Analysis-I

Teacher

Dr. Bhairaba Kumar Majhi

Category

Core Courses

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Course Name : Analysis-I

Code(Credit) : CUTM-1513(3-0-3)

Course Objectives:

Analysis extends and refines calculus; it encompasses differentiation, integration, measure, limits, infinite series, and analytic functions, primarily in the context of real and complex number systems.

In much of analysis, the emphasis is not on finding explicit solutions to specific problems, but rather on determining which problems can be solved and what general properties solutions may share

Learning Outcomes:

Upon successful completion of this course, students will be able to:

Describe the real line as a complete, ordered field,

Use the definitions of convergence as they apply to sequences, series, and functions,

Determine the continuity, differentiability, and integrability of functions defined on subsets of the real line,

Produce rigorous proofs of results that arise in the context of real analysis.

Course Syllabus:

Module-I   (5 hr+0 hr+4 hr)

Field Structure and Order Structure; Bounded and unbounded Sets, Supremum, Infimum;Completeness in the Set of Real numbers; Archimedean Property of Real numbers; Inequalities and Metric property of R

project 1: A study on completeness property of R and its application.

project 2: A Report on Some Inequalities of Real Numbers.

Module-II (5 hr+0 hr+6 hr)

Neighborhood of a point, Interior point of a set, Open Sets; Some Important Theorems on open Sets; Limit point of a Set, Closed Sets, Closure of a Set, Dense Set; Some Useful Theorems on Open Set and Closed Sets; Blozanoweirstrass theorem for Sets.

Project 3 : A study on some Useful Theorems on Limit point of a Sets.

Project 4:  A Study on  some Useful Theorems on Open Set and Closed Sets.

Project 5:  A study on the Blozanoweirstrass theorem for Sets.

Module-III   (6 hr+0 hr+6 hr)

Real Sequences and Theorems on Convergence of  Sequence;Limit point  a Sequence and Bolzano-Weierstrass theorem for Sequences; Convergent and Non-Convergent Sequences; Cauchy’s general Principle of Convergence and Cauchy Sequence;Some Important theorems on Real Sequences; Monotonic Sequences and Sub sequences.

Project 6: A Report on Cauchy’s general Principle of Convergence and Cauchy Sequence.

Project 7: A Report on Convergent and Non-Convergent Sequences.

Project 8: A study on Some Important Theorems of Real Sequences.

Module-IV   (5 hr+0 hr+6 hr)

Infinite Series and Some Preliminary Theorems; Positive term Series and Condition for Convergence ; Geometric Series and Comparison Series for Convergence of Infinite Series; Comparison Tests for Positive Term Series ; Cauchy’s Root test and D’ Alembert's Ratio Test.

Project 9:  A study on Some useful theorems of Infinite Series.

Project 10: A report on Comparison Test for Positive term Series.

Project 11: A Report on Cauchy’s Root test and D’ Alembert's Ratio Test.

Module-V   (3 hr+0 hr+4 hr)

Alternating Series and Leibnitz Test ; Absolute and Conditional Convergence; Solving problems on Absolute and Conditional Convergence.

Project 12: A Report on Alternating Series and Leibnitz Test.

Project 13: A Report on Absolute and Conditional Convergence.

Module-VI   (4 hr+0 hr+4 hr)

Limit and Continuity of Functions; Discontinuities and Types of Discontinuity;Uniform continuity and related Theorems; Differentiability of Real functions.

Project 14: A Report  on Finding Limit and continuity of Functions.

Project 15: A Report  on Differentiability of Real functions.

Module-VII (4 hr+0 hr+6 hr)

The Derivative and Higher Order Differentiation's; Darboux's Theorem and Roll's Theorem;Lagrange's Mean value theorems and Cauchy's Mean value Theorem ;Taylor’s Theorem with Reminder.

Project 16: A Report on Darboux's Theorem  and  Roll's Theorem .

Project 17: A Report on Lagrange's Mean value theorems and Cauchy's Mean value Theorem.

Project 18 : A study on Taylor's Theorem with Remainder and its Importance.

Text Book:

Mathematical Analysis (Wiley Eastern) : S.C. Malik and S.Arora (4 th Edition)

Chapters: 1 (except 4.3 and 4.4),  2, 3, 4 (upto Art.5 and Art 10.1, 10.2),  5, 6,

Reference Books:

Fundamentals of Real Analysis :S.L.Gupta& Nisha Rani

Mathematical Analysis-II : Sharma &Vasistha

Fundamental of Mathematical Analysis :G. Das&S.Pattanayak

Session Plan:

Session 1:

Field Structure and Order Structure

https://www.youtube.com/watch?v=1CnGQ-whizk

Session 2:

Session 3:

Completeness in the Set of Real numbers

https://www.youtube.com/watch?v=h7jxY0ksYZ8

Session 5:

Inequalities and Metric property of R.

https://www.youtube.com/watch?v=lj765RaxreE

Session -(6 & 7):

Project 1:

A study on completeness property of R and its application.

Session-( 8 & 9):

Project 2:

A Report on Some Inequalities of Real Numbers.

