Showing Page:
1/34
1
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 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
Showing Page:
2/34
2
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.
Showing Page:
3/34
3
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
Showing Page:
4/34
4
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
Showing Page:
5/34
5
Proper and improper normal subgroups
Factor groups
Isomorphism theorems
Auto orphism group of a group
Commentator 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
Slow Theorems
Double cossets
Cauchy’s theorem for Aeolian and non-Aeolian group
Slow theorems (with proofs)
Applications of Slow 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 Homomorphism’s
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
Space L( X, Y) of all linear transformations
Matrices and Linear Operators
Matrix representation of a linear operator
Showing Page:
6/34
6
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. Her stein, Topics in Algebra, (Xerox Publishing Company, 1964)
3. G. Birkhoff and S. McLane, A Survey of Modern Algebra, (Macmillan, 1964)
4. Seymour Lipchitz, 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. Farleigh, 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
Infinite Series
Power series, Derived series, Radius of convergence
Taylor series and Laurent series
Conformal Representation
Transformation, conformal transformation
Linear transformation
Mobius transformations
Complex Integration
Complex integrals
Cauchy-Groused theorem
Showing Page:
7/34
7
Cauchy’s integral formula and their consequences
Lowville’s theorem
Moreira’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 binomial
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
Principal, Gaussian, Mean, Geodesic and normal curvatures
Gauss and Weingarten equations
Gauss and Codazzi equations
Recommended Books
1. H. S. Kasana, Complex Variables: Theory and Applications, (Prentice Hall, 2005)
2. M. R. Spiegel, Complex Variables, (McGraw Hill Book Company, 1974)
3. J. W. Brown, R. V. Churchill, Complex Variables and Applications, (McGraw
Hill, 2009)
4. Louis L. Pennies, Elements of Complex Variables, (Holt, Line hart and
Winston, 1976)
5. W. Kaplan, Introduction to Analytic Functions, (Addison-Wesley, 1966)
Showing Page:
8/34
8
6. R. S. Mill man and G.D. Parker, Elements of Differential Geometry,
(Prentice- Hall, 1977)
7. E. Kreyzig, Differential Geometry, (Dover Publications, 1991)
8. M. M. Lipchitz, Schism’s Outline of Differential Geometry, (McGraw Hill,
1969)
9. D. Somasundaram, Differential Geometry, (Narosa Publishing House, 2005)
Paper IV: Mechanics
NOTE: Attempt any FIVE questions selecting at least TWO questions from each
section.
Section-I (4/7)
Vector Integration
Line integrals
Surface area and surface integrals
Volume integrals
Integral Theorems
Green’s theorem
Gauss divergence theorem
Stoke’s theorem
Curvilinear Coordinates
Orthogonal coordinates
Unit vectors in curvilinear systems
Arc length and volume elements
The gradient, Divergence and curl
Special orthogonal coordinate systems
Tensor Analysis
Coordinate transformations
Einstein summation convention
Tensors of different ranks
Contra variant, Covariant and mixed tensors
Symmetric and skew symmetric tensors
Addition, Subtraction, Inner and outer products of tensors
Contraction theorem, Quotient law
The line element and metric tensor
Christ of fell symbols
Section-II (3/7)
Non Inertial Reference Systems
Accelerated coordinate systems and inertial forces
Rotating coordinate systems
Velocity and acceleration in moving system: Carioles, Centripetal and
transverse acceleration
Dynamics of a particle in a rotating coordinate system
Planar Motion of Rigid Bodies
Showing Page:
9/34
9
Introduction to rigid and elastic bodies, Degrees of freedom, Translations,
Rotations, instantaneous axis and center of rotation, Motion of the center of mass
Euler’s theorem and Chase’s theorem
Rotation of a rigid body about a fixed axis: Moments and products of inertia of
various bodies including hoop or cylindrical shell, circular cylinder, spherical
shell
Parallel and perpendicular axis theorem
Radius of gyration of various bodies
Motion of Rigid Bodies in Three Dimensions
General motion of rigid bodies in space: Moments and products of inertia,
Inertia matrix
The momental ellipsoid and equimomental systems
Angular momentum vector and rotational kinetic energy
Principal axes and principal moments of inertia
Determination of principal axes by diagonal zing the inertia matrix
Euler Equations of Motion of a Rigid Body
Force free motion
Free rotation of a rigid body with an axis of symmetry
Free rotation of a rigid body with three different principal moments
Euler’s Equations
The Eulerian angles, Angular velocity and kinetic energy in terms of Euler
angles, Space cone
Motion of a spinning top and gyroscopes- steady precession, Sleeping top
Paper V: Topology & Functional Analysis
NOTE: Attempt any FIVE questions selecting at least TWO questions from each
section.
