M.A./M.Sc. (FINAL) MATHEMATICS Syallbus 2012-13

Paper - I: Analysis and Advanced Calculus

This paper is divided into FIVE Units. TWO questions will be set from each Unit. Candidates are required to attempt FIVE questions in all taking ONE question from each Unit. All questions carry equal marks.

Normed linear spaces. Quotient space of normed linear spaces and its completeness. Banach spaces and examples. Bounded linear transformations. Normed linear space of bounded linear transformations. Weak convergence of a sequence of bounded linear transformations. Equivalent norms. Basic properties of finite dimensional normed linear spaces and compactness. Reisz Lemma. Multilinear mapping.

Open mapping theorem. Closed graph theorem. Uniform boundness theorem. Continuous linear functionals. Hahn-Banach theorem and its consequences. Embedding and Reflexivity of normed spaces. Dual spaces with examples. Inner product spaces. Hilbert space and its properties.

Orthogonality and Functionals in Hilbert Spaces. Phythagorean theorem, Projection theorem, Orthonormal sets, Bessel's inequality, Complete orthonormal sets, Parseval's identity, Structure of a Hilbert space, Riesz representation theorem, Reflexivity of Hilbert spaces. Adjoint of an operator on a Hilbert space. Self-adjoint, Positive, Normal and Unitary operators and their properties.

Projection on a Hilbert space. Invariance. Reducibility. Orthogonal projections. Eigen values and eigen vectors of an operator. Spectrum of an operator. Spectral theorem. Derivatives of a continuous map from an open subset of Banach space to a Banach space. Rules of derivation. Derivative of a composite, Directional derivative. Mean value theorem and its applications. Partial derivatives and Jacobian Matrix.

Continuously differentiable maps. Higher derivatives. Taylor's formula. Inverse function theorem. Implicit function theorem. Primitives and integral. Differentiation under the integral sign. Riemann integral of function of real variable with values in normed linear space. Existence and uniqueness of solutions of the type x' = ƒ (t, x) ordinary differential equation.

Paper - II: Continuum Mechanics

Cartesian Tensors, Index notation and transformation laws of Cartesian tensors. Addition, Subtraction and Multiplication of cartesian tensors, Gradient of a scalar function, Divergence of a vector function and Curl of a vector function using the index notation. - identity. Conservative vector field and concept of a scalar potential function. Stokes, Gauss and Green's theorems.

Continuum approach, Classification of continuous media, Body forces and surface forces. Components of stress tensor, Force and Moment equations of equilibrium. Transformation law of stress tensor. Stress quadric. Principal stress and principal axes. Stress invariants and stress deviator. Maximum shearing stress.

Lagrangian and Eulerian description of deformation of flow. Comoving derivative, Velocity and Acceleration. Continuity equation. Strain tensors. Linear rotation tensor and rotation vector, Analysis of relative displacements. Geometrical meaning of the components of the linear strain tensor, Properties of linear strain tensors. Principal axes, Theory of linear strain. Linear strain components. Rate of strain tensors. The vorticity tensor. Rate of rotation vector and vorticity, Properties of the rate of strain tensor, Rate of cubical dilation.

Law of conservation of mass and Eulerian continuity equation. Reynolds transport theorem. Momentum integral theorem and equation of motion. Kinetic equation of state. First and the second law of thermodynamics and dissipation function. Applications (Linear elasticity and Fluids) - Assumptions and basic equations. Generalized Hook's law for an isotropic homogeneous solid.

Compatibility equations (Beltrami-Michell equations). Classification of types of problems in linear elasticity. Principle of superposition, Strain energy function, Uniqueness theorem, p-p relationship and work kinetic energy equation, Irrotational flow and Velocity potential, Kinetic equation of state and first law of Thermodynamics. Equation of continuity. Equations of motion. Vorticity-stream surfaces for inviscid flow, Bernoullis equations. Irrotational flow and velocity potential. Similarity parameters of fluid flow.

Paper - III: Fluid Dynamics

Irrotation motion in two-dimensions. General motion of a cylinder in two-dimensions. Motion of a circular cylinder, Coaxial cylinders. Blasius theorem. Motion of an elliptic cylinder. Kinetic energy of rotating elliptic cylinder. Motion of a liquid in rotating cylinder, Joukowski theorem.

