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 Arcs- Cauchy’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

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 polynomial – Invariant subspaces – Simultaneous triangulation –Simultaneous diagonalization – Direct sum decompositions – Vector spaces Bases and dimension 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

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  –

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

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

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  

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 –

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

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 and cubic Spline approximation – Single-step methods: Euler’s method – Taylor    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.

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-normal- Curvature 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

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 theorem – Applications: The timetabling problem – Independent       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.

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 co- ordinates – 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.