommended Books

1. Walter Rudin, Principles of Mathematical Analysis, (3rd Ed. 1976)
2. T.M. Apostal, Mathematical Analysis,( 2nd Ed. Addison Wesley, 1974)
3. W. Kaplan, Advanced calculus, (5th Ed. Addison Wesley, 2002)
4. R.L. Rabenstein, Elements of Ordinary differential equations, (Academic Press, 1984)
5. Robert G. Bartle and Donald R. Sherbert, Introduction to Real Analysis,( 3rdEd.1999)
6. James Stewart, Calculus, (8th Ed)

ASSESSMENT CRITERIA

Sessional: 20 (Presentation / Assignment 10, Attendance 05, Quiz 05)

Mid-Term Exam:   30

Final-Term Exam: 50

Key Dates and Time for Class Meeting

Monday to Tuesday: 11:00am -12:30pm (M.Sc Regular)

Monday:                  2:00pm -3:30pm (M.Sc Self Support)

Tuesday:                 3:30pm - 5:00pm (M.Sc Self Support)   

Commencement of Classes: March 02, 2020 (Monday)

Mid Term Examination: April 27 to May 04, 2020 (Monday to Monday)

Final Term Examination: Jun, 2020

Declaration of Result: July, 2020