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MATHEMATICS, UNIT 1: REAL ANALYSIS, Ordered sets – Fields – Real field – The extended real number system – The complex, field- Euclidean space - Finite, Countable and uncountable sets - Limits of functions, - Continuous, , functions – Continuity and compactness – Continuity and, , connectedness – Discontinuities - Monotonic functions - Equi-continuous families of, functions, Stone – Weierstrass theorem – Cauchy sequences – Some special, sequences – Series - Series of nonnegative terms – The number e – The root and ratio, tests – Power series – Summation by parts – Absolute convergence - Addition and, multiplication of series - Rearrangements, The Derivative of a Real Function – Mean, Value Theorem - The Continuity of Derivatives - L'Hospital's Rule – Derivatives of, Higher Order - Taylor's Theorem – Differentiation of Vector valued functions – Some, Special Functions - Power Series – The Exponential and Logarithmic functions – The, Trigonometric functions - The algebraic completeness of the complex field – Fourier, series – The Gamma function - The Riemann – Stieltjes Integral – Definition and, Existence of the Integral – Properties of the Integral - Integration and Differentiation, – Integration of Vector – valued functions – Rectifiable curves., UNIT 2: COMPLEX ANALYSIS, Spherical representation of complex numbers – Analytic functions – Limits and, continuity - Analytic Functions – Polynomials – Rational functions – Elementary, Theory of Power series-Sequences – Series – Uniform Convergence – Power series Abel's limit functions – Exponential and Trigonometric functions – Periodicity – The, Logarithm - Analytical Functions as Mappings - Conformality - Arcs and closed, curves - Analytic functions in Regions – Conformal mapping - Length and area Linear transformations - Linear group – Cross ratio – symmetry - Oriented Circles, – Families of circles – Elementary conformal mappings – Use of level curves – Survey, of, , Elementary mappings – Elementary Riemann surfaces – Complex Integration -, , Fundamental Theorems - Line Integrals – Rectifiable Arcs - Line Integrals as ArcsCauchy's Theorem for a rectangle and in a disk-Cauchy's Integral Formula - Index, of point with respect to a closed, , curve - The Integral formula - Higher order, , derivatives – Local properties of analytic functions - Taylor's Theorem – Zeros and, Poles – Local mapping – Maximum Principle – The General form, , of Cauchy's, , Theorem – Chains and Cycles – Simple connectivity Homology – General statement

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of Cauchy's theorem – Proof of Cauchy's theorem – LocalIy exact differentials Multiply connected regions – Calculus of residues - Residue Theorem – Argument, Principle - Evaluation of definite Integrals – Harmonic Functions – Definition and, basic properties – Mean - value, , Property - Poisson's formula - Schwarz's Theorem, , – Reflection Principle – Weierstrass’s theorem – Taylor’s series – Laurent series., UNIT 3: ALGEBRA, Another counting principle - Sylow's theorems – Direct products – Finite abelian, groups, Polynomial rings – Polynomials over the rational field – Polynomial rings over, commutative rings - Extension fields – Roots of polynomials – More about roots – The, element of Galois theory – Finite fields - Wedderbum's theorem on finite division, rings – Theorem of Frobenius – The algebra of polynomials - Lagrange Interpolation, – Polynomial ideals – The prime factorization of a polynomial –Commutative rings –, Determinant functions – Permutations and the uniqueness of determinant Classical adjoint of a matrix – Inverse of an invertible matrix using determinants Characteristic, , values, , –, , Annihilating, , Simultaneous, , triangulation, , polynomial, , –Simultaneous, , –, , Invariant, , diagonalization, , decompositions – Vector spaces Bases and dimension, , –, , subspaces, Direct, , –, , sum, , Subspaces – Matrices and, , linear maps – Rank nullity theorem – Inner product spaces – Orthonormal basis –, Gram – Schmidt