CSIR NET Mathematics Syllabus 2024, Download PDF

CSIR NET Mathematical Sciences Syllabus: Part B and Part C

unit 1 Analysis Elementary set theory, finite, countable and uncountable sets, real number system, Archimedean property, supremacy, infinity. Sequences and series, convergence, limsup, limif. Bolzano Weierstrass theorem, Heine Borel theorem Continuity, uniform continuity, variation, mean value theorem. Sequences and series of functions, uniform convergence. Riemann sums and Riemann integrals, improper integrals. linear algebra Vector spaces, subspaces, linear dependence, basis, dimensions, algebra of linear transformations Algebra of matrices, rank, and determinant of matrices, linear equations. Eigenvalues ​​and eigenvectors, Cayley-Hamilton theorem. Matrix representation of linear transformations. Change of base, canonical form, diagonal form, triangular form, Jordan form. Internal product space, vertical base. Quadratic Forms, Reduction and Classification of Quadratic Forms unit 2 complex analysis Algebra of complex numbers, complex plane, polynomials, power series,
Transcendental functions such as exponential, trigonometric and hyperbolic functions Analytical functions, Cauchy-Riemann equation. Contour integral, Cauchy’s theorem, Cauchy’s integral formula, Liouville’s theorem, maximum
Modulus theory, Schwarz lemma, open mapping theorem. Taylor series, Laurent series, calculation of residues. Conformal mapping, Möbius transformation. algebra Permutation, Combination, Pigeon-hole Principle, Inclusion-Exclusion Principle,
Disturbance. Fundamental theorem of arithmetic, divisibility in Z, congruences, Chinese remainder theorem, Euler’s Ø-function, primitive roots. Groups, subgroups, normal subgroups, quotient groups, isomorphism, cyclic groups, permutation groups, Cayley’s theorem, square equations and Sylow’s theorem. Rings, ideals, prime and maximum ideals, quotient rings, unique factor domains, prime ideal domains, Euclidean domains. Topology: Basis, dense sets, subspace and product topology, separation principle, connectivity and compactness. unit 3 Ordinary Differential Equations (ODE): Existence and uniqueness of solutions of initial value problems for first-order ordinary differential equations, singular solutions of first-order ODEs, and systems of first-order ODEs. A general theory of homogeneous and non-homogeneous linear ODEs, variation of parameters, Sturm–Liouville boundary value problem, Green’s function. Partial differential equation (PDE) 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 constants
Coefficients, method of separation of variables for Laplace, heat and wave equations. numerical analysis Numerical solution of algebraic equations, method of iteration and Newton-Raphson method, rate of convergence, solution of systems of linear algebraic equations using Gauss elimination and Gauss-Seidel methods, finite difference, Lagrange, Hermite and spline interpolation, numerical differentiation and Numerical solutions of ODEs using integration, Picard, Euler, modified Euler and
Ranj-Kutta methods. calculation of variations Functional variations, Euler-Lagrange equations, necessary and sufficient conditions for extrema. Variational methods for boundary value problems in ordinary and partial differential equations. linear integral equation Linear integral equations of the first and second types of Fredholm and Volterra type, solutions with different kernels. Characteristic numbers and eigenfunctions, resolving kernels. classical mechanics Generalized coordinates, Lagrange’s equations, Hamilton’s canonical equations, Hamilton’s
Theory and principle of minimum action, two-dimensional motion of rigid bodies, Euler’s dynamic equations for the motion of a rigid body around an axis, theory of small oscillations. unit 4 Descriptive Statistics, Exploratory Data Analysis Finite and countable state spaces, classification of states, finite behavior of n-phase transition probabilities, Markov chains with stationary distributions, Poisson and birth-and-death processes. Standard discrete and continuous univariate distributions. Sampling distributions, standard errors and asymptotic distributions, distributions of order statistics, and limits. Estimation methods, properties of estimators, confidence intervals. Tests of Hypotheses: The Most Powerful and Equally Most Powerful Test, Likelihood Ratio Test. Analysis of discrete data and chi-square test of goodness of fit. Large sample testing. Simple non-parametric tests for one and two sample problems, rank correlation, and tests for independence, a priori Bayesian inference. Simple random sampling, stratified sampling and systematic sampling. The probability is proportional to the sample size. Ratio and Regression Methods. Hazard functions and failure rates, sensing and life testing, series and parallel systems.