Section-I (3/7)
Topology
Definition and examples
Open and closed sets
Showing Page:
10/34
10
Subspaces
Neighborhoods
Limit points, Closure of a set
Interior, Exterior and boundary of a set
Bases and Sub-bases
Base and sub bases
Neighborhood bases
First and second axioms of accountability
Separable spaces, Lindel of spaces
Continuous functions and homeomorphism
Weak topologies, Finite product spaces
Separation Axioms
Separation axioms
Regular spaces
Completely regular spaces
Normal spaces
Compact Spaces
Compact topological spaces
Count ably compact spaces
Sequentially compact spaces
Connectedness
Connected spaces, Disconnected spaces
Totally disconnected spaces
Components of topological spaces
Section-II (4/7)
Metric Space
Review of metric spaces
Convergence in metric spaces
Complete metric spaces
Completeness proofs
Dense sets and separable spaces
No-where dense sets
Baire category theorem
Normed Spaces
Normed linear spaces
Banach spaces
Convex sets
Quotient spaces
Equivalent norms
Linear operators
Linear functional
Finite dimensional normed spaces
Showing Page:
11/34
11
Continuous or bounded linear operators
Dual spaces
Inner Product Spaces
Definition and examples
Orthonormal sets and bases
Annihilators, Projections
Hilbert space
Linear functional on Hilbert spaces
Reflexivity of Hilbert spaces
M.Sc. Part II Papers
Paper I: Advanced Analysis
NOTE: Attempt any FIVE questions selecting at least TWO questions from each
section.
Section-I (3/7)
Advanced Set Theory
Equivalent Sets
Countable and Uncountable Sets
The concept of a cardinal number
The cardinals
o
and c
Addition and multiplication of cardinals
Cartesian product, Axiom of Choice, Multiplication of cardinal numbers
Order relation and order types, Well ordered sets, Transfinite induction
Addition and multiplication of ordinals
Statements of Zorn’s lemma, Maximality principle and their simple implications
Section-II (4/7)
Measure Theory
Outer measure, Lebesgue Measure, Measureable Sets and Lebesgue measure, Non
measurable sets, Measureable functions
The Lebesgue Integral
The Riemann Integral, The Lebesgue integral of a bounded function
The general Lebesgue integral
General Measure and Integration
Measure spaces, Measureable functions, Integration, General convergence
theorems
Signed measures, The Lp-spaces, Outer measure and measurability
The extension theorem
The Lebesgue Stieltjes integral, Product measures
Showing Page:
12/34
12
Paper II: Methods of Mathematical Physics
NOTE: Attempt any FIVE questions selecting at least TWO questions from each
section.
