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Course Title: Real Analysis |
Full Marks: 100 |
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Course No: Math Ed 423 |
Pass Marks: 35 |
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Nature of Course: Theory |
Period Per Week: 6 |
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Level: Bachelor Degree |
Total Period: 150 |
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Year: Second |
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1. Course Description:
This course is designed to provide students with the fundamental concepts of
Real Analysis. It deals with a rigorous development of the subject which
includes the background of the knowledge of differential and integral calculus.
The course focuses on the subject matter specially based on real line only. It
covers foundation properties of real numbers, topological framework of real
numbers, real sequences, infinite series of real numbers, limits, continuity,
derivability and Riemann integrability of real functions.
2. General Objectives:
The general objectives of this course are as follows:
·
To acquaint the students with the
axiomatic structure of real number system.
·
To familiarize the students with the
various properties of open and closed sets.
·
To identify the features of real
sequences and their convergence.
·
To apply
different tests for the convergence of infinite series.
·
To develop a deeper understanding of the
properties of infinite series in the students with the arbitrary terms and
infinite product.
·
To make students able to have a deeper
understanding of limits of functions.
·
To acquaint with the properties of
continuous functions defined on an interval.
·
To familiarize the students with the
properties of uniform continuity, non uniform continuity of functions and
monotonic functions.
·
To enable the students in understanding
the derivable and non-derivable functions at a point or on an interval.
·
To apply the concept of derivative in
expanding real functions in finite or infinite series.
·
To make the students able to apply
concept of derivative in examining extreme values of real functions.
·
To familiarize the students with the
Darboux sums of real functions on a closed interval.
·
To acquaint the students with the
properties of Riemann integrability of functions.
3. Specific Objectives and Contents
|
Objectives |
Contents |
|
·
Describe
different system of real numbers ·
Describe
algebraic structure of real numbers. ·
Explain
the properties of real numbers. ·
Describe
order axioms of real numbers and their properties ·
Explain
and prove properties of supremum and infimum of a set of numbers. ·
Explain
completeness property Archimedian property and Dedekind property of real
numbers. ·
Explain
absolute value of a real number and its properties. |
Unit 1. Real Numbers (12) 1.1 System of real
numbers 1.2 Algebraic
structure of R 1.3 Order axioms
and properties of R 1.4 Absolute value
of a real number and its properties. 1.5 Boundedness of
subsets of R 1.6 Completeness
axioms in R 1.7 Archimedian property 1.8 Dedekind's
construction of the set of real numbers 1.9 Representation
of real numbers on a line. |
|
·
Expkain
neighbourhood of a point ·
Prove
properties of open sets in R ·
Explain
and prove properties of interior of a set. ·
Explain
limit point of a set ·
Prove
properties of limit points of a set and derived set ·
Prove
properties of closed sets. ·
Prove
properties of compact sets connected sets and perfect sets. |
Unit-II. Open and Closed
Sets (15) 2.1 Open and closed
intervals 2.2 Neighbourhoods 2.3 Interior points
and interior of a set 2.4 Open sets 2.5 Limit points of
a set 2.6
Bolzano-Weirstrass's theorem 2.7 Closed sets 2.8 Covering of a
set 2.9 Compact sets 2.10 Cantor sets 2.11 Connectedness |
|
·
Explain
convergence of a sequence ·
Prove
properties of convergent sequence of
real numbers ·
Prove
properties of a Cauchy sequence. ·
Explain
non-convergent sequences and their properties ·
Prove
properties of monotonic sequences ·
Describe
subsequences and prove properties of subsequences ·
Explain
uniform convergence of a sequence ·
Discuss
the concept of Cesaro summability for sequences |
Unit-III
Real Sequences (20) 3.1 Convergent
sequences 3.2 Cauchy Sequence 3.3 Cauchy's
Criterion for convergence 3.4 Non-convergent
sequences 3.5 Cauchy's first
and second theorems on limit 3.6 Monotonic
sequences, monotone convergence theorem, Cantor's intersection theorem 3.7 Subsequences 3.8 Uniform convergence 3.9 Summability of
sequences |
|
·
Explain
the conditions for the convergence of an infinite series ·
Establish
different tests for the convergence of infinite series ·
Explain
the convergence of an infinite series with positive and negative terms. ·
Explain
absolute convergence and conditionally convergent of a series ·
Prove
properties of the re-arrangement of the terms of an infinite series. ·
Prove
the properties of the convergence of an infinite product ·
To
discuss the Cesaro summability of series. |
Unit-IV.