D'Alembert's paradox. Motion of a sphere. Vortex motion, Vortex line, Vortex tube, rectilinear vortices, Karman vortex street.

Viscosity, Analysis of stress and rate of strain. Navier-Stokes equations of motion and equation of energy. Vorticity and circulation. Dynamical similarity, inspection and dimensional analysis. Buckinghem -theorem. Physical importance of non-dimensional parameters. Reynolds number, Froude number, Mach number, Prandtl number, Grashoff number. Some exact solutions of Navier-Stokes equations. Plane Couette flow, Plane Poiseuille flow, Generalized plane Couette flow.

Hagen-Poiseuille flow, Flow in tubes of uniform cross-sections. Flow between two concentric cylinders, Stagnation point flow. Flow due to a rotating disc. Flow due to a plane wall suddenly set in motion (Stokes' first problem). Flow due to an oscillating plane wall (Stokes second problem). Starting flow in a pipe.

Theory of very slow motion- Stokes flow past a sphere, Oseen's flow past a sphere, Lubrication theory. Concept of Boundary layer, Derivation of Prandtl boundary layer equations in 2D flows and Boundary layer on a flat plate.

Paper - IV: Boundary Layer Theory

Derivation of boundary layer equations for tw -dimensional flow. Boundary layer along a flat plate (Blasius-Topfer solution). Characteristic boundary layer parameters. Similar solutions. Exact solution of the steady state boundary layer equations in two-dimensional flow. Flow past a wedge. Flow along the wall of a convergent channel. Boundary layer separation. Flow past a symmetrically placed cylinder (Blasius series solution). Gortler new series method.

Plane free jet, Circular jet, Plane wall jet. Prandtl-Mises transformation and its application of plane free jet. Axially symmetrical boundary layers on bodies at rest. Boundary layers on a body of revolution. Mangler's transformation.

Three-dimensional boundary layers - Boundary layer flow on yawed cylinder. Growth of three-dimensional boundary layer on a rotating disc impulsively set in motion. Unsteady boundary layers - Method of successive approximations, Boundary layer growth after impulsive start of motion and in accelerated motion, Boundary layer for periodic flow (Pulsatile pressure gradient).

Approximate methods for the solution of the boundary layer equations. Karman momentum integral equation. Karman-Pohlhausen method and its application. Waltz-Thwaites method. Energy integral equation. Derivation of two-dimensional thermal boundary layer equation for flow over a plane wall

Forced convection in a laminar boundary layer on a flat plate, Crocco's first and second integrals. Reynolds analogy. Temprature distribution in the spread of a jet - (i) Plane free jet, (ii) Circular jet (iii) Plane wall jet. Free convection from a heated vertical plate. Thermal-energy integral equation. Approximate solution of the Pohlhausan's problem of free convection from a heated vertical plate.

Paper - V: Mathematical Programming

Separating and supporting hyperplane theorems. Revised Simplex method for linear programming problem (LPP), Bounded variable problem. Convex function.

Integer programming. Gomory's algorithm for the all integer programming problem, Branch and Bound technique. Quadratic forms. Lagrange function and multiplier.

Non-linear programming problem (NLPP) and its fundamental ingredients, Necessary and Sufficient conditions for saddle points. Kuhn-Tucker theorem. Convex separable programming algorithm.

Kuhn-Tucker conditions for optimization for NLPP. Quadratic Programming, Wolf's method. Beale's method.Duality in quadratic Programming.

Paper - VI: Mathematical Theory of Statistics

Sample space -Combination of events. Statistical independence, Conditional probability. Bay's repeated trials. Random variable, Distribution function, Probability function. Density function, Mathematical expectation, Generating function, Continuous probability distribution, Characteristic function. Fourier's inversion, Chebyshev and Kolmogrovea inequality. Weak and strong laws of large numbers

Normal hypergeometric, rectangular, Negative Binomial Beta, Gamma and Cauchy's distribution. Methods of least squares and curve fitting, Correlation and Regression coefficients, Association of Attributes.

