Mathematical Analysis

Dr. Manju K Menon
Mathematics
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Mathematical  Analysis

 

Course Code: MM5CRT01                      Coordinator: Dr. Manju K. Menon

 

Course Credit: 4

 

Semester: V                                                                            Class: BSc Mathematics

 

Course Type: Core

 

Department: Mathematics

 

 

 

COURSE OBJECTIVES

 

To make students understand the Real number system and explore the basic terminologies in Real Analysis which is one of the main core subjects in Mathematics.

 

Learning Outcomes

 

After the completion of this course the student will be able to

  • Determine the basic properties of subsets of the real numbers
  • Describe the real line as a complete, ordered field
  • Describe about sequences, limit of a sequence and its applications
  • Identify the nature of sequences such as convergence, divergence, etc.
  • Explain about infinite series and its nature
  • Choose tests and analyze the convergence of a series
  • Discuss about the absolute convergence of a series

Syllubus

MODULE I: REAL NUMBERS

Finite and Infinite Sets, The Algebraic and Order Properties of R, Absolute Value and Real

Line, The Completeness Property of R, Applications of the Supremum Property, Intervals.

Chapter 1: Section 1.3 and Chapter 2 : Sections 2.1, 2.2,2.3,2.4,2.5

 

MODULE II: SEQUENCES

Sequences and their Limits, Limit Theorems, Monotone Sequences, Subsequences and the

Bolzano- Weierstrass Theorem, The Cauchy Criterion, Properly Divergent Sequences.

Chapter 3 : Sections 3.1,3.2,3.3,3.4, 3.5,3.6

 

MODULE III: SERIES

Introduction to Series, Absolute Convergence, Tests for Absolute convergence, Tests for

Non absolute Convergence

Chapter 3 : Section 3.7, Chapter 9 : Sections 9.1,9.2,9.3

 

MODULE IV: LIMITS

Limits of Functions, Limit Theorems, Some Extensions of the Limit Concept

Chapter 4 : Sections 4.1,4.2,4.3

 

COURSE OUTLINE

No Topics Learning Outcomes Hours
1  

Finite and Infinite Sets

 

The Algebraic and Order Properties of R

 

Absolute Value and Real Line

 

The Completeness Property of R

 

Applications of the Supremum Property

 

Intervals.

 

 

·         Understands the concept of Subsets of realnumbers and its basic properties

·         Algebraic Properties of R

·         Order Properties of R

 

·         Absolute Value

·         The Properties like OC,  of R and i

·         Supremum, Infimum and its properties

·         Intervals and Nesterd interval properties

·         R is countable

 

 

 

 

6

 

 

2

 

 

2

 

 

1

 

3

 

 

5

 

 

5

 

 

3

Seminar will be given to 1/4 th of students
2 Sequences and their Limits

 

Limit Theorems

 

Monotone Sequences

 

Subsequences and the

Bolzano- Weierstrass Theorem

 

The Cauchy Criterion

 

Properly Divergent Sequences.

 

·         Have an idea about Sequences and their Limits

·         Learns Theorems on limits

·         Understands Monotone sequences and their properties

·         Have a clearcut idea about subsequences

·         Understands Bolzano- Weierstrass Theorem

·         Understands Cauchy Criterion and its uses

·         Have an idea about Convergence/ divergence of sequences

 

3

 

 

3

 

 

 

3

 

 

 

 

2

 

 

3

 

 

3

 

 

 

5

Seminar will be given to ¼ th of students
Internal I
3 Introduction to Series

 

Absolute Convergence

 

Tests for Absolute convergence

 

Tests for Non absolute Convergence

 

·         Understands the difference between sequence and series

·         Understands properties of Series

·         Differentiate between convergence and absolute convergence and understands whether a series is absolute convergent

·         Convergence and divergences tests

·         Given a series, they could test for the convergence/divergence

 

 

1

 

 

 

 

3

 

 

 

5

 

 

 

 

 

 

 

10

 

 

4

Seminar will be given to ¼ th of students students
4 Limits of Functions

 

Limit Theorems

 

Some Extensions of the Limit Concept.