orthonormalization process – Eigen spaces – Algebraic and, Geometric multiplicities – Cayley – Hamilton theorem – Diagonalization – Direct sum, decomposition - Invariant direct sums – Primary decomposition theorem – Unitary, matrices and their properties - Rotation matrices - Schur, Diagonal and Hessenberg, forms and Schur decomposition - Diagonal, , and the general cases – Similarity, , Transformations and change of basis – Generalised eigen vectors - Canonical basis, – Jordan canonical form – Applications to linear differential equations, , -Diagonal, , and the general cases - An error correcting code – The method of least squares –, Particular solutions of non-homogeneous differential equations with constant, coefficients - The Scrambler transformation., UNIT 4: TOPOLOGY, Topological spaces – Basis for a topology – Product topology on finite Cartesian, products –Subspace topology – Closed sets and Limit points – Continuous functions, - Homeomorphism - Metric Topology – Uniform limit theorem – Connected spaces, – Components – Path components - Compact spaces – Limit point compactness -

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Local compactness – Countability axioms -T1-spaces – Hausdorff spaces Completely regular spaces – Normal spaces – Urysohn lemma - Urysohn, metrization theorem - Imbedding theorem - Tietze extension theorem - Tychonoff, theorem., UNIT 5: MEASURE THEORY AND FUNCTIONAL ANALYSIS, MEASURE THEORY : Lebesgue Outer Measure – Measurable Sets – Regularity –, Measurable Functions – Boreland Lebesgue Measurablity – Abstract Measure Outer Measure – Extension of a Measure – Completion of a Measure – Integrals, of simple functions – Integrals of Non Negative Functions – The Generallntegral, – Integratiion of Series – Riemann and Lebesgue Integrals – Legesgue, Differentiation Theorem – Integration and Differentiation – The Lebesgue Set –, Integration with respect to a general measure Convergence in Measure – Almost, Uniform convergence - Signed, , measures and Hahn Decomposition - Radon-, , Nikodym Theorem and its applications- Measurability in a product space – The, Product measure and Fubini's Theorem., FUNCTIONAL, , ANALYSIS: Banach spaces – Continuous linear transformations -, , The Hahn-Banach theorem – The natural imbedding of N in N** - The open, mapping theorem - Closed, , graph theorem - The conjugate of an operator –, , Uniform boundedness theorem - Hilbert, , Spaces – Schwarz inequality –, , Orthogonal complements – Orthonormal sets - Bessel's Inequality – Gram –, Schmidt, , orthogonalization, , process, , –, , The, , conjugate, , space, , H*-, , Riesz, , representation theorem – The adjoint of an operator - Self-adjoint operators –, Normal and, , unitary operators – Projections – Matrices – Determinants and the, , spectrum of an operator - spectral theorem – Fixed point theorems and some, applications to analysis., UNIT 6: DIFFERENTIAL EQUATIONS ORDINARY DIFFERENTIAL EQUATIONS:, Second order homogeneous equations – Initial value problems – Linear dependence, and independence – Formula for Wronskian - Non-homogeneous equations of order, two - Homogeneous, , and, , non-homogeneous equations of order n – Annihilator, , method to solve a non - homogeneous, , equation – Initial value problems for the, , homogeneous equation – Solutions of the homogeneous, , equations – Wronskian, , and linear independence – Reduction of the order of a homogeneous equation –, Linear, , equation with regular singular points – Euler equation, , - Second order

