Syllabus MSc Mathematics PU

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Syllabus MSc Mathematics PU

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Syllabus for M.Sc. Mathematics (Morning/Evening)

Programmed

2 years M.Sc. Mathematics programmed consists of two parts namely Part-I and Part II.

The regulation, Syllabi and Courses of Reading for the M.Sc. (Mathematics) Part-I and

Part-II (Regular Scheme) are given below.

Regulations

The following regulations will be observed by M.Sc. (Mathematics) regular students

i. There are a total of 1200 marks for M.Sc. (Mathematics) for regular students as is

the case with other M.Sc. subjects.

ii. There are five papers in Part-I and six papers in Part-II. Each paper carries 100

marks.

iii. There is a Viva Voce Examination of M.Sc. Part II. The topics of Viva Voce

Examination shall be:

Analysis (Real, Complex and Functional)

Algebra and Topology

Mechanics

M.Sc. Part-I

The following five papers shall be studied in M.Sc. Part-I:

Paper I Real Analysis

Paper II Algebra

Paper III Complex Analysis and Differential Geometry

Paper IV Mechanics

Paper V Topology and Functional Analysis

Note: All the papers of M.Sc. Part-I given above are compulsory.

M.Sc. Part-II

In M.Sc. Part-II examinations, there are six written papers. The following three papers

Are compulsory. Each paper carries 100 marks.

Paper I Advanced Analysis

Paper II Methods of Mathematical Physics

Paper III Numerical Analysis

Optional Papers

A student may select any three of the following optional courses:

Paper IV-VI option (i) Mathematical Statistics

Paper IV-VI option (ii) Computer Applications

Paper IV-VI option (iii) Group Theory

Paper IV-VI option (iv) Rings and Modules

Paper IV-VI option (v) Number Theory

Paper IV-VI option (vi) Fluid Mechanics

Paper IV-VI option (vii) Quantum Mechanics

Paper IV-VI option (viii) Special Theory of Relativity and Analytical Mechanics

Paper IV-VI option (ix) Electromagnetic Theory

Paper IV-VI option (x) Operations Research

Paper IV-VI option (xi) Theory of Approximation and Splines

Paper IV-VI option (xii) Advanced Functional Analysis

Paper IV-VI option (xiii) Solid Mechanics

Paper IV-VI option (xiv) Theory of Optimization

Note: The students who opt for Computer Applications paper shall have to pass in

both the theory and practical parts of the examinations.

Detailed Outline of Courses

M.Sc. Part I Papers

Paper I: Real Analysis

NOTE: Attempt any FIVE questions selecting at least TWO questions from each

section.

Section-I (4/7)

Real Number System

• Ordered sets, Fields, Completeness property of real numbers

• The extended real number system, Euclidean spaces

Sequences and Series

• Sequences, Subsequences, Convergent sequences, Cauchy sequences

• Monotone and bounded sequences, Bolzano Weierstrass theorem

• Series, Convergence of series, Series of non-negative terms, Cauchy condensation

test

• Partial sums, The root and ratio tests, Integral test, Comparison test

• Absolute and conditional convergence

Limit and Continuity

• The limit of a function, Continuous functions, Types of discontinuity

• Uniform continuity, Monotone functions

Differentiation

• The derivative of a function

• Mean value theorem, Continuity of derivatives

• Properties of differentiable functions.

Functions of Several Variables

• Partial derivatives and differentiability, Derivatives and differentials of composite

functions

• Change in the order of partial derivative, Implicit functions, Inverse functions,

Jacobeans

• Maxima and minima, Lagrange multipliers

Section-II (3/7)

The Riemann-Stilettoes Integrals

• Definition and existence of integrals, Properties of integrals

• Fundamental theorem of calculus and its applications

• Change of variable theorem

• Integration by parts

Functions of Bounded Variation

• Definition and examples

• Properties of functions of bounded variation

Improper Integrals

• Types of improper integrals

• Tests for convergence of improper integrals

• Beta and gamma functions

• Absolute and conditional convergence of improper integrals

Sequences and Series of Functions

• Definition of point-wise and uniform convergence

• Uniform convergence and continuity

• Uniform convergence and integration

• Uniform convergence and differentiation

Recommended Books

1. W. Rodin, Principles of Mathematical Analysis, (McGraw Hill, 1976)

2. R. G. Bartle, Introduction to Real Analysis, (John Wiley and Sons, 2000)

3. T. M. Apostil, Mathematical Analysis, (Addison-Wesley Publishing Company,

1974)

4. A. J. Kosmala, Introductory Mathematical Analysis, (WCB Company , 1995)

5. W. R. Parzynski and P. W. Zips, Introduction to Mathematical Analysis,

(McGraw Hill Company, 1982)

6. H. S. Gaskell and P. P. Narayan swami, Elements of Real Analysis, (Prentice

Hall, 1988)

Paper II: Algebra (Group Theory and Linear Algebra)

NOTE: Attempt any FIVE questions selecting at least TWO questions from each

section.

Section-I (4/7)

Groups

• Definition and examples of groups

• Subgroups lattice, Lagrange’s theorem

• Cyclic groups

• Groups and symmetries, Cayley’s theorem

Complexes in Groups

• Complexes and cosset decomposition of groups

• Centre of a group

• Normalizer in a group

• Centralizer in a group

• Conjugacy classes and congruence relation in a group

Normal Subgroups

• Normal subgroups

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Syllabus for M.Sc. Mathematics (Morning/Evening) Programmed 2 years M.Sc. Mathematics programmed consists of two parts namely Part-I and Part II. The regulation, Syllabi and Courses of Reading for the M.Sc. (Mathematics) Part-I and Part-II (Regular Scheme) are given below. Regulations The following regulations will be observed by M.Sc. (Mathematics) regular students i. There are a total of 1200 marks for M.Sc. (Mathematics) for regular students as is the case with other M.Sc. subjects. ii. There are five papers in Part-I and six papers in Part-II. Each paper carries 100 marks. iii. There is a Viva Voce Examination of M.Sc. Part II. The topics of Viva Voce Examination shall be: a) Analysis (Real, Complex and Functional) b) Algebra and Topology c) Mechanics M.Sc. Part-I The following five papers shall be studied in M.Sc. Part-I: Paper I Paper II Paper III Paper IV Paper V Real Analysis Algebra Complex Analysis and Differential Geometry Mechanics Topology and Functional Analysis Note: All the papers of M.Sc. Part-I given above are compulsory. M.Sc. Part-II In M.Sc. Part-II examinations, there are six written papers. The following three papers Are compulsory. Each paper carries 100 marks. Paper I Paper II Paper III Advanced Analysis Methods of Mathematical Physics Numerical Analysis 1 Optional Papers A student may select any three of the following optional courses: Paper IV-VI option (i) Paper IV-VI option (ii) Paper IV-VI option (iii) Paper IV-VI option (iv) Paper IV-VI ...
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