Here is CSIR UGC NET SYLLABUS 2016 for Mathematics or Mathematical Sciences.
CSIR (Council of Scientific & Industrial Research) conducts NET (National Eligibility Test) exam two times every year for Eligibility for Lectureship and Junior Research Fellowship in Mathematics. Those candidates who are preparing for CSIR NET Exam in subject of Mathematics (Maths) should have the syllabus of CSIR UGC NET SYLLABUS of MATHEMATICS. So to help those candidates, here is CSIR UGC NET SYLLABUS 2016 for MATHEMATICS (MATHEMATICAL SCIENCES) with PDF also given at bottom of post.

UNIT – 1
Real number system as a complete ordered field, Elementary set theory, finite, countable and uncountable sets, Lebesgue measure, Lebesgue integral. Functions of several variables, directional derivative, partial derivative, derivative as a linear transformation, Metric spaces, compactness, connectedness. Normed Linear Spaces, Spaces of Continuous functions as examples.
Archimedean property, supremum, infimum. Sequences and series, convergence, limsup, liminf. Bolzano Weierstrass theorem, Heine Borel theorem. Continuity, uniform continuity, differentiability, mean value theorem. Sequences and series of functions, uniform convergence. Riemann sums and Riemann integral, Improper Integrals. Monotonic functions, types of discontinuity, functions of bounded variation
Linear Algebra: Vector spaces, subspaces, linear dependence, basis, dimension, algebra of linear transformations. Algebra of matrices, rank and determinant of matrices, linear equations. Eigenvalues and eigenvectors, Change of basis, canonical forms, diagonal forms, triangular forms, Jordan forms. Inner product spaces, orthonormal basis, Cayley-Hamilton theorem. Matrix representation of linear transformations. Quadratic forms, reduction and classification of quadratic forms
UNIT – 2
Complex Analysis:
 Algebra of complex numbers, the complex plane, polynomials, Power series, transcendental functions such as exponential, trigonometric and hyperbolic functions. Analytic functions, Cauchy-Riemann equations. Contour integral, Cauchy’s theorem, Cauchy’s integral formula, Taylor series, Laurent series, calculus of residues. Conformal mappings, Mobius transformations, Liouville’s theorem, Maximum modulus principle, Schwarz lemma, Open mapping theorem
Algebra: Permutations, combinations, inclusion-exclusion principle, derangements, pigeon-hole principle
Fundamental theorem of arithmetic, divisibility in Z, congruence, Chinese Remainder Theorem, Euler’s Ø- function, primitive roots, prime and maximal ideals, Cayley’s theorem, class equations, Sylow theorems. Rings, ideals, quotient rings, unique factorization domain, principal ideal domain, Euclidean domain. Polynomial rings and irreducibility criteria. Fields, finite fields, field extensions, Groups, subgroups, normal subgroups, quotient groups, homomorphisms, cyclic groups, permutation groups
UNIT – 3
Ordinary Differential Equations (ODEs):
Existence and Uniqueness of solutions of initial value problems for first order ordinary differential equations, singular solutions of first order ODEs, Sturm-Liouville boundary value problem, Green’s function, system of first order ODEs. General theory of homogenous and non-homogeneous linear ODEs, variation of parameters

Partial Differential Equations (PDEs):
Lagrange and Charpit methods for solving first order PDEs, Method of separation of variables for Laplace, Heat and Wave equations, Cauchy problem for first order PDEs. Classification of second order PDEs, General solution of higher order PDEs with constant coefficients,
Numerical Analysis :
Numerical solutions 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, Numerical solutions of ODEs using Picard, Euler, modified Euler and Runge-Kutta methods, Finite differences, Lagrange, Hermite and spline interpolation, Numerical differentiation and integration

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, resolvent kernel.
 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.

Classical Mechanics:
Generalized coordinates, 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, theory of small oscillations.
UNIT – 4
Descriptive statistics, exploratory data analysis.
Sample space, discrete probability, independent events, 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, Markov, Jensen). , stationary distribution. Standard discrete and continuous univariate distributions. Sampling distributions. Standard errors and asymptotic distributions, distribution of order statistics and range, Modes of convergence, weak and strong laws of large numbers, Central Limit theorems (i.i.d. case). Markov chains with finite and countable state space, classification of states, limiting behaviour of n-step transition probabilities

Methods of estimation. Properties of estimators. Confidence intervals. Tests of hypotheses: most powerful and uniformly most powerful tests, 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, rank correlation and test for independence. Elementary Bayesian inference.Gauss-Markov models, estimability of parameters, Best linear unbiased estimators, tests for linear hypotheses and confidence intervals. Analysis of variance and covariance. Fixed, random and mixed effects models. Simple and multiple linear regression. Elementary regression diagnostics. Logistic regression.

Multivariate normal distribution, Wishart distribution and their properties. Distribution of quadratic forms. Inference for parameters, partial and multiple correlation coefficients and related tests. Data reduction techniques: Principle component analysis, Discriminant analysis, Cluster analysis, Canonical correlation. Connected, complete and orthogonal block designs, Simple random sampling, stratified sampling and systematic sampling. Probability proportional to size sampling. Ratio and regression methods. Completely randomized, randomized blocks and Latin-square designs, BIBD. 2K factorial experiments: confounding and construction. Series and parallel systems, hazard function and failure rates, censoring and life testing. Steady-state solutions of Markovian queuing models: M/M/1, M/M/1 with limited waiting space, M/M/C, M/M/C with limited waiting space, M/G/1, Linear programming problem. Simplex methods, duality. Elementary queuing and inventory models.