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Indian Institute of Technology Bombay

Basic Real Analysis

Indian Institute of Technology Bombay and NPTEL via Swayam

Overview

This s first course in Real Analysis. The aim of the course is to over the basic concepts like Real line, Topological concepts of real line, differentiation and integration with applications. Introduction to calculus of several variables.


INTENDED AUDIENCE : Any discipline, with proper exposure to Calculus.
PREREQUISITES : Basic Calculus exposure
INDUSTRY SUPPORT : All

Syllabus

COURSE LAYOUT

Week 1 : Review of sequences and series of real numbers.
Week 2 : Tests for convergence of Series. Limit superior and limit inferior.
Week 3 : Cauchy sequences and completeness of R.
Week 4 : Basic notions of Metric Spaces with emphasis on Rn. Connectedness, Compactness, and Heine Borel Theorem.
Week 5 : Continuity and Uniform continuity.
Week 6 : Monotone functions and functions of bounded variation.
Week 7 : Derivatives. Mean Value Theorem and applications
Week 8 : Riemann Stieltjes integral. Riemann`s Criterion for integrability. Improper integrals and the Gamma function.
Week 9 : Sequences and series of functions. Uniform convergence.
Week 10 : Functions of several variables: Directional derivative, partial derivative, total derivative,
Week 11 : Mean Value Theorem, Taylor`s Theorem and applications to Maxima/Minima and convexity.
Week 12 : Double and triple integrals. Statement of Fubini's Theorem and change of variable formula (without proofs) with illustrations.

TEACHING ASSISTANT
S. Venkitesh

S. Venkitesh is a final year Ph.D. student at IIT Bombay, with research interests in Combinatorics and Theoretical Computer Science, working with Prof. Srikanth Srinivasan. He has co-authored 3 research papers till date. His current research focuses on problems in Boolean function and circuit complexity, and theory and algorithms for linear codes, which have an inherent combinatorial/algebraic flavour. Two key methods often employed in such research are the polynomial method and the probabilistic method.He obtained his Masters’ degree in Mathematics from IIT Bombay (with Third rank) and Bachelors’ degree in Mathematics from University of Mumbai (with First rank). He has so far been a Teaching Assistant in 7 Mathematics courses of NPTEL and 10 courses at the Department of Mathematics, IIT Bombay. He also frequently accompanies the faculty of the Department of Mathematics, as a resource person for mathematics workshops at colleges in Mumbai.

Taught by

Prof. I. K. Rana

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