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Syllabus for M.Sc. Mathematics
Programme
2 years M.Sc. Mathematics programme 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
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
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
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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.
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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,
Jacobians
Maxima and minima, Lagrange multipliers
Section-II (3/7)
The Riemann-Stieltjes Integrals
Definition and existence of integrals, Properties of integrals
Fundamental theorem of calculus and its applications
Change of variable theorem
Integration by parts
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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. Rudin, Principles of Mathematical Analysis, (McGraw Hill, 1976)
2. R. G. Bartle, Introduction to Real Analysis, (John Wiley and Sons, 2000)
3. T. M. Apostol, Mathematical Analysis, (Addison-Wesley Publishing Company,
1974)
4. A. J. Kosmala, Introductory Mathematical Analysis, (WCB Company , 1995)
5. W. R. Parzynski and P. W. Zipse, Introduction to Mathematical Analysis,
(McGraw Hill Company, 1982)
6. H. S. Gaskill and P. P. Narayanaswami, Elements of Real Analysis, (Printice 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 coset decomposition of groups
Centre of a group
Normalizer in a group
Centralizer in a group
Conjugacy classes and congruence relation in a group
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Normal Subgroups
Normal subgroups
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Proper and improper normal subgroups
Factor groups
Isomorphism theorems
Automorphism group of a group
Commutator subgroups of a group
Permutation Groups
Symmetric or permutation group
Transpositions
Generators of the symmetric and alternating group
Cyclic permutations and orbits, The alternating group
Generators of the symmetric and alternating groups
Sylow Theorems
Double cosets
Cauchy’s theorem for Abelian and non-Abelian group
Sylow theorems (with proofs)
Applications of Sylow theory
Classification of groups with at most 7 elements
Section-II (3/7)
Ring Theory
Definition and examples of rings
Special classes of rings
Fields
Ideals and quotient rings
Ring Homomorphisms
Prime and maximal ideals
Field of quotients
Linear Algebra
Vector spaces, Subspaces
Linear combinations, Linearly independent vectors
Spanning set
Bases and dimension of a vector space
Homomorphism of vector spaces
Quotient spaces
Linear Mappings
Mappings, Linear mappings
Rank and nullity
Linear mappings and system of linear equations
Algebra of linear operators
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Space L( X, Y) of all linear transformations
Matrices and Linear Operators
Matrix representation of a linear operator
Change of basis
Similar matrices
Matrix and linear transformations
Orthogonal matrices and orthogonal transformations
Orthonormal basis and Gram Schmidt process
Eigen Values and Eigen Vectors
Polynomials of matrices and linear operators
Characteristic polynomial
Diagonalization of matrices Recommended Books
1. J. Rose, A Course on Group Theory, (Cambridge University Press, 1978)
2. I. N. Herstein, Topics in Algebra, (Xerox Publishing Company, 1964)
3. G. Birkhoff and S. Maclane, A Survey of Modern Algebra, (Macmillan, 1964)
4. Seymour Lipschutz, Linear Algebra, (McGraw Hill Book Company, 2001)
5. Humphreys, John F. A Course on Group Theory, (Oxford University Press, 2004)
6. P. M. Cohn, Algebra, (John Wiley and Sons, 1974)
7. J. B. Fraleigh, A First Course in Abstract Algebra, (Pearson Education, 2002)
Paper III: Complex Analysis and Differential Geometry
NOTE: Attempt any FIVE questions selecting at least TWO questions from
each section. Section-I (4/7)
The Concept of Analytic Functions
Complex numbers, Complex planes, Complex functions
Analytic functions
Entire functions
Harmonic functions
Elementary functions: Trigonometric, Complex exponential, Logarithmic and
hyperbolic functions
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Infinite Series
Power series, Derived series, Radius of convergence
Taylor series and Laurent series Conformal Representation
Transformation, conformal transformation
Linear transformation
Möbius transformations
Complex Integration
Complex integrals
Cauchy-Goursat theorem
Cauchy’s integral formula and their consequences
Liouville’s theorem
Morera’s theorem
Derivative of an analytic function
Singularity and Poles
Review of Laurent series
Zeros, Singularities
Poles and residues
Cauchy’s residue theorem
Contour Integration
Expansion of Functions and Analytic Continuation
Mittag-Leffler theorem
Weierstrass’s factorization theorem
Analytic continuation
Section-II (3/7)
Theory of Space Curves
Introduction, Index notation and summation convention
Space curves, Arc length, Tangent, Normal and binormal
Osculating, Normal and rectifying planes
Curvature and torsion
The Frenet-Serret theorem
Natural equation of a curve
Involutes and evolutes, Helices
Fundamental existence theorem of space curves
Theory of Surfaces
Coordinate transformation
Tangent plane and surface normal
The first fundamental form and the metric tensor
The second fundamental form
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