CSIR NET Mathematics Syllabus 2022: Download Syllabus PDF

CSIR NET Mathematical Science Syllabus: Part A (General Aptitude)

Graphical Analysis & Data Interpretation

Pie-Chart

Line & Bar Chart

Graph

Mode, Median, Mean

Measures of Dispersion

Table

Reasoning

Puzzle

Series Formation

Clock and Calendar

Direction and Distance

Coding and Decoding

Ranking and Arrangement

Numerical Ability

Geometry

Proportion and Variation

Time and Work

HCF and LCM

Permutation and Combination

Compound and Simple Interest

CSIR NET Mathematical Science Syllabus: Part B & Part C

Unit 1

Analysis

Elementary set theory, finite, countable, and uncountable sets, Real number system, Archimedean property, supremum, infimum.

Sequence 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.

Linear Algebra

Vector spaces, subspaces, linear dependence, basis, dimension, algebra of linear transformation

Algebra of matrices, rank, and determinant of matrices, linear equations.

Eigenvalues and eigenvectors, Cayley-Hamilton theorem.

Matrix representation of linear transformations. Change of basis, canonical forms, diagonal forms, triangular forms, Jordan forms.

Inner product spaces, orthonormal basis.

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, Liouville’s theorem, Maximum
modulus principle, Schwarz lemma, Open mapping theorem.

Taylor series, Laurent series, calculus of residues.

Conformal mappings, Mobius transformations.

Algebra

Permutations, combinations, pigeon-hole principle, inclusion-exclusion principle,
derangements.

Fundamental theorem of arithmetic, divisibility in Z, congruences, Chinese Remainder Theorem, Euler’s Ø- function, primitive roots.

Groups, subgroups, normal subgroups, quotient groups, homomorphisms, cyclic groups, permutation groups, Cayley’s theorem, class equations, and Sylow theorems.

Rings, ideals, prime and maximal ideals, quotient rings, unique factorization domain, principal ideal domain, Euclidean domain.

Topology: basis, dense sets, subspace and product topology, separation axioms, connectedness, and compactness.

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, and the system of first-order ODEs.

A general theory of homogenous and non-homogeneous linear ODEs, variation of parameters, Sturm-Liouville boundary value problem, Green’s function.

Partial Differential Equations (PDEs)

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.

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, Finite differences, Lagrange, Hermite, and spline interpolation, Numerical differentiation and integration, Numerical solutions of ODEs using Picard, Euler, modified Euler and
Runge-Kutta methods.

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.

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.

Classical Mechanics

Generalized coordinates, Lagrange’s equations, Hamilton’s canonical equations, Hamilton’s
principle and the 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

Markov chains with finite and countable state space, classification of states, limiting behavior of n-step transition probabilities, stationary distribution, Poisson, and birth-and-death processes.

Standard discrete and continuous univariate distributions. sampling distributions, standard errors and asymptotic distributions, distribution of order statistics, and range.

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.

Simple random sampling, stratified sampling, and systematic sampling. Probability is proportional to size sampling. Ratio and regression methods.

Hazard function and failure rates, censoring and life testing, series and parallel systems.

Best Books for CSIR NET Mathematical Science Syllabus

Mathematical Analysis

S C Malik

Schaum's Outline of Linear Algebra, Sixth Edition

Seymour Lipschutz and Marc Lipson

Algebra: Abstract and Modern

Swamy and Murthy

Paper Sections

Questions Provided

Marks for Each Section

Marks Per Question

Negative Marking 

Total time

Part A

20

30

2

0.5

3 hours

Part B

40

75

3

0.75

Part C

60

95

4.75

0

Total Marks

200