By Sanjay P. K. |
National Institute of Technology, Calicut

Learners enrolled: 648

Weeks Weekly Lecture Topics (Module Titles)

1 Day 1 Module 1: Introduction to Riemann integration, Darboux sums.

Day 2 Module 2: Inequalities for upper and lower Darboux sums.

Day 3 Module 3: Darboux integral

Day 4 Interaction based on the three modules covered

Day 5 Subjective Assignment

2 Day 1 Module 4: Cauchy criterion for integrability

Day 2 Module 5: Riemann’s definition of integrability

Day 3 Module 6: Equivalence of definitions.

Day 4 Interaction based on the three modules covered

Day 5 Subjective Assignment

3 Day 1 Module 7: Riemann integral as a sequential limit

Day 2 Module 8: Riemann integrability of monotone functions and continuous functions

Day 3 Module 9: Further examples of Riemann integral of functions

Day 4 Interaction based on the three modules covered

Day 5 Subjective Assignment

4 Day 1 Module 10: Algebraic properties of Riemann integral

Day 2 Module 11: Monotonicity and additivity properties of Riemann integral

Day 3 Module 12: Approximation by step functions

Day 4 Interaction based on the three modules covered

Day 5 Subjective Assignment

5 Day 1 Module 13: Mean value theorem for integrals

Day 2 Module 14: Fundamental Theorem of Calculus (first form)

Day 3 Module 15: Fundamental Theorem of Calculus (second form)

Day 4 Interaction based on the three modules covered

Day 5 Subjective Assignment

6 Day 1 Module 16: Improper integrals of Type-1.

Day 2 Module 17: Improper integrals of Type-2 and mixed type.

Day 3 Module 18: Gamma and beta functions

Day 4 Interaction based on the three modules covered

Day 5 Subjective Assignment

7 Day 1 Module 19: Pointwise convergence of a sequence of functions

Day 2 Module 20: Uniform convergence

Day 3 Module 21: Uniform norm

Day 4 Interaction based on the three modules covered

Day 5 Subjective Assignment

8 Day 1 Module 22: Cauchy criterion for uniform convergence

Day 2 Module 23: Uniform converegnce and continuity

Day 3 Module 24: Uniform convergence and integration

Day 4 Interaction based on the three modules covered

Day 5 Subjective Assignment

9 Day 1 Module 25: Uniform convergence and differentiation

Day 2 Module 26: Review of infinite series

Day 3 Module 27: Absolute convergence

Day 4 Interaction based on the three modules covered

Day 5 Subjective Assignment

10 Day 1 Module 28: Infinite series of functions

Day 2 Module 29: Weierstrass M-test

Day 3 Module 30: Theorems on the continuity and differentiability of the sum function of a series of functions;

Day 4 Interaction based on the three modules covered

Day 5 Subjective Assignment

11 Day 1 Module 31: Limit superior and limit inferior of a numerical sequence.

Day 2 Module 32: Limit inferior, limit superior and convergence

Day 3 Module 33: Properties of limit superior and Limit inferior

Day 4 Interaction based on the three modules covered

Day 5 Subjective Assignment

12 Day 1 Module 34: Power series and its radius of convergence

Day 2 Module 35: Convergence of power series

Day 3 Module 36: Differentiation and integration of power series

Day 4 Interaction based on the three modules covered

Day 5 Subjective Assignment

13 Day 1 Module 37: Convergence of a power series at the endpoints, Abel's Theorem.

Day 2 Module 38: Weierstrass approximation Theorem

Day 3 Module 39: Proof of Weierstrass approximation theorem

Day 4 Interaction based on the three modules covered

Day 5 Subjective Assignment

1. K.A. Ross, Elementary Analysis, The Theory of Calculus, Undergraduate Texts in Mathematics, Springer (SIE), Indian reprint, 2004.

2. R.G. Bartle D.R. Sherbert, Introduction to Real Analysis, 3rd Ed., John Wiley and Sons (Asia) Pvt. Ltd., Singapore, 2002.

3. Charles G. Denlinger, Elements of Real Analysis, Jones & Bartlett (Student Edition), 2011.

Sanjay P. K. has 19 years of experience in teaching both undergraduate level and post graduate level. Taught ‘Real Analysis’, Complex Analysis’, ‘Topology’ and ‘Linear Algebra’ for M.Sc. propgramme at NIT Calicut. Awarded Ph. D.. from the Indian Institute of Science in 2012.

Selected for support under the Mathematical Research Impact Centric Support (MATRICS) scheme of the Science and Engineering Research Board (SERB) in 2018.

Assessment/Assignment marks will be considered for Internal Marks and will carry 30 percent for overall Result.

End Term Exam- will have 100 questions and will carry 70 percent of overall Result.

*All students, who obtain 40% marks in in-course assessment and 40% marks in end-term proctored exam separately, will be eligible for certificate and credit transfer.

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