Section-I (4/7)
Sturm Lowville Systems
Some properties of Sturm-Lowville equations
Regular, Periodic and singular Sturm-Lowville systems and its applications
Series Solutions of Second Order Linear Differential Equations
Series solution near an ordinary point
Series solution near regular singular points
Series Solution of Some Special Differential Equations
Hyper geometric function F(a, b, c; x) and its evaluation
Series solution of Bessel equation
Expression for J
n
(X) when n is half odd integer, Recurrence formulas for J
n
(X)
Orthogonally of Bessel functions
Series solution of Legendre equation
Introduction to PDEs
Review of ordinary differential equation in more than one variables
Linear partial differential equations (PDEs) of the first order
Cauchy’s problem for quasi-linear first order PDEs
PDEs of Second Order
PDEs of second order in two independent variables with variable coefficients
Cauchy’s problem for second order PDEs in two independent variables
Boundary Value Problems
Laplace equation and its solution in Cartesian, Cylindrical and spherical polar
coordinates
Dirichlet problem for a circle
Poisson’s integral for a circle
Wave equation
Heat equation
Section-II (3/7)
Fourier Methods
The Fourier transform
Fourier analysis of generalized functions
The Laplace transform
Green’s Functions and Transform Methods
Expansion for Green’s functions
Transform methods
Closed form of Green’s functions
Variation Methods
Euler-Lagrange equations
Integrand involving one, two, three and n variables
Necessary conditions for existence of an extreme of a functional
Showing Page:
13/34
13
Constrained maxima and minima
Paper III: Numerical Analysis
NOTE: Attempt any FIVE questions selecting at least TWO questions from each
section.
Section-I (4/7)
Error Analysis
Errors, Absolute errors, Rounding errors, Truncation errors
Inherent Errors, Major and Minor approximations in numbers
The Solution of Linear Systems
Gaussian elimination method with pivoting, LU Decomposition methods,
Algorithm and convergence of Jacobi iterative Method, Algorithm and
convergence of Gauss Seidel Method
Eigenvalue and eigenvector, Power method
The Solution of Non-Linear Equation
Bisection Method, Fixed point iterative method, Newton Rap son method, Secant
method, Method of false position, Algorithms and convergence of these methods
Difference Operators
Shift operators
Forward difference operators
Backward difference operators
Average and central difference operators
Ordinary Differential Equations
Euler’s, Improved Euler’s, Modified Euler’s methods with error analysis
Showing Page:
14/34
14
Runge-Kutta methods with error analysis
Predictor-corrector methods for solving initial value problems
Finite Difference, Collocation and variation methods for boundary value
problems
Section-II (3/7)
Interpolation
Lagrange’s interpolation
Newton’s divided difference interpolation
Newton’s forward and backward difference interpolation, Central difference
interpolation
Hermit interpolation
Spline interpolation
Errors and algorithms of these interpolations
Numerical Differentiation
Newton’s Forward, Backward and central formulae for numerical differentiation
Numerical Integration
Rectangular rule
Trapezoidal rule
Simpson rule
Boole’s rule
Weddle’s rule
Gaussian quadrature formulae
Errors in quadrature formulae
Newton-Cotes formulae
Difference Equations
Linear homogeneous and non-homogeneous difference equations with constant
coefficients
Showing Page:
15/34
15
Paper (IV-VI) option (i): Mathematical Statistics
NOTE: Attempt any FIVE questions selecting at least TWO questions from each
section.
Section-I (3/7)
Probability Distributions
The postulates of probability
Some elementary theorems
Addition and multiplication rules
Bayes’ rule and future Bayes’ theorem
Random variables and probability functions
Discrete Probability Distributions
Uniform, Bernoulli and binomial distribution
Hyper geometric and geometric distribution
Negative binomial and Poisson distribution
Continuous Probability Distributions
Uniform and exponential distribution
Gamma and beta distributions
Normal distribution
Mathematical Expectations
Moments and moment generating functions
Moments of binomial, Hyper geometric, Poisson, Gamma, Beta and
normal distributions
Section-II (4/7)
Functions of Random Variables
Distribution function technique
Transformation technique: One variable, Several variables
Moment-generating function technique
Sampling Distributions
The distribution of mean and variance
The distribution of differences of means and variances
The Chi-Square distribution
The t distribution
The F distribution
Regression and Correlation
Linear regression
Showing Page:
16/34