1. Course Description
This course deals with the additional topics of real analysis. It provides a rigorous development of the techniques of analysis dealing with the topics such as convergence of improper integrals, sequences and series of functions, function of several variables, multiple integrals, metric spaces and approximation methods of calculating roots of equations and definite integrals. Besides these , this course also introduces metric space and numerical methods which would be foundations for the study of higher mathematical concepts .
2. General Objectives
The general objectives of this course are as follows:
1. To make the students understand the convergence of the improper integrals and uniform convergence of the sequence of the functions.
2. To provide the students with the knowledge of basic properties of power series
3. To help students understand the limit, continuity, differentiability, chain rule and extreme values of the functions defined on Rn
4. To familiarize students with the use of Lagrange’s method of multipliers to find the stationary points in implicit functions.
5. To help students
to analyze the properties of double and triple integrals
6. To acquaint students with the basic features of metric space
7. To familiarize the students with the approximate values of the roots of the equations and find the solution of definite integrals.
3. Contents
|
Contents |
|
Unit I: Improper Integral (18) 1.1 Improper
integrals and their convergence 1.2 Comparison
test 1.3 General
test for convergence 1.4 Absolute
convergence 1.5 Abel’s test 1.6 Dirichlet’s test Unit II: Sequence and Series of Functions (18) 2.1 Pointwise and uniform convergence sequence of functions 2.2 Cauchy criterion for uniform convergence 2.3 Tests for uniform convergence of sequences and series of functions 2.4 Properties
of uniformly convergent sequences and series 2.5 Dini’s integration Unit III: Power Series
(16) 3.1 Basic concepts of power series 3.2 Basic theorems on power series 3.3 Cauchy Hadamard
theorem 3.4 Differentiation theorem 3.5 Multiplication theorem
and Taylor’s series Unit IV: Functions of Several Variables (26) 4.1 Sets and functions in Rn 4.2 Limiting values of functions of several variables 4.3 Continuous functions
of several variables Partial derivatives 4.4 Directional derivatives and differentials of a function
of several variables 4.5 Partial derivatives of higher orders including Schwartz theorem
and Young’s theorem 4.6 The chain rule 4.7 Taylor’s theorem 4.8 Extreme values
of functions of two and three variables Unit V: Implicit Functions
(10) 5.1 Concept of implicit
functions 5.2 Existence theorem 5.3 Derivative of implicit
functions 5.4 Jacobian and its properties UnitVI: Multiple
Integrals (28) 6.1
Line integrals over the plane
curves 6.2 Double integral over rectangle 6.3 Conditions of integrability 6.3 Properties of integrable functions 6.4 Fubini’s theorem 6.5 Lebnitz theorem 6.6 Double integral over a region 6.7 Green’s theorem and its deductions 6.8 Double integrals in polar form 6.9 Surface area 6.10 Triple integral over a parallelepiped 6.11 Triple integrals in cylindrical
and spherical coordinates Unit VII: Metric Spaces (18)
Metric space and examples
Open balls, closed balls
Open sets and closed sets
Closure of a set
Boundary of a set
Diameter of a set
Subspaces of a metric space
Continuous mapping on a metric space Cauchy sequence Complete metric space Compact metric spaceUnit VIII: Numerical Methods
(16)
Rounding off errors Truncation error Rounding off errors in basic computational process
Difference of a polynomial, locating, evaluating and correcting mistakes in difference table
Linear interpolation Approximate roots of algebraic and transcendental
equations by bisection method, false position method,
Newton- Raphson method |
4 .Instructional Techniques
Because of the theoretical nature of the course, teacher-centered instructional techniques will be dominant in the teaching learning process. The teacher will adopt the following techniques
· Lecture with illustration
·
Discussion
· · Demonstration
·
Question-answer
Specific Instructional Techniques
Unit-wise specific
instructional techniques are suggested as follows :
|
Units |
Specific Instructional Techniques |
|
Unit I |
Individual assignment and group
discussion |
|
Unit II |
Problem solving and presentation |
|
Unit III |
Discussion and assignment |
|
Unit IV |
Group works |
|
UnitV |
Individual discussion and
assignment |
|
Unit VI |
Problem solving and presentation |
|
Unit VII |
Group discussion and
assignment |
|
Unit VIII |
Discussion and project
work |
5.Evaluation
The office of the Controller of the Examination will conduct the annual examination at the end of the academic session to evaluate the students’ performance. The types, number and marks of the objective and subjective questions will be as follows:
|
Types of questions |
Total number of questions |
Number of questions and marks allocated |
Total marks |
|
Group A: Multiple choice items |
20 questions |
20 x 1 mark |
20 |
|
Group B: Short answer questions |
8 with 3 ‘or’ questions |
8 x 7 marks |
56 |
|
Group C: Long answer questions |
2 with 1 ‘or’ question |
2 x 12 marks |
24 |
5. Recommended Books and References
Recommended Books
Mallik, S. C. & Arora, S. (1992)
Mathematical analysis.
New Delhi: New Age
International (P.) Limited Publishers ( I-VII)
Sastry, S. S. (1990). Introductory methods of numerical analysis. New Delhi: Prentice Hall of India(VIII)
Reference Books
Bhattarai B.N. (2074)Advanced calculus , Kathmandu : Cambridge Publication
Goldberg, R. R. (1970).
Methods
of real analysis. New Delhi:
Oxford and IBH Publishing Co. Pvt. Ltd.
Delhi: S. Chanda & Company Ltd.
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