Interpolation- Introduction, Newton-Gregory theorem. Newton's, Lagrange's, Gauss's and Striling's formulae. Index numbers- Introduction, Price relatives, Quantity relatives, Value relatives, Link and Chain relatives. Aggregate methods, Fisher's ideal Index. Change of the base of the index numbers. Elementary sampling theory. Distribution of means of samples for Binomial. Cauchy, rectangular and normal population. Exact distributions of x², t, z and F. Statistics in samples from a normal population, their simple properties and applications.

Test of significance and difference between two means and two standard deviations for large samples with modification for small samples and taken from normal population. Analysis of variance, Simple cases (One criteria and two criteria of classification).

Elementary Statistical theory of Estimation of efficient, Fisher's criteria for the estimator, Consistant, Efficient and Sufficient estimator, Method of maximum likelihood. Maximum Likelihood Estimator, Other methods of estimation. Methods of moments, Minimum variance, Minimum Chi-square and Least Squares.

Paper - VII: Combinatorics and Graph Theory

Combinatorics- Counting of sets and multisets. Binomial and multinomial numbers. Unordered selection with repetitions, Selection without repetition. Counting objects and functions. Functions and the Pigeonhole principle. Inclusion and exclusion principle.

Discrete numeric functions and combinatorial problems. Generating functions and recursions. Power series and their algebraic properties. Homogeneous and non-homogeneous linear recursions.

Graphs- Basic terminology, Simple graphs, Multi graphs and Weighted graphs. Walk and connectedness. Paths and circuits. Shortest path in weighted graphs, Eulerian paths and circuits. Hamiltonian paths and circuits Traveling salesman problem, operations on graphs. Trees- Trees, Rooted trees, Paths lengths in rooted trees, spanning trees, minimum spanning trees.

Cut sets- Cut-sets, Cut vertices. Fundamental cut sets, Connectivity and seperativity. Net work flows, Max-flow min-cut theorem. Plannar Graphs- Combinatorial and geometric graphs, Kuratowski's graphs. Euler's formula. Detection of planarity. Geometric dual. Thickness and Crossing number.

Graph Colouring. Vertex colouring, Edge colouring and Map colouring. Chromatic number. Chromatic polynomials, The four and five colour theorems. Digraphs- binary relations, Directed graphs and Directed trees, Arborescence, Polish notation method, Tournaments.
Counting of Labeled Trees- Cayley's theorem. Counting methods, Polya's theory.

Paper-VIII: Integral Transforms and Integral Equations

Laplace transform- Definition and its properties. Rules of manipulation. Laplace transform of derivatives and integrals. Properties of inverse Laplace transform. Convolution theorem. Complex inversion formula.

Fourier transform - Defiition and properties of Fourier sine, cosine and complex transforms. Convolution theorem. Inversion theorems. Fourier transform of derivatives.Mellin transform- Definition and elementary properties. Mellin transforms of derivatives and integrals. Inversion theorem. Convolution theorem.

Infinite Hankel transform- Definition and elementary properties. Hankel transform of derivatives. Inversion theorem. Parseval Theorem.Solution of ordinary differential equation with constant and variable coeffifients by laplace transform application to the solution of simple boundary value problem by Laplace, Fourier and infinite Hanbel transforms.

Linear integral equation - Defination and classificatin Conversion of initial and boundary value problems to an integral equation.Eigen value and Eigen function.Sulation of homogenous and general Fredholm integral equations of second kind with separable kernels. Sulation of Fredholm and Volterra integral equations of second kind by method of sucessive substitutions and successive approximation.Resovent kernal and its result.Condition of uniform convergence and uniqueness of series solution.

Solultion of volterra integral equations of second kind with convolution type kernels by laplace transform.Solution of singular integral equation by fourier transform.Integral equations. Fundamentals prperties of eigan value and eigen functions for symmetric kernels-Expansion in eigen function and bilinear form.Hilbert-Schmidt theorem.Solulatin of scmidt theorem.Classical Fredholm integral equation of second kind by using Fredholm theorm. Paper- IX: Relativity and Cosmology

Teaching : 6 Hours per Week
Examination : Common for Regular/Non-collegiate Candidates

3 Hrs. duration

Theory Paper

Max. Marks 100

Note :

Unit 1:

Relative Character of space and time, Principle of Relativity and its postulates, Derivation of special Lorentz transformation equations, Composition of Parallel velocities, Lorentz-Fitzgerald contraction formula, Time dilation, Simultaneity, Relativistic transformation formulae for velocity, Lorentz contraction factor, Particle acceleration, Velocity of light as fundamental velocity, Relativistic aberration and its deduction to Newtonian theory.