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equations, , with regular singular points – Solutions and properties of Legendre and, , Bessel's equation – Equations with variables separated – Exact equations – Method, of successive approximations - Lipschitz condition – Convergence of the successive, approximations., PARTIAL DIFFERENTIAL EQUATIONS:, Integral surfaces passing through a given curve – Surfaces orthogonal to a given, system of, , surfaces – Compatible system of equations - Charpit's method –, , Classification of second order, , Partial Differential Equations – Reduction to, , canonical form - Adjoint operators - Riemann's, , method- One-dimensional wave, , equation – Initial value problem - D'Alembert's solution – Riemann – Volterra, solution – Vibrating string - Variables Separable solution – Forced vibrations, , -, , Solutions of non-homogeneous equation – Vibration of a circular membrane –, Diffusion equation -, , Solution of diffusion equation in cylindrical and spherical, , polar coordinates by method of, , Separation of variables - Solution of diffusion, , equation by Fourier transform – Boundary value, , problems – Properties of, , harmonic functions - Green's function for Laplace equation - The methods of images, - The eigen function method., UNIT 7: MECHANICS AND CONTINUM MECHANICS MECHANICS:, The Mechanical system - Generalized coordinates – Constraints - Virtual work – and, Energy Momentum derivation of Lagrange's equations – Examples – Integrals of the, motion Hamilton's principle - Hamilton's equations – Other variational principle Hamilton principle function – Hamilton – Jacobi equation – Separability – Differential, forms and generating functions – Special transformations – Lagrange and Poisson, brackets., CONTINUM MECHANICS:, Summation convention - Components of a tensor -, , Transpose of a tensor -, , Symmetric and anti-symmetric tensor – Principal values and directions - Scalar, invariants - Material and spatial descriptions - Material derivative – Deformation –, Principal strain – Rate of deformation - Conservation of mass – Compatibility, conditions – Stress vector and tensor – Components of a stress tensor – Symmetry, – Principal stresses – Equations of motion, - Boundary conditions –

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Isotropic solid - Equations of infinitesimal theory – Examples of elastodynamics, elastostatics –, Equations of hydrostatics - Newtonian fluid – Boundary conditions – Stream lines, examples of laminar flows – Vorticity vector - Irrotational flow., UNIT 8: MATHEMATICAL STATISTICS AND NUMERICAL METHODS, MATHEMATICAL STATISTICS:, Sampling, , distributions – Characteristics of good estimators – Method of moments, , – Maximum likelihood estimation – Interval estimates for mean, variance and, proportions- Type I and type II errors – Tests based on Normal, t, and F distributions, for testing of mean, variance and proportions - Tests for independence of attributes, and goodness, , of fit – Method of, , least, , squares - Linear regression – Normal, , regression analysis- Normal correlation analysis - Partial and multiple correlation, – Multiple linear regression – Analysis of variance - One-way and, , two-way, , classifications – Completely randomized design – Randomized block design – Latin, square, , design – Covariance matrix - Correlation matrix – Normal density function, , – Principal components – Sample variation by principal components – Principal, components by graphing., NUMERICAL METHODS:, Direct methods : Gauss elimination method – Error analysis – Iterative methods :, Gauss-Jacobi and Gauss-Seidel – Convergence considerations – Eigen value, Problem : Power method - Interpolation: Lagrange's and Newton's interpolation –, Errors in interpolation – Optimal points for interpolation – Numerical differentiation, by finite differences - Numerical integration: Trapezoidal, Simpson's and Gaussian, quadratures - Error in quadratures – Norms of functions – Best approximations:, Least squares polynomial approximation – Approximation with Chebyshev, polynomials – Piecewise linear, methods: Euler's method - Taylor, , and cubic Spline approximation - Single-step, series, , method – Runge – Kutta method of, , fourth order – Multistep methods : Adams-Bashforth and Milne's methods – Linear, two point BVPs: Finite difference method-Elliptic equations: Five point finite, difference formula in rectangular region - truncation error; One-dimensional, parabolic equation: Explicit and Crank-Nicholson schemes; Stability of the above, schemes - One-dimensional hyperbolic equation: Explicit scheme.

