The CSIR NET Mathematical Science Syllabus 2026 helps candidates plan preparation for one of the five subjects offered in the Joint CSIR-UGC NET examination. It covers analysis, algebra, topology, differential equations, numerical methods, probability and statistics. Understanding how the four units are organised makes it easier to choose suitable books, practise questions and manage revision.
This guide explains the syllabus in student-friendly language, provides the official PDF for viewing and downloading, and outlines the Mathematical Sciences examination pattern. Whether you specialise in pure mathematics, applied mathematics or statistics, start by identifying the topics relevant to your background and the requirements of your exam session.
CSIR NET Mathematical Science Syllabus PDF – Preview and Download
Read the four-unit syllabus in the PDF viewer below, or save a copy for offline study. The file is linked from the CSIR-HRDG Mathematical Sciences examination page.
Download Mathematical Sciences Syllabus PDF ↗If your browser cannot display the preview, use the PDF link above to open the document directly.
CSIR NET Mathematical Science Exam Pattern
Mathematical Sciences is a single computer-based paper lasting three hours, with a maximum of 200 marks. It has three parts, not three separate papers. Part A tests general aptitude, while Parts B and C assess mathematical understanding and application.
| Section | Questions given | Attempt | Marks |
|---|---|---|---|
| Part A – General Aptitude | 20 | 15 | 30 |
| Part B – Subject MCQs | 40 | 25 | 75 |
| Part C – Analytical, multiple-correct | 60 | 20 | 95 |
| Total maximum marks | 200 | ||
According to the CSIR-HRDG Mathematical Sciences scheme, Parts A and B carry 25% negative marking for incorrect answers. Part C has no negative marking, but a question earns credit only when all correct options—and no incorrect options—are selected. Check the instructions issued for your examination session before attempting the paper.
CSIR NET Mathematical Sciences Syllabus 2026 – Unit-Wise Topics
The established syllabus groups subject-specific content into four units. Unit 1 provides foundations common to candidates. Units 2 and 3 are especially relevant to mathematics candidates, while Unit 4 concentrates on statistics and operations research. These units guide study planning; they do not replace the official PDF.
Unit 1: Analysis and Linear Algebra
Begin with the mathematical foundations used throughout the paper. Clear proofs and regular problem-solving are important here because later topics rely on these concepts.
Analysis
- Sets and real numbers: countable and uncountable sets, completeness, Archimedean property, supremum and infimum.
- Sequences and series: convergence, limit superior and inferior, Bolzano–Weierstrass and Heine–Borel theorems.
- Calculus: continuity, uniform continuity, differentiability, mean value theorem and sequences of functions.
- Integration: Riemann and improper integrals, functions of bounded variation, Lebesgue measure and integral.
- Multivariable analysis: partial and directional derivatives, inverse and implicit function theorems.
- Metric and normed spaces: compactness, connectedness and spaces of continuous functions.
Linear Algebra
- Vector spaces, subspaces, basis, dimension, linear independence and transformations.
- Matrices, determinants, rank, linear equations, eigenvalues and eigenvectors.
- Cayley–Hamilton theorem, change of basis, diagonal and Jordan forms.
- Inner-product spaces, orthonormal bases and classification of quadratic forms.
Unit 2: Complex Analysis, Algebra and Topology
This unit develops abstract reasoning through functions of a complex variable, algebraic structures and topological spaces. Focus on the conditions under which major theorems apply.
Complex Analysis
- Complex plane, analytic functions, Cauchy–Riemann equations and elementary transcendental functions.
- Contour integration, Cauchy theorem and integral formula, Liouville theorem, maximum modulus principle and Schwarz lemma.
- Taylor and Laurent expansions, residue calculus, conformal maps and Möbius transformations.
Algebra
- Combinatorics: permutations, combinations, pigeonhole principle, inclusion–exclusion and derangements.
- Number theory: divisibility, congruences, Chinese remainder theorem, Euler function and primitive roots.
- Group theory: subgroups, normal and quotient groups, homomorphisms, Sylow theorems and class equations.
- Rings, ideals, integral domains, polynomial irreducibility, fields, extensions and Galois theory.
Topology
- Topological bases, dense sets, subspace and product topologies.
- Separation axioms, compactness and connectedness.
Unit 3: Differential Equations and Applied Mathematics
Unit 3 connects theory to modelling and computation. Practise setting up equations, choosing solution methods and interpreting the results, rather than memorising procedures alone.