Unit 2:

Variation of mass with velocity, Equivalence of mass and energy, Transformation formulae for mass, Momentum and energy, Problems on conservation of mass, Momentum and energy, Relativistic Lagrangian and Hamiltonian, Minkowski space, Space-like, Time-like and Light-like intervals, Null cone, Relativity and Causality, Proper time, World line of a particle.

Unit 3:

Principles of Equivalence and General Covariance, Geodesic postulate, Mach's principle, Newtonian approximation of equation of motion, Einstein's field equation for matter and empty space, Reduction of Einstein's field equation to Poisson's equation, Schwarzschild exterior metric, its isotropic form and singularity, Relativistic differential equation for orbit of the planet.

Unit 4:

Three crucial tests in general Relativity and their detailed descriptions, Analogues of Kepler's laws in General Relativity, Trace of Einstein tensor and energy-momentum tensor for perfect fluid, proof of its expression for perfect fluid, Schwarzs child interior metric and boundary conditions, Radar Echodelay (Fourth test).

Unit 5:

Lorentz invariance of Maxwell's equations and their tensor form, Lorentz force on charged particle, Energy-momentum tensor for electromagnetic field, Reissner-Nordstrom metric for spherically charged particle. Cosmology - Einstein�s field equation with cosmological term, static cosmological models (Einstein and de-Sitter) and their physical and geometrical properties. Red Shift in non-static form of de-Sitter line-element. Einstein-space, Hubble's law, Weyl's postulate.

Paper - X: Industrial Mathematics

Teaching : 6 Hours per Week
Examination : Common for Regular/Non-collegiate Candidates

3 Hrs. duration

Theory Paper

Max. Marks 100

Note :

Unit 1:

Partial differential equations and techniques of solution. Finite difference methods for solving PDE. Application to problems of industry with special reference to Fluid Mechanics. Operational Techniques.

Unit 2:

Linear Programming problems. Computational procedure of Simplex method, Two-phase Simplex method, Big-M-method, Revised Simplex method, Duality in linear programming, Duality and Simplex method.

Unit 3:

Assignment models. Mathematical formulation, Hungarian method. Travelling Salesman problem. Transportation models. Mathematical formulation. Initial basic feasible solution. Degeneracy and unbalanced transportation problems.

Unit 4:

Inventory Models. EOQ models with and without shortages. EOQ models with constraints.

Unit 5:

Replacement and Reliability models. Replacement of items that deteriorate, Replacement of items that fail completely.
Reliability Theory - Coherent structure, Reliability of systems of independent components, Bounds on system reliability, Shapes of the system reliability function, Motion of aging, Parametric families of life distribute with Monotone failure rate.

Paper - XI: Magnetohydrodynamics

Teaching : 6 Hours per Week
Examination : Common for Regular/Non-collegiate Candidates

3 Hrs. duration

Theory Paper

Max. Marks 100

Note :

Unit 1:

Maxwell electromagnetic field equations. Constitutive equations of fluid motion, Stokes hypothesis. Maxwell stress tensor. Fundamental equations of Magnetofluid-dynamics. Magnetofluiddynamic approximations. Magnetic field equation, Frozen in fluid, Alfven transverse waves. MHD boundary conditions. Inspection and Dimensional analysis,-products.

Unit 2:

Reynolds number, Mach number, Prandtl number, Magnetic Reynolds number, Magnetic pressure number, Hartmann number, Magnetic parameter, Magnetic Prandtl number and Nusselt number. Hartmann plane Poiseuille flow and plane Couette flow including temperature distribution. MHD flow in a tube of rectangular cross-section. MHD pipe flow. MHD flow in annular channel. MHD flow between two rotating coaxial cylinders.