Infinite Series (25) 4.1 Meaning of an
infinite series 4.2 Sequence of
partial sums of a series 4.3 Convergence of
an infinite series 4.4 Cauchy's
general principle of convergence of series 4.5 Series of
positive terms 4.6 Different tests
of convergence of series (comparison test, P-test, D-Alembert's,ratio test,
root test, Rabee's test, Kummer's test, logarithmic ratio test) 4.7 Series of
positive and negative terms 4.8 Alternative
series and Leibnitz' test 4.9 Absolute
convergent and conditionally convergent of series 4.10 Arbitrary
series and infinite products, (Dirichlet's theorems), Abel's theorem) 4.11 Grouping of
terms of a series 4.12 Re-arrangement
of terms of series 4.13 Infinite
product and its convergence 4.14 Summability of
series |
|
·
Explain
types of functions ·
Discuss
boundedness of functions ·
Prove
properties of limits of functions ·
Explain
|
Unit-V
Functions and Limits (10) 5.1 Types of
functions 5.2 Boundedness of
function 5.3 Monotonic
functions 5.4 Limit of
functions 5.5 Properties of
limits of functions 5.6 One sided
infinite limits |
|
·
Explain
the concept of continuity of a function at a point and on intervals ·
Prove
theorems on algebra of continuous functions ·
Classify
discontinuous functions at a point ·
Prove
theorems on properties of continuous functions ·
Prove
the theorems on properties of monotonic functions ·
Prove
theorems on properties of uniform continuity of functions |
Unit-VI
Continuity of Functions (18) 6.1 Continuous
functions 6.2 Continuity of
intervals and sets 6.3 Properties of
continuous functions 6.4 Discontinuous
functions 6.5 Sign preserving
theorem, intermediate value theorem, Bolzano theorem and fixed point theorem 6.6 Continuity of
inverse functions 6.7 Continuity of monotonic
functions 6.8 Uniform
continuity of functions 6.9 Lipschitz
functions |
|
·
Explain
the concept of derivative ·
Establish
the relation between continuity and derivability ·
Prove
properties of derivability in relation to algebraic compostions ·
Prove
Darboux theorem and its consequences ·
Establish
and illustrate mean value theorems ·
Prove
Taylor's series (finite and infinite) and Maclaurin's series ·
Discuss
extreme values of functions ·
Discuss
and prove various types of intermediate forms and their properties |
Unit-VII
Derivability (22) 7.1 Derivative of a
function 7.2 Derivative of
inverse function 7.3 Darboux theorem 7.4 Mean value
theorems (Rolle's theorem, Lagrange's theorem, Cauchy's theorem) 7.5 Deductions from
mean value theorems 7.6 Generalized
mean value theorems 7.7 Taylor
Polynomial 7.8 Power series
representation of functions 7.9 Extreme values
of a functions 7.9 Indeterminate
forms 7.10 L-Hospital's
rule. |
|
·
Establish
relationship between lower and upper Darboux sums of a bounded function on a closed interval ·
Compute
lower and upper Darboux sums of functions with a partition defined on the
domain of definition of the function ·
Establish
the properties of lower and upper Riemann integrals ·
Establish
the necessary and sufficient condition for integrability of a function ·
Prove
the properties of integrable functions ·
Establish
mean value theorems of integral calculus ·
Establish
fundamental theorem of integral calculus ·
Establish
the techniques of evaluating the definite integrals, by integration by parts and change of variable
of integration |
Unit-VIII
Riemann Integral (28) 8.Partitions 8.2 Lower and upper
Darboux sums and properties of Darboux sums. 8.3 Upper and lower
integrals 8.4 Riemann
Integral 8.5 Necessary and
sufficient condition for integrability 8.6 Properties of
integrable functions 8.7 Mean value
theorems (Mean value theorems of integral calculus, generalized mean value
theorem) 8.8 Bonnet's and
Weierstrass' mean value theorem. 8.9 Continuity and
derivability of integrable functions 8.10 Fundamental
theorems of integral calculus 8.11 Integration by
parts 8.12 Change of
variable |
4. Instructional Techniques:
The nature of this course being theoretical, teacher
centered teaching technique will dominate the teaching learning process. The teaching techniques are referred
as follows:
4.1 General instructional techniques
·
Lecture with illustrations
·
Discussion
·
Inquiry and question answer
·
Demonstration
4.2 Specific Instructional
techniques
·
Individual and group work presentation
for illustrations and exercise for all units
5. Evaluation:
Students will be evaluated on the basis of written classroom test in between and at the end of academic
session, the classroom participation, presentation
of the reports, and other activities. The scores obtained will be used only for feedback purposes. The
office of the Controller of Examination will
conduct annual examination at the end of the academic session to evaluate students' performance. The types
of questions and marks of each type of
questions will be as follows:
|
Types
of questions |
Total
questions to be asked |
No.
of questions to be answered and marks to be allocated |
Full
Marks |
|
Multiple
choice items |
20
questions |
20 |
20 |
|
Short
answer questions |
8
(with 3 alternatives) |
8 |
56 |
|
Long
answer questions |
2
(with 1 alternatives) |
2 |
24 |
6. Recommended and Reference
Books.
6.1
Recommended Books
Gupta, S. L. & Gupta, N. R. (1993). Fundamentals of real analysis. New
Delhi, Vikas Publishing House Pvt. Ltd.
Maskey S.M. (2007), Principles of real analysis (second ed.). Kathmandu : Ratna Pustak Bhandar (For Units
I-VIII).
Pandey U. N. (2003). Real analysis ( Fourth revised ed.2015 ) Kathmandu : Vidyarthi
Prakashan Pvt. Ltd. ( For units I - VIII).
6.2
References
Apstol, T. M. (1997). Mathematical analysis. Tokyo: Addison Wesly Publishing Company
Bartle, R. G. & Sherbert, D. R. (1982). Introduction
to real analysis. New York: John Wiley and Sons
Bhattarai, B. N. & Shrestha, B. K.,(2072). A textbook of real analysis (revised
edition) Kathmandu: Shuvakamana Prakashan Pvt. Ltd.
Jain, P. K. & Kaushik S.K. (2001). An
introduction to real analysis, New Delhi: S. Chand and Comp. Ltd.
Narayan, S. (1971). A course of mathematical analysi., Delhi: S. Chand and Com. Ltd.
Rudin W. (194), Principles
of mathematical analysis, Newyork; Mc. Graw Hill
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