NET Maths Syllabus - Important Dates

The table below contains CSIR NET exam dates and other event-related dates:

CSIR NET Important Dates

CSIR NET Maths Syllabus - Units-wise

The table below contains the units from where the topics and the chapters will be mentioned in the syllabus.

NET syllabus For Maths

UNIT I - CSIR NET Maths Syllabus

Analysis

Elementary set theory, finite, countable and uncountable sets, Real number system as a complete ordered field, 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, Lebesgue measure, Lebesgue integral.

Functions of several variables, directional derivative, partial derivative, derivative as a linear transformation, inverse and implicit function theorems.

Metric spaces, compactness, connectedness. Normed linear Spaces. Spaces of continuous functions as examples.

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, 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 II - CSIR NET Syllabus For Maths

Complex Analysis

1) Algebra of complex numbers, power series, the complex plane, trigonometric and hyperbolic functions, polynomials, transcendental functions such as exponential.

2) Analytic functions, Cauchy-Riemann equations. Cauchy’s integral formula, Contour integral, Cauchy’s theorem, Schwarz lemma, Liouville’s theorem, Maximum modulus principle, Open mapping theorem.

3) Taylor series, calculus of residues, Laurent series, Mobius transformations, Conformal mappings.

4) Algebra: Permutations, inclusion-exclusion principle, combinations, pigeon-hole principle, derangements.

5) Fundamental theorem of arithmetic, congruences, divisibility in Z, Chinese Remainder Theorem, Euler’s Ø- function, primitive roots. normal subgroups, Groups, subgroups, homomorphisms, quotient groups, cyclic groups, class equations, permutation groups, Cayley’s theorem, Sylow theorems.

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

7) Polynomial rings and irreducibility criteria. Fields, finite fields, field extensions, Galois Theory.

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

UNIT III - CSIR NET Maths Syllabus

• Ordinary Differential Equations (ODEs)

Existence and uniqueness of solutions of initial value problems for first order ordinary differential equations, system of first order ODEs, singular solutions of first order ODEs. General theory of homogeneous and non-homogeneous linear ODEs, Sturm-Liouville boundary value problem, variation of parameters, Green’s function.

• Partial Differential Equations (PDEs)

Cauchy problem for first order PDEs, Lagrange and Charpit methods for solving first order PDEs, General solution of higher order PDEs with constant coefficients, Classification of second order PDEs, Method of separation of variables for Laplace, Heat and Wave equations.

• 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, modified Euler and Runge-Katta methods.

• Calculus of Variations

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

• Linear Integral Equations

Solutions with separable kernels, Linear integral equation of the first and second kind of Fredholm and Volterra type. Characteristic numbers and eigenfunctions, resolvent kernel.

• Classical Mechanics

Hamilton’s canonical equations, Lagrange’s equations, Generalised coordinates, Hamilton’s principle and principle of least action, Euler’s dynamical equations for the motion of a rigid body about an axis, Two-dimensional motion of rigid bodies, theory of small oscillations.

Read More:

• CSIR UGC NET Syllabus 2025

• CSIR UGC NET Exam Pattern 2025

• CSIR UGC NET Question Papers 2025

• CSIR UGC NET Exam Dates

UNIT IV - CSIR Maths Syllabus For NET

• 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).

• 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, limiting behaviour of n-step transition probabilities, classification of states, stationary distribution, Poisson and birth-and-death processes.

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

• 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, tests for linear hypotheses, estimability of parameters, best linear unbiased estimators, confidence intervals, Analysis of variance and covariance. Fixed, random and mixed effects models.

• Elementary regression diagnostics, Simple and multiple linear regression, 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: Principal component analysis, Discriminant analysis, Cluster analysis,

• Canonical

• Correlation

CSIR NET Maths Paper Pattern

According to the CSIR NET exam pattern, the test is held for five subjects - Physical Science, Chemical Sciences, Earth Sciences, Life Sciences and Mathematical Sciences. Candidates have to select one of their desired subjects. The test will be held for a total of 200 marks for which the candidates will be given a total of 3 hours. The test is held in online mode. The table below contains the NET Exam pattern for all five subjects for which the test is held.

CSIR NET Exam Pattern 2025