Unit 3:

MHD flow near a stagnation point. MHD Reyleigh's flow. MHD Stoke's flow past a sphere, MHD Oseen's flow past a sphere. MHD boundary layer flow past a flat plate in an aligned magnetic flow. Wilson's numerical solution technique.

Unit 4:

MHD boundary layer flow past a flat plate in a transverse magnetic field. modified Rossow's method of solution. MHD plane free jet flow. Wave and theory of characteristics, Equation of the characteristics, Characteristic surfaces, MHD characteristic equations. MHD waves.

Unit 5:

Friedriches diagrams. Dispersion relation. MHD shock waves. Generalized Hugoniot condition. Compressive nature of MHD shocks. MHD shock wave classification. MHD shock stability.

Paper- XII: Numerical Analysis

Teaching : 6 Hours per Week
Examination : Common for Regular/Non-collegiate Candidates

3 Hrs. duration

Theory Paper

Max. Marks 100

Note :

Unit 1:

Iterative methods - Theory of iteration method, Acceleration of the convergence, Chebyshev method, Muler's method, Methods for multiple and complex roots. Newton-Raphson method for simultaneous equations, Convergence of iteration process in the case of several unknowns.

Unit 2:

Solution of polynomial equations - Polynomial equation, Real and complex roots, Synthetic division, the Birge-Vieta, Bairstow and Graeffe's root squaring method. System of simultaneous Equations (Linear)- Direct method, Method of determinant, Gauss-Jordan, LU-Factorizations-Doolitte's, Crout's and Cholesky's. Partition method. Method of successive approximate-conjugate gradient and relaxation methods.

Unit 3:

Eigen value problems- Basic properties of eigen values and eigen vector, Power methods, Method for finding all eigen values of a matrix. Jacobi, Given's and Rutishauser method. Complex eigen values. Curve Fitting and Function Approximations - Least square error criterion. Linear regression. Polynomial fitting and other curve fittings, Approximation of functions by Taylor series and Chebyshev polynomials.

Unit 4:

Numerical solution of Ordinary differential Equations - Taylor series Mathod, Picard method, Runge-Kutta methods upto fourth order, Multistep method (Predictor-corrector strategies), Stability analysis - Single and Multistep methods.

Unit 5:

BVP's of ordinary differential Equations - Boundary value problems (BVP's), Shooting methods, Finite difference methods. Difference schemes for linear boundary value problems of the type y" =f(x,y),y" =f(x,y,y') and y^IV=f(x,y).

Paper - XIII: Computer Applications

Teaching: 4 Hours per Week for Theory Paper.
Examination: Common for Regular/Non-collegiate Candidates.

2:30 Hrs. duration

Theory Paper

Max. Marks 70

Note:

This paper is divided into FOUR Units. TWO questions will be set from each Unit. Candidates are required to attempt FOUR questions in all taking ONE question from each Unit. All questions carry equal marks.

Unit 1:

Introduction to computers, Computer organization, Input-output devices, Memory system. Hardware and software. Operating system.

Unit 2:

Computer languages, System software and application software. Algorithms and flow charts. Programming languages and problem solving on computers.

Unit 3:

Programming in C - Constants and variables. Arithmetic expressions, Input-output, Conditional statements, Implementing loops in programs.

Unit 4:

Defining and manipulating arrays, Processing character strings, functions. Files in C. Simple computer programming.

Practical

Teaching: 2 Hours per week

Examination: 2 Hours duration

Max.Marsk: 30

Simple C Programming of problems of numerical analysis, Solution of quadratic equations, Mean and standard deviation, Fitting of curves, Correlation coefficient, Applications into matrices, Sorting of numerical character string data etc.

Distribution of Marks:

Two Practical - 10 Marks each	= 20 Marks
Practical Record	= 05 Marks
Viva-Voce	= 05 Marks
Total Marks	= 30 Marks

Note:

Each candidate is required to appear in the Practical examination to be conducted by internal and external examiners. External examiner will be appointed by the University through BOS and internal examiner will be appointed by the Head of the Department/Principal of the College.
Each candidate has to prepare his/her practical record.
Each candidate has to pass in Theory and Practical examinations separately.

M.A./M.Sc. (FINAL) MATHEMATICS Syallbus 2012-13

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