Session 10:

Real Sequence, Limit Points of a sequence Limit inferior and superior

https://www.youtube.com/watch?v=X-7KTO2-gjg&t=207s

https://www.youtube.com/watch?v=ADVZOhMDmQI

 

Session 11:

Some Important Theorems on open Sets.

https://www.youtube.com/watch?v=97vXMfhtWd0

Session 12:

Limit point of a Set, Closed Sets, Closure of a set, Dense Set.

https://www.youtube.com/watch?v=q6698vWK3ts&t=127s

Session 13:

Some Useful Theorems on Open Set and Closed Sets.

https://www.youtube.com/watch?v=uMreZL8X1Pk

 

Session 14:

Blozanoweirstrass theorem for Sets.

 

Session-( 15 & 16):

Project 3:

A study on some Useful Theorems on Limit point of a Sets.

Session -(17 & 18):

Project 4:

A Study on  some Useful Theorems on Open Set and Closed Sets.

Session -(19 & 20):

Project 5:

A study on the Blozanoweirstrass theorem for Sets.

Session 21:

Real Sequences and Theorems on Convergence of  Sequence

 

Session 22:

Limit point  a Sequence and Bolzano-Weierstrass theorem for Sequences.

https://www.youtube.com/watch?v=s8g8rUCfdtA

https://www.youtube.com/watch?v=3EEdowen-qI

Session 23:

Convergent and Non-Convergent Sequences.

https://www.youtube.com/watch?v=L-JqHo4-W4k

Session 24:

Cauchy’s general Principle of Convergence and Cauchy Sequence.

https://www.youtube.com/watch?v=Q-tKlgx3ddw&t=21s

https://www.youtube.com/watch?v=jD5s2jvnH1k

Session 25:

Some Important theorems on Real Sequences.

 

Session -(27 & 28):

Project 6:

A Report on Cauchy’s general Principle of Convergence and Cauchy Sequence.

Session -(29 & 30):

Project 7:

A Report on Convergent and Non-Convergent Sequences.

Session -(31 & 32):

Project 8:

A study on Some Important Theorems of Real Sequences.

 

Session 33:

Infinite Series and Some Preliminary Theorems

 

Session 34:

Positive term Series and Condition for Convergence

 

Session 35:

Geometric Series and Comparison Series for Convergence of Infinite Series

 

Session 36:

Comparison Tests for Positive Term Series

 

Session 37:

Cauchy’s Root test and D’ Alemberts Ratio Test.

 

Session -(38 & 39):

Project 9:

A study on Some useful theorems of Infinite Series.

Session -(40 & 41):

Project 10:

A report on Comparison Test for Positive term Series.

Solving Problems on Limit and continuity of functions

https://www.youtube.com/watch?v=GMyQRRmUSWY&list=PL476FB76FF4883C39&index=5

Session -(42 & 43):

Project 11:

A Report on Cauchy’s Root test and D’ Alembert's Ratio Test.

Discontinuities and Types of Discontinuity

https://www.youtube.com/watch?v=kWcsZ9AANSs

Session 45:

Absolute and Conditional Convergence;

https://www.youtube.com/watch?v=FPK6LO1iiXc&t=125s

Session 46:

Solving problems on Absolute and Conditional Convergence.

https://www.youtube.com/watch?v=EdvguTClZrE

Session -(47 & 48):

Project 12:

Solving problems on Uniform continuity

Solving problems on Uniform continuity

https://www.youtube.com/watch?v=Fs6em4tg-Yo

Session-(49 & 50):

Project 13:

Solving problems on Taylor’s Theorem

Solving problems on Taylor’s Theorem

https://www.youtube.com/watch?v=Jk9xMY4mPH8

Session 51:

Limit and Continuity of Functions

https://www.youtube.com/watch?v=9brk313DjV8&t=980s

Session 52:

Discontinuities and Types of Discontinuity.

https://www.youtube.com/watch?v=1rMY4Xckd64

 

Session 53:

Session 54:

Differentiability of Real functions.

https://www.youtube.com/watch?v=xuAiQOzIkWY

Session -(55 &  56):

Project 14:

A Report  on Finding Limit and continuity of Functions.

Session -(57 &  58):

Project 15:

A Report  on Differentiability of Real functions.

Session 60:

Darboux's Theorem and Roll's Theorem

 

https://www.youtube.com/watch?v=LbR-T6_jvj0

 

Session 61:

Lagrange's Mean value theorems and Cauchy's Mean value Theorem.

https://www.youtube.com/watch?v=cTp-VaCU7XU

https://www.youtube.com/watch?v=VxH89U2E1bQ

Session 62:

Taylor’s Theorem with Reminder.

https://www.youtube.com/watch?v=DP_pGQaNGdw

Session-( 63 & 64):

Project 16 :

A Report on Darboux's Theorem  and  Roll's Theorem .

 

Session -(65 & 66):

Project 17:

A Report on Lagrange's Mean value theorems and Cauchy's Mean value Theorem.

Session -(67 & 68):

Project 18:

A study on Taylor's Theorem with Remainder and its Importance.

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Recent Comments

    Our Main Teachers

    My Self Dr.Bhairaba Kumar Majhi,M,Sc,B.Ed,M.Tech(CSE),Ph.D(Mathematics ) ,Asst.Professor And HOD Department of Mathematics School Of Applied Sciences,Presently working at CUTM,Bolangir Campus.I have 16 Years of Experience in Teaching at Different Levels Of UG,PG & Engineering Mathematics.