MATHEMATICS
Course title: Advanced Calculus Full Marks:100
Course No.: Math. Ed.446 Pass Marks: 35
Nature of the course: Theory Periods per week: 6 Level: B. Ed. Total periods:150
Year:Fourth
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:
- To make the students understand the convergence of the
improper integrals and uniform convergence of the sequence of the functions.
- To provide the students with the knowledge of basic
properties of power series
- To help students understand the limit, continuity, differentiability, chain
- rule
and extreme values of the functions defined on Rn
- To familiarize students with the use of Lagrange’s
method of multipliers to find the stationary points in implicit functions.
- To help students to analyze the properties of double and triple integrals
- To acquaint students with the basic features of metric space
- To familiarize the students with the approximate values
of the roots of the equations and
find the solution of definite integrals.
3. Specific Objectives and Contents
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Specific Objectives |
Contents |
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· Distinguish between proper and improper integral with examples Discuss the convergence of the given improper integrals Test the convergence of the given improper integral by |
Unit I: Improper Integral (18) Improper integrals and their convergence Comparison test General test for convergence Absolute convergence Abel’s test |
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comparision test, M- test, Abel and
Dirichlet test ·
Test the absolute convergence for the given improper integrals |
1.6 Dirichlet’s test |
|
·
Define the pointwise convergence of sequences and of functions with
examples ·
State and test the uniform convergence of sequences
of functions and series ·
State the Cauchy criterion
for uniform convergence ·
Apply Abel’s test and Dirichlet’s test for uniform
convergence ·
Prove the properties of uniform convergence of
sequences and series ·
Deal uniform convergence with differentiation and integration. |
Unit II:
Sequence and Series of Functions (18)
Pointwise and uniform convergence
sequence of functions
Cauchy criterion for uniform
convergence
Tests for uniform convergence of
sequences and series of functions
Properties of uniformly convergent
sequences and series
Dini’s integration
Uniform convergence and integration,
Uniform convergence and differentiation |
|
·
Explain the basic features of power series ·
State and
prove basic theorems of power series ·
Prove Cauchy Hadamard
theorem and Abel’s theorem ·
Determine the radius
of convergence of a given power series. |
Unit III: Power Series (16)
Basic concepts of power series
Basic theorems on power series
Cauchy Hadamard theorem
Differentiation theorem
Multiplication theorem and Taylor’s
series
Abel’s theorem |
|
·
Define sets and functions in Rn ·
Determine the limit and continuity of the real valued functions in R 2 and R 3 ·
Define partial derivatives of the functions of several variables and
derive related results ·
Explain the differentiability of the functions of several variables and the
sufficient |
Unit IV:
Functions of Several Variables (26)
Sets and functions in Rn
Limiting values of functions of several
variables
Continuous functions of several
variables
Partial derivatives
Directional derivatives and differentials of a function of several variables
Partial derivatives of higher orders |
|
condition of
differentiability ·
Prove the necessary and sufficient conditions for extreme values of
functions of two variables. ·
State Taylor’s formula for functions of two variables |
including Schwartz theorem and Young’s theorem
The chain rule
Taylor’s theorem
Extreme values of functions of two
and three variables |
|
·
Explain implicit function with examples ·
State the existence theorem ·
Define Jacobian of the functions of the several variables ·
List the properties of Jacobian of a function and
solve related problems ·
Determine the stationary point of a given function
under given conditions ·
Apply Lagrange’s method of multiplier to find the stationary points |
Unit V: Implicit Functions (10)
Concept of implicit functions
Existence theorem
Derivative of implicit functions
Jacobian and its properties
Stationary points and Lagrange’s
method of multipliers |
|
·
Define line integrals over the plane curves ·
Explain the properties of line integrals ·
Define double integral over the rectangle ·
Prove the Fubini’s theorem and other related theorems ·
Solve the problems of double integrals over a region ·
State and prove Green’s theorem and its deductions ·
Determine
the area of closed regions through
Green’s theorem ·
Define double integral in polar form and solve the related problems ·
State triple integral with examples ·
Evaluate triple integrals in cylindrical and spherical |
UnitVI: Multiple Integrals (28)
Line integrals over the plane curves
Double integral over rectangle
Conditions of integrability
Properties of integrable functions
Fubini’s theorem
Lebnitz theorem
Double integral over a region
Green’s theorem and its deductions
Double integrals in polar form
Surface area
Triple integral over a parallelepiped
Triple integrals in cylindrical and
spherical coordinates
Evaluation of triple integrals in cylindrical and spherical coordinates. |
|
forms |
|
|
·
Define metric space with |
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 sequences
Complete metric space
Compact metric space |
|
examples |
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|
·
Define ball, open set ,closed |
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set and subspace of a metric |
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space and prove related |
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|
theorems |
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·
Prove the theorems of |
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continuous mappings on |
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metric space |
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·
State and prove the |
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theorems on complete |
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metric space and compact |
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|
metric space |
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·
List the possible sources of |
Unit VIII: Numerical
Methods (16)
Rounding off errors
Truncation errors
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
Integration by Simpson rule and trapezoidal rule |
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human errors in |
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|
computations |
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·
Distinguish between |
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rounding off error and |
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truncation error |
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·
Construct the difference |
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table for given polynomial |
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·
Construct the equation of |
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the polynomial and |
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interpolate the values of the |
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function from the |
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difference table |
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·
Find the approximate |
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solutions of the algebraic |
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and transcendental |
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equations by bisection, false |
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position,and Newton- |
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Raphson methods |
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·
Find the integration of the |
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given function by Simpson |
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rule and trapezoidal rules |
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.
General instructional Techniques
4.
·
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
David, V. W. (1996). Advanced calculus. New Delhi: Prentice Hall of India
Goldberg, R. R. (1970). Methods of real analysis. New Delhi: Oxford and IBH Publishing Co.
Pvt. Ltd.
. Narayan, S. & Raisinghania, M. D. (2009). Elements of real analysis (10th
Ed.). New Delhi: S.
Chanda & Company Ltd.
Pahari, N. P. (2063). A textbook of mathematical analysis. Kathmandu:
Sukunda Pustak Bhawan.
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