TRB Lecturers in Government Polytechnic Colleges Mathematics Syllabus 2017. This is the final syllabus and remember it is enough to read just this portion to write TRB exam for Lecturers in Govt Polytechnic Colleges. 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 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 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 - 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 - RadonNikodym 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 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 – 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 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-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 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 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.