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UNIT 9: DIFFERENTIAL GEOMETRY AND GRAPH THEORY DIFFERENTIAL, GEOMETRY:, Representation of space curves - Unique parametric representation of a space curve, - Arc-length – Tangent and osculating plane – Principal normal and bi-normalCurvature and torsion – Behaviour of a curve near one of its points – The curvature, and torsion of a curve as the intersection of two surfaces – Contact between curves, and, , surfaces – Osculating circle and Osculating sphere – Locus of centres of, , spherical curvature – Tangent surfaces, involutes and evolutes – Intrinsic equations, of space curves – Fundamental existence theorem – Helices – Definition of a surface, - Nature of points on a surface – Representation of a surface – Curves on surfaces, – Tangent plane and surface normal – The general surfaces of revolution – Helicoids, – Metric on a surface - Direction coefficients on a surface – Families of curves –, Orthogonal trajectories - Double family of curves – Isometric correspondence –, Intrinsic properties - Geodesics and their differential equations – Canonical geodesic, equations – Geodesics on surface revolution – Normal property of geodesics –, Differential equations of geodesics using normal property – Existence theorems –, Geodesic parallels – Geodesic curvature, , - Gauss – Bonnet theorem – Gaussain, , curvature - Surfaces of constant curvature., GRAPH THEORY:, Graphs and, , subgraphs: Graphs and simple graphs – Graph isomorphism -, , Incidence and adjacency matrices – Subgraphs – Vertex degrees – Path and, Connection cycles – Applications : The shortest path problem - Trees: Trees – Cut, edges and bonds - Cut vertices - Cayley's formula – Connectivity : Connectivity –, Blocks – Euler tours and Hamilton cycles: Euler, , tours - Hamilton cycles -, , Applications: The Chinese postman problem – Matchings : Matchings – Matching, and coverings in bipartite graphs - Perfect matchings – Edge colourings : Edge, chromatic number - Vizing's, Independent, , theorem - Applications: The timetabling problem -, , sets and cliques : Independent sets-Ramsey's theorem - Turan's, , theorem – Vertex colourings :, , Chromatic number - Brook's theorem - Hajos', , conjecture – Chromatic polynomials - Girth and chromatic number – Planar graphs, : Plane and planar graphs – Dual graphs - Euler's formula – Bridges - Kuratowski's, Theorem - The Five color theorem and the four color conjecture – Non Hamiltonian, planar graphs.

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UNIT-10:, , MATHEMATICAL, , PROGRAMMING, , AND, , FLUID, , DYNAMICS, , MATHEMATICAL PROGRAMMING:, Linear programming : Formulation and graphical solutions – Simplex method –, Transportation and Assignment problems – Advanced linear programming : Duality, - Dual simplex method – Revised simplex method - Bounded variable technique Integer programming : Cutting plane algorithm – Branch and bound technique –, Applications of integer programming – Non-linear programming: Classical, optimization theory Unconstrained problems - Constrained problems - Quadratic, programming -, , Dynamic programming : Principle of optimality – Forward and, , backward recursive equations – Deterministic dynamic programming applications., FLUID DYNAMICS:, Kinematics of fluids in motion : Real and ideal fluids – Velocity – Acceleration –, Streamlines - Pathlines - Steady and unsteady flows – Velocity potential - Vorticity, vector - Local and particle rates of change - Equation of continuity – Conditions at a, rigid boundary – Equations of motion of a fluid : Pressure at a point in a fluid –, Boundary conditions of two inviscid immiscible fluids - Euler's equations of motion, - Bernoullt's equation - Some potential theorems – Flows involving axial symmetry –, Two dimensional flows : Two-dimensional flows – Use of cylindrical polar coordinates - Stream function, complex potential for, , two-dimensional flows,, , irrotational, incompressible flow – Complex potential for standard two-dimensional, flows – Two dimensional image systems – Milne – Thomson circle theorem - Theorem, of Blasius – Conformal transformation and its applications : Use of conformal, transformations - Hydro-dynamical aspects of conformal mapping – Schwarz, Christoffel transformation – Vortex rows – Viscous flows : Stress - Rate of strain –, Stress analysis – Relation between stress and rate of strain-Cofficient of viscosity Laminar flow – Navier – Stokes equations of motion – Some problems in viscous flow.