š PG TRB Mathematics ā Unit-wise Syllabus
š Unