TN TRB Assistant Professor Syllabus

Introduction

The TN TRB Assistant Professor Mathematics Syllabus provides a comprehensive outline for aspirants preparing for the Tamil Nadu Teachers Recruitment Board (TRB) examination. This syllabus includes essential topics such as algebra, real analysis, complex analysis, topology, differential equations, linear algebra, and mathematical statistics. It is designed to assess the candidate’s depth of understanding, problem-solving ability, and logical reasoning skills. By going through the syllabus carefully, candidates can plan their preparation effectively and focus on the mathematical areas most relevant to teaching and research roles in higher education.

TN TRB Assistant Professor Syllabus – MATHEMATICS

UNIT – 1

Algebra Permutations, Combinations, Applications of classical number theoretic properties, Groups, Counting Principles, Cayley’s theorem, Permutation groups, Sylow’s theorems, Direct Products, Polynomial Rings, Vector spaces, Inner Product Spaces, Orthonormal bases, Modules, Fields, Roots of polynomials, Elements of Galois theory, Splitting fields, Degrees of Splitting fields of polynomials, Solvable groups, Linear transformations, Matrix representation, Canonical forms, Determinants, Cayley Hamilton theorem and Applications, Hermitian, Unitary and Normal Transformations, Quadratic Forms, Finite fields, Wedderburn’s Theorem on Finite division rings, Frobenius Theorem, Integral Quaternions and Four Square Theorem.

UNIT – 2

Real Analysis Real number system as a complete ordered field, Sequences and series, convergence, limit supremum and limit Infinimum, Euclidean space, Accumulation points, Bolzano Weierstrass theorem, Heine-Borel theorem, Metric spaces, Compactness, Connectedness, Continuity, Uniform continuity, Differentiability, Mean value theorem for derivatives, Sequences and series of functions, Uniform convergence and continuity, Cauchy condition for Uniform convergence, Riemann-Steiltjes sums and integrals, Riemann sums and integrals, Improper Integrals, Monotonic functions, types of discontinuity, Functions of bounded variation, Measure and Integration, Lebesgue measure, Lebesgue integral, Functions of several variables, Directional derivative, Partial derivative, derivative as a linear transformation, Inverse and Implicit function theorems.

UNIT – 3

Complex Analysis Algebra of complex numbers, transcendental functions such as exponential, trigonometric and hyperbolic functions, Analytic functions, Cauchy-Riemann equations, Contour integral, Cauchy’s theorem, Cauchy’s integral formula, Residues and Poles, Calculus of residues, Applications of Residues, Liouville’s theorem, Maximum modulus principle, Rouche’s theorem, Schwarz lemma, Open mapping theorem, Power series, Taylor series, Laurent series, Conformal mappings, Mobius transformations and simple applications of complex integration.

UNIT – 4

Topology Topological spaces, Basis for a Topology, subbasis, Order topology, Product topology, Subspace topology, Closed sets, Limit points, Hausdorff spaces, Continuous functions, Homeomorphisms, Metric Topology, Convergence of functions, Uniform convergence, Connected spaces, Compactness, Countability axioms, Separation axioms and Normal spaces. Functional Analysis Normed linear Spaces, Banach Spaces, Continuous linear transformations, Hahn Banach theorem, Open mapping theorem, Closed graph theorem, Conjugate of an operator, Hilbert spaces, Orthogonality, Orthonormal bases, Conjugate space of a Hilbert space, Adjoint of an operator, Self-adjoint operators, Normal operators, Unitary operators, Projections, Holders and Minkowski inequalities, Spectral theorem for operators on a finite dimensional Hilbert space, Banach Algebra, Regular and singular elements, Topological divisors of zero, Spectrum and spectral radius.

UNIT – 5

Ordinary Differential Equations Existence and uniqueness of solutions of initial value problems for first order ordinary differential equations, singular solutions of first order ODEs, system of first order ODEs, General theory of homogenous and non-homogeneous linear ODEs, Variation of parameters, Sturm-Liouville boundary value problem and Green’s function. Application- Numerical solutions of ODEs using Picard, Euler, modified Euler and Runge-Kutta methods.

UNIT – 6

Partial Differential Equations Lagrange and Charpit methods for solving first order PDEs, Cauchy problem for first order PDEs, Classification of second order PDEs, General solution of higher order PDEs with constant coefficients, Method of separation of variables for Laplace, Heat and Wave equations, Applications of Laplace and Fourier Transforms.

UNIT – 7

Classical Mechanics Equation of motions, Constraints, Generalized coordinates, Holonomic Systems, Non-Holonomic Systems, Virtual Work, D’ Alembert’s Principle, Lagrange’s equations, Hamilton’s canonical equations, Hamilton’s Principle and Principle of least action, Two-dimensional motion of rigid bodies, Euler’s dynamical equations for the motion of a rigid body about an axis and theory of small oscillations. Calculus of Variations Variation of a functional, Euler-Lagrange equation, Necessary and sufficient conditions for extrema, Variational methods for boundary value problems in ordinary and partial differential equations.

UNIT – 8

Differential Geometry Curves in spaces, Serret- Frenet formulas, Locus of center of curvature, Spherical curvature, Intrinsic equation, Helices, Spherical indicatrix surfaces, Envelope, Edge of regression, Developable surfaces associated to a curve, First and Second fundamental forms, Lines of curvature, Gaussian curvature, Euler’s theorem, Dupin’s Indicatrix, Surface of revolution, Conjugate systems, Asymptotic lines, Isometric lines and Geodesics. Linear Integral Equations Linear integral equation of the first and second kind of Fredholm and Volterra type, Solutions with separable kernels, Characteristic numbers and eigenfunctions and resolvent kernel.

UNIT – 9

Operations Research Linear programming, Revised simplex method, Duality problems, Degeneracy procedure, Integer programming, Non-linear programming, Convex programming, Dynamic programming, Game theory, Queuing theory, Single server and Multi server models, Erlang service distributions, Mathematical theory of inventory control, Optimal inventory policies in Deterministic models, Storage models, Replacement theory, Markovian decision models in replacement theory, Reliability, Failure rates, System reliability and Network analysis.

UNIT – 10

Mathematical Statistics Probability Theory, Bayes theorem, Random variables and distribution functions (univariate and multivariate), expectation and moments, Independent random variables, marginal and conditional distributions, Characteristic functions, Probability inequalities(Tchebyshef and Markov), Simple, partial and multiple correlation, Regression Analysis, rank correlation, Weak laws of large numbers, Central Limit theorem, Discrete and continuous sampling distributions, standard errors, Methods of estimation, properties of estimators, confidence intervals, Tests of hypotheses, likelihood ratio tests, Analysis of discrete data and chi-square test of goodness of fit, Large sample tests, Simple nonparametric tests for one and two sample problems, test for independence of attributes, confidence intervals and Analysis of variance.

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Conclusion

The Mathematics syllabus for the TN TRB Assistant Professor exam acts as a structured guide for candidates aiming to qualify for teaching positions in Tamil Nadu’s Government Colleges. With focused preparation and strong conceptual clarity, aspirants can perform confidently in the exam and strengthen their academic foundation. This syllabus not only supports exam readiness but also encourages analytical thinking, precision, and a deeper appreciation for the applications of mathematics in various fields.

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TN TRB Assistant Professor Syllabus – MATHEMATICS