Ordinary Differential Equations
- Existence and uniqueness of initial-value solutions; first-order systems and singular solutions.
- Homogeneous and nonhomogeneous linear equations, variation of parameters, Sturm–Liouville problems and Green functions.
Partial Differential Equations
- Lagrange and Charpit methods; Cauchy problems and classification of second-order equations.
- Separation of variables for Laplace, heat and wave equations; constant-coefficient PDEs.
Numerical Analysis
- Iteration and Newton–Raphson methods, convergence rates, Gauss elimination and Gauss–Seidel iteration.
- Finite differences, interpolation, numerical differentiation and integration.
- Picard, Euler, modified Euler and Runge–Kutta methods for ODEs.
Calculus of Variations and Integral Equations
- Functionals, Euler–Lagrange equation, extremum conditions and variational boundary-value methods.
- Fredholm and Volterra equations, separable kernels, eigenfunctions and resolvent kernels.
Classical Mechanics
- Generalized coordinates, Lagrange and Hamilton equations, least-action principle.
- Rigid-body motion, Euler equations and small oscillations.
Unit 4: Statistics, Probability and Operations Research
Unit 4 is particularly important for candidates specialising in statistics. It combines probability theory, statistical modelling, sampling, experimental design and optimisation.
Probability and Stochastic Processes
- Descriptive statistics, exploratory data analysis, conditional probability and Bayes theorem.
- Random variables, moments, characteristic functions, probability inequalities and laws of large numbers.
- Central limit theorem, Markov chains, Poisson processes and birth–death processes.
Statistical Inference and Regression
- Common distributions, sampling distributions, estimation, confidence intervals and hypothesis tests.
- Likelihood ratio tests, chi-square tests, nonparametric tests and elementary Bayesian inference.
- Gauss–Markov models, ANOVA/ANCOVA, linear and logistic regression, multivariate normal and Wishart distributions.
- Principal component analysis, discriminant analysis, clustering and canonical correlation.
Sampling, Experimental Design and Optimization
- Simple random, stratified, systematic and probability-proportional-to-size sampling.
- Randomized block and Latin-square designs, BIBD and factorial experiments.
- Reliability and survival analysis, censoring and failure rates.
- Linear programming, simplex method, duality, queueing and inventory models.
How to Prepare for CSIR NET Mathematical Sciences
A practical study plan should balance conceptual learning with timed question practice. The best approach depends on your academic background, but a few habits are useful across the syllabus.
- Map the syllabus: Tick off topics you already know and mark those needing deeper study.
- Strengthen Unit 1: Review analysis and linear algebra regularly because they support many advanced problems.
- Choose a pathway: Allocate additional time to Units 2–3 for mathematics or Unit 4 for statistics, while respecting the official instructions.
- Write solutions: Work through proofs, calculations and counterexamples without immediately looking at answers.
- Practise previous-year questions: Group problems by topic to identify recurring methods and knowledge gaps.
- Take timed mock tests: Practise selection of questions and accuracy under the three-hour limit.
- Review errors: Keep a notebook of wrong assumptions, missed conditions and computational mistakes.
- Revise systematically: Use short revision cycles for definitions, standard results and techniques.
Suggested Study Sequence
Start with real analysis and linear algebra, then move to complex analysis and algebraic structures. Mathematics-focused candidates can follow with ODEs, PDEs and numerical analysis; statistics-focused candidates can devote more time to probability, inference and stochastic processes. Reserve the final stage for mixed practice and full-length mock tests.
When solving Part C questions, carefully check every option. Because the section uses multiple-correct questions with no partial credit, understanding why an option is false can be as valuable as recognising the correct statements.
Frequently Asked Questions
How many units are in the Mathematical Sciences syllabus?
The established CSIR-HRDG syllabus contains four units: Analysis and Linear Algebra; Complex Analysis, Algebra and Topology; Applied Mathematics; and Statistics and related topics.
Is General Aptitude included?
Yes. Part A is common to CSIR NET subjects and includes general science, quantitative reasoning, analysis and research aptitude.
Can I download the Mathematical Sciences syllabus for free?
Yes. Use the official PDF link near the top of this page to read or save the document.
Does Part C have negative marking?
The CSIR-HRDG Mathematical Sciences examination scheme states that Part C has no negative marking. However, it also states that credit is awarded only when all correct options are selected without marking an incorrect option.