UGC NET Mathematics Syllabus 2026 – Complete Guide, Paper 1 & Paper 2 Topics, Exam Pattern
UGC NET Mathematics is one of the most sought-after papers among candidates who want to pursue a career in teaching or research at the university level. Whether you are targeting an Assistant Professor position or a Junior Research Fellowship (JRF), understanding the full syllabus is the first and most important step. This guide gives you the complete, updated UGC NET Mathematics syllabus for 2026, unit-wise topic breakdown, exam pattern, preparation strategy, and important resources — all in one place.
UGC NET Mathematics Exam Pattern 2026
The UGC NET exam has two papers — Paper 1 (General/Teaching & Research Aptitude) and Paper 2 (subject-specific). For Mathematics, Paper 2 is entirely based on advanced mathematical topics. Both papers are conducted on the same day.
UGC NET Mathematics Paper 2 Syllabus – Unit Wise
The Mathematics Paper 2 syllabus is divided into 10 units. Each unit covers a specific branch of mathematics. Topics below are aligned with the latest NTA-prescribed syllabus.
Unit 1 – Real Analysis
Real analysis forms the foundation for higher mathematics. This unit tests your understanding of sequences, series, continuity, and integration — all at a rigorous theoretical level.
- Finite, countable and uncountable sets; Real number system; Sequences and series of real numbers; Monotone sequences; Cauchy sequences; Subsequences; Bolzano–Weierstrass theorem
- Limits, continuity and differentiability; Mean value theorems (Rolle's, Lagrange's, Cauchy's); Taylor's theorem with remainder
- Functions of several variables: partial derivatives, Jacobians, maxima and minima
- Riemann integral; Improper integrals; Uniform convergence; Power series
- Fourier series: convergence, Dirichlet's conditions; Lebesgue measure and integration (basic concepts)
Unit 2 – Complex Analysis
Complex Analysis builds on real analysis and introduces functions of complex variables, which are crucial for advanced mathematics and physics.
- Complex numbers; Algebraic and geometric representation; Modulus, argument, conjugate
- Analytic functions; Cauchy–Riemann equations; Harmonic functions
- Contour integration; Cauchy's theorem and integral formula; Taylor and Laurent series
- Singularities: removable, poles, essential; Residue theorem; Evaluation of real integrals
- Conformal mappings; Bilinear (Möbius) transformations
Unit 3 – Algebra
Abstract Algebra is a major unit covering groups, rings, and fields — concepts that underpin all of modern mathematics.
- Groups: subgroups, normal subgroups, cosets, Lagrange's theorem, quotient groups, homomorphisms, isomorphisms
- Permutation groups; Cayley's theorem; Sylow theorems; Direct products
- Rings: ideals, quotient rings, integral domains, fields; Ring homomorphisms; Polynomial rings
- Field extensions; Finite fields; Galois theory (basic concepts)
- Vector spaces: linear dependence, basis, dimension, linear transformations; Eigenvalues and eigenvectors
Unit 4 – Linear Algebra
- Matrices and systems of linear equations; Row echelon form; Rank; Determinants
- Vector spaces and subspaces; Linear maps; Null space; Range; Rank–nullity theorem
- Inner product spaces; Gram-Schmidt orthogonalization; Orthogonal and unitary matrices
- Bilinear forms; Quadratic forms; Positive definite matrices
- Eigenvalues, eigenvectors; Characteristic polynomial; Diagonalization; Cayley–Hamilton theorem
Unit 5 – Topology
- Topological spaces; Open and closed sets; Interior, exterior, boundary; Closure; Dense sets
- Basis; Sub-basis; Product topology; Subspace topology; Quotient topology
- Continuous functions; Homeomorphisms; Open and closed maps
- Compactness: finite intersection property; Tychonoff's theorem; Connectedness; Path-connectedness
- Separation axioms: T0, T1, T2, T3, T4 spaces; Urysohn's lemma; Metrization
Unit 6 – Ordinary and Partial Differential Equations
- ODEs: order, degree; Separable equations; Exact equations; Linear equations (1st & higher order); Bernoulli's equation
- Systems of ODEs; Phase plane analysis; Stability; Series solutions; Special functions (Legendre, Bessel)
- PDEs: formation; Classification; Wave equation; Heat equation; Laplace equation; Method of characteristics
- Boundary value problems; Sturm–Liouville problems; Eigenfunction expansions
Unit 7 – Calculus of Variations and Integral Equations
- Functionals; Euler-Lagrange equations; Extremals; Brachistochrone problem; Geodesics
- Constrained optimization; Isoperimetric problems
- Volterra and Fredholm integral equations; Neumann series; Resolvent kernel
- Symmetric kernels; Eigenvalues of integral equations; Hilbert-Schmidt theorem
Unit 8 – Mechanics and Fluid Dynamics
- Generalized coordinates; Lagrange's equations; Hamilton's equations; Conservation laws
- Rigid body dynamics; Moment of inertia; Euler's equations of motion
- Fluid kinematics; Continuity equation; Euler's and Bernoulli's equation; Navier-Stokes equation (basics)
- Potential flow; Stream functions; Vorticity; Irrotational flow
Unit 9 – Numerical Analysis and Computer Programming
- Error analysis; Floating point representation; Numerical solution of algebraic equations (bisection, Newton-Raphson, secant methods)
- Interpolation: Newton's, Lagrange's; Numerical differentiation and integration (Trapezoidal, Simpson's rules)
- Numerical solution of ODEs: Euler, Runge-Kutta methods
- Computer programming basics: flow charts, algorithms; Programming in C (basics)
