CSIR NET Mathematical Science Syllabus 2023

Main Topics

Sub-topics

UNIT-1

1. Analysis

1. Elementary Set Theory-Finite, Countable, Uncountable sets
2. Real number system as a complete ordered field
3. Archimedean Property
4. Supremum & Infimum
5. Balzano Weierstrass theorem
6. Heine Borel Theorem
7. Continuity, uniform continuity, Differentiability, mean value theorem
8. Sequences and series of functions, uniform convergence, Mean value theorem
9. Sequences and series of functions, uniform convergence
10. Reimann sums and Reimann integral, Improper Integrals
11. Monotonic functions, types of discontinuity, functions of bounded variation, Lebesgue measure, Lebesgue integral
12. Functions of several integrals, directional derivatives, partial derivatives, derivative as a Linear Lebesgue integral
13. Functions of several variables, directional derivative, partial derivative, derivative as a linear transformation, inverse and implicit function theorems
14. Metric spaces, compactness, connectedness, Normal linear Spaces, Spaces of continuous functions as examples

2. Linear Algebra

1. Vector Spaces, Sub-spaces, linear dependence, basis, dimension, algebra of linear transformations
2. Algebra of matrices, rank and determinant of matrices, linear equations
3. Eigen values and eigen vectors, Cayley-Hamilton theorem
4. Matrix representation of linear transformations, Change of basis, Canonical forms, Diagonal forms, Triangular forms, Jordan forms
5. Inner product spaces, Orthonormal basis
6. Quadratic forms, reduction and classification of quadratic forms

UNIT-2

1. Complex Analysis

1. Algebra of complex numbers, the complex plane
2. Polynomials
3. Power series
4. Transcendental functions such as exponential, Trigonometric and hyperbolic functions
5. Analytic functions
6. Cauchy-Riemann equations
7. Contour Integral, Cauchy’s integral formula, Liouville theorem, maximum modulus principle, Schwarz Lemma, Open mapping theorem
8. Taylor series, Laurent series, Calculus of residues
9. Conformal mappings, mobius transformations

2. Algebra

1. Permutations, combinations, Pigeon-hole principle, inclusion-exclusion principle, derangements
2. Fundamental theorem of arithmetic, divisibility in Z, congruences, Chinese Remainder Theorem, Euler’s function, Primitive roots
3. Groups, sub-groups, normal subgroups, quotient, homomorphisms, cyclic groups, permutation groups, Cayley’s theorem, class equation, Sylow theorems
4. Rings, ideals, prime and maximal ideals, quotient rings, unique factorization domain, permutation groups, Principle Ideal domain, Euclidean domain.
5. Polynomials rings and irreducibility criteria
6. Fields, Finite fields, Fields extensions, Galois Theory

3. Topology

1. Basis, dense sets, subspace and product topology, separation axioms, connectedness and compactness.

UNIT-3

1. Ordinary Differential Equations (ODEs)

1. 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

2. Partial Differential Equations (PDEs)

1. Lagrange and Charpit methods for solving first order PDEs, Cauchy’s problem for first order PDEs.
2. Classification of second order PDEs, General solution of higher order PDEs with constant coefficients, methods of separation variables for Laplace, Heat and Wave equation

3. Numerical Analysis

1. Numerical solutions of algebraic equations
2. Method of iteration and Newton-Raphson method
3. rate of convergence
4. Solution of systems of linear algebraic equations using Gauss elimination and Gauss-Seidel methods
5. Finite differences
6. Lagrange, Hermite and spline interpolation
7. Numerical differentiation and integration
8. Numerical solutions of ODEs USING Picard
9. Euler
10. Modified Euler and Runge-Kutta methods

4. Calculus of Variations

1. Variation of a functional, Euler-Lagrange equation, Necessary and sufficient conditions for extrema
2. Variational methods for boundary value problems in ordinary and partial differential equations

5. Linear Integral Equations

1. Linear integral equation of the first and second kind of Fredholm and Volterra type
2. Solution with separable kernels
3. Characteristic numbers and eigen functions
4. Resolvent kernel

6. Classical Mechanics

1. Generalized coordinates, Lagrange’s equations
2. Hamilton’s principle and principle of least action
3. Two-dimensional motion of rigid bodies
4. Euler’s dynamical equations for the motion of a rigid body about an axis
5. Theory of small oscillations

UNIT-4

1. Descriptive Statistics, Exploratory data analysis

1. Sample space, discrete probability, independent events, Bayes theorem
2. Random variables and distribution functions
3. Independent random variables, marginal and conditional distributions.
4. Characteristic functions- Probability inequalities
5. Modes of convergence, weak & strong laws of large numbers, central limit theorems
6. Markov chains with finite and countable state space, classification of states, limiting behaviour of n-step transition probabilities, stationery distribution, Poisson and birth-and-death processes
7. Standard discrete and continuous univariate distributions, sampling distribution, standard errors and asymptotic distributions, distributions of order statistics and range
8. 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
9. Simple nonparametric tests for one and two sample problems, rank correlation and test for independence, Elementary Bayesian inference
10. Gauss-Markov models, estimability of parameters, best linear unbiased estimators, confidence intervals, tests for linear hypotheses, analysis of variance and co-variance, fixed, random and mixed effects models, simple and multiple linear regression, elementary regression diagnostics, Logistic regression
11. 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
12. Simple random sampling, stratified sampling and systematic sampling, probability proportional to size sampling, ratio and regression methods
13. Completely randomized designs, randomized block designs, connectedness and orthogonality of block designs, BIBD 2k factorial experiments: confounding and construction
14. Hazard function and failure rates, censoring and life testing, series and parallel systems
15. Linear programming problem, simplex methods, duality, elementary questing and inventory models, 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.

Parts

Subjects

Tot QN

Attempt QN

Marks

Negative marks

Part ‘A’

• General science
• Quantitative Analysis & Reasoning
• Research Aptitude

20

15

2

0.5

Part ‘B’

• Mathematical Science (core MCQs)

40

25

3

0.75

Part ‘C’

• Mathematical Science

(Application type)

60

20

4.75

nil

Download CSIR NET Mathematical Science Syllabus 2023

Category

JRFs

LS

UR

48.38

43.543

EWS

42.75

38.475

OC

40.5

36.450

SC

32.38

29.142

ST

27.13

25.00

PwD

25

25.000