Real Analysis I G9JKY3BRCR

URI	`m2://res/G9JKY3BRCR`
Kind	course
Creators	Jaikrishnan J (instructor)
Date	2020-09-16
Scope	in

Expressions

e1: edition

Language	en
Date	2020-09-16

m1: page, IIT Palakkad (record). nptel 111106142
m2: page, 2020-09-16 (record). nptel 111106142

Contents

part 1: Week 1
session 1: 1.1 WEEK 1 INTRODUCTION
session 2: 1.2 Why study Real Analysis
session 3: 1.3 Square root of 2
session 4: 1.4 Wason's selection task
session 5: 1.5 Zeno's Paradox
session 6: 2.1 Basic set theory
session 7: 2.2 Basic logic
session 8: 2.3 Quantifiers
session 9: 2.4 Proofs
session 10: 2.5 Functions and relations
session 11: 3.1 Axioms of Set Theory
session 12: 3.2 Equivalence relations
session 13: 3.3 What are the rationals
session 14: 3.4 Cardinality
part 2: Week 2
session 15: WEEK 2 INTRODUCTION
session 16: 4.1 Field axioms
session 17: 4.2 Order axioms
session 18: 4.3 Absolute value
session 19: 5.1 The completeness axiom
session 20: 5.2 Nested intervals property
session 21: 6.1 NIP+APâ‡’ Completeness
session 22: 6.2 Existence of square roots
session 23: 6.3 Uncountability of the real numbers
session 24: 6.4 Density of rationals and irrationals
part 3: Week 3
session 25: WEEK 3 INTRODUCTION
session 26: 7.1 Motivation for infinite sums
session 27: 7.2 Definition of sequence and examples
session 28: 7.3 Definition of convergence
session 29: 7.4 Uniqueness of limits
session 30: 7.5 Achilles and the tortoise
session 31: 8.1 Deep dive into the definition of convergence
session 32: 8.2 A descriptive language for convergence
session 33: 8.3 Limit laws
session 34: 9.1 Subsequences
session 35: 9.2 Examples of convergent and divergent sequences
session 36: 9.3 Some special sequences-CORRECT
part 4: Week 4
session 37: 10.1 Monotone sequences
session 38: 10.2 Bolzano-Weierstrass theorem
session 39: 10.3 The Cauchy Criterion
session 40: 10.4 MCT implies completeness
session 41: 11.1 Definition and examples of infinite series
session 42: 11.2 Cauchy tests-Corrected
session 43: 11.3 Tests for convergence
session 44: 11.4 Erdos_s proof on divergence of reciprocals of primes
session 45: 11.5 Resolving Zeno_s paradox
session 46: 12.1 Absolute and conditional convergence
session 47: 12.2 Absolute convergence continued
session 48: 12.3 The number e
session 49: 12.4 Grouping terms of an infinite series
session 50: 12.5 The Cauchy product
part 5: Week 5
session 51: WEEK 5 - INTRODUCTION
session 52: 13.1 The role of topology in real analysis
session 53: 13.2 Open and closed sets
session 54: 13.3 Basic properties of adherent and limit points
session 55: 13.4 Basic properties of open and closed sets
session 56: 14.1 Definition of continuity
session 57: 14.2 Deep dive into epsilon-delta
session 58: 14.3 Negating continuity
session 59: 15.1 The functions x and x2
session 60: 15.2 Limit laws
session 61: 15.3 Limit of sin x_x
session 62: 15.4 Relationship between limits and continuity
session 63: 15.5 Global continuity and open sets
session 64: 15.6 Continuity of square root
session 65: 15.7 Operations on continuous functions
part 6: Week 6
session 66: 16.1 Language for limits
session 67: 16.2 Infinite limits
session 68: 16.3 One sided limits
session 69: 16.4 Limits of polynomials
session 70: 17.1 Compactness
session 71: 17.2 The Heine-Borel theorem
session 72: 17.3 Open covers and compactness
session 73: 17.4 Equivalent notions of compactness
session 74: 18.1 The extreme value theorem
session 75: 18.2 Uniform continuity
part 7: Week 7
session 76: 19.1 Connectedness
session 77: 19.2 Intermediate Value Theorem
session 78: 19.3 Darboux continuity and monotone functions
session 79: 20.1 Perfect sets and the Cantor set
session 80: 20.2 The structure of open sets
session 81: 20.3 The Baire Category theorem
session 82: 21.1 Discontinuities
session 83: 21.2 Classification of discontinuities and monotone functions
session 84: 21.3 Structure of set of discontinuities
part 8: Week 8
session 85: WEEK 8 & 9 - INTRODUCTION
session 86: 22.1 Definition and interpretation of the derivative
session 87: 22.2 Basic properties of the derivative
session 88: 22.3 Examples of differentiation
session 89: 23.1 Darboux_s theorem
session 90: 23.2 The mean value theorem
session 91: 23.3 Applications of the mean value theorem
session 92: 24.1 Taylor's theorem NEW
session 93: 24.2 The ratio mean value theorem and L_Hospital_s rule
part 9: Week 9
session 94: 25.1 Axiomatic characterisation of area and the Riemann integral
session 95: 25.2 Proof of axiomatic characterization
session 96: 26.1 The definition of the Riemann integral
session 97: 26.2 Criteria for Riemann integrability
session 98: 26.3 Linearity of integral
session 99: 27.1 Sets of measure zero
session 100: 27.2 The Riemann-Lebesgue theorem
session 101: 27.3 Consequences of the Riemann-Lebesgue theorem
part 10: Week 10
session 102: WEEK 10 & 11 - INTRODUCTION
session 103: 28.1 The fundamental theorem of calculus
session 104: 28.2 Taylor's theorem-Integral form of remainder
session 105: 28.3 Notation for Taylor polynomials
session 106: 28.4 Smooth functions and Taylor series
session 107: 29.1 Power series
session 108: 29.2 Definition of uniform convergence
part 11: Week 11
session 109: 31.1 The exponential function
session 110: 31.2 The inverse function theorem
session 111: 31.3 The Logarithm
session 112: 32.1 Trigonometric functions
session 113: 32.2 The number Pi
session 114: 32.3 The graphs of sin and cos
session 115: 33.1 The Basel problem
part 12: Week 12
session 116: 34.1 Improper integrals
session 117: 34.2 The Integral test
session 118: 35.1 Weierstrass approximation theorem
session 119: 35.2 Bernstein Polynomials
session 120: 35.3 Properties of Bernstein polynomials
session 121: 35.4 Proof of Weierstrass approximation theorem
part 13: Live Session
session 122: Live Session 04-10-2020
session 123: Live Session 10-11-2020

Catalogue records

nptel: 111106142 (manifestation)
nptel_video: 111106142 (manifestation)