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.

# CSIR UGC NET SYLLABUS OF MATHEMATICS (MATHEMATICAL SCIENCES)

Here is CSIR UGC NET SYLLABUS 2017 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 2017 for MATHEMATICS (MATHEMATICAL
SCIENCES) with PDF also given at bottom of post.

Below is CSIR UGC NET SYLLABUS 2017 FOR MATHEMATICS (MATHEMATICAL
SCIENCES)

UNIT – 1

Analysis:

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.

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.

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