·         Understands limit concepts

·         Studies theorems on limits

·         Understands some extensions of limits

4

 

 

 

4

 

 

4

 

Seminar will be given to ¼ th of students students
Internal II

 

TEACHING SCHEDULE

TOPICS HOURS DATES DESCRIPTION
 

Module I

 

Subsets of Real numbers and its basic properties

 

6 July 15, 16, 17, 20, 21, 22  
Algebraic Properties of R 2 July 23, 24  
Order Properties of R 2 July 27, 28  
Absolute Value 1  July  29  
The Order Completeness and other Properties of R 3 August 3, 4, 5  
Supremum, Infimum and its properties 5 August 10-14  
Intervals and Nesterd interval properties 5 August 17-21  
R is countable 3 August 25-27  
Discussion on Important Ideas   September 7  
Unit Test   September 8  
 

Module II

 

Have an idea about Sequences and their Limits and studies the Theorems  

6

 

September 9 -11, 14-16

 
Monotone sequences and their properties 3 September 17, 18, 22  
Subsequences and Bolzano- Weierstrass Theorem

 

5 September 23 -25, 28-29  
Cauchy Criterion and its uses 3 October 1-2  
Convergence and Divergence 8 October 5-14  
 

Module III

 

Introduction to Series 4 October 15, 16 Two hrs each
 Absolute Convergence 5 October 19-21  
 Tests for Absolute convergence 8 October 22-26  
Tests for Non absolute Convergence 6 October 27-30  
 

Module IV

 

Limits of Functions

 

4 November 2-4  
Limit Theorems 4 November 5-7  
Some Extensions of the Limit Concept. 4 November 9-13  

 

PEDAGOGY

After the course the students will understand the Basic Concepts in Mathematical Analysis. The tutorials will focus on the learning and understanding of concepts. This course is a stepping stone to study the Real Analysis.

EVALUATION STRATEGY

The internal evaluation is based on the students attendance, seminar, Internal Assessments and assignment marks.

EVALUATION SCHEME

Sl. No Components Weightage
1 END TERM EXAM 80
2 ATTENDANCE 5
3 INTERNAL MARKS 10
4 SEMINARS/ ASSIGNMENT/ VIVA 5

 

TEXT BOOK :

Introduction to Real Analysis – Robert G Bartle and Donald R Sherbert (3rd Edition) John Wiley & Sons, In. 2007

 

REFERENCE BOOKS:

  1. Richard R Goldberg – Methods of real Analysis, 3rd edition , Oxford and IBM

Publishing Company (1964)

  1. Shanti Narayan – A Course of Mathematical Analysis, S Chand and Co. Ltd ( 2004)
  2. Elias Zako – Mathematical Analysis Vol 1, Overseas Press, New Delhi ( 2006)
  3. J.M Howie – Real Analysis, Springer 2007.
  4. K.A Ross- Elementary – Real Analysis, Springer, Indian Reprints.
  5. S.C Malik, Savitha Arora – Mathematical Analysis, Revised Second Edition

WEB REFERENCE:

FACULTY DETAILS

Website: stpauls.ac.in

Email: [email protected]

Mobile: 9846335837

SEMINARS/ASSIGNMENTS

Topics will be announced in the online/offline class during the course.

 

 

 

 

  • REAL NUMBERS

    Finite and Infinite Sets, The Algebraic and Order Properties of R, Absolute Value and Real Line, The Completeness Property of R, Applications of the Supremum Property, Intervals.

    No items in this section
  • SEQUENCES

    Sequences and their Limits, Limit Theorems, Monotone Sequences, Subsequences and the Bolzano- Weierstrass Theorem, The Cauchy Criterion, Properly Divergent Sequences.

    No items in this section
  • SERIES

    Introduction to Series, Absolute Convergence, Tests for Absolute convergence, Tests for nonabsoute Convergence

    No items in this section
  • LIMITS

    Limits of Functions, Limit Theorems, Some Extensions of the Limit Concept.

    No items in this section
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Dr. Manju K Menon
Dr. Manju K. Menon joined St. Paul's College, Kalamassery in 2005. She was an NBHM Research Fellow in 2002-2005 and completed her Ph D in Graph Theory under the guidance of Prof. A. Vijayakumar, CUSAT in 2010. She has 10 publications to her credit and successfully completed a Minor Research Project Funded by UGC. She was the convener of two UGC Sponsored National Seminars and a Compact Course organised in association with the Ramanujan Mathematical Society. She is currently the Controller of Examinations of St. Paul's College, Kalamassery. She is a life member of RMS, IMS and KMA. She is also a resource person in many National and International Seminars including the Third Indo Taiwan International Conference on Discrete Mathematics held at Taiwan in 2013. She is a research guide in Mathematics in Maharajas College, Ernakulam affiliated to Mahathma Gandhi University, Kottayam.

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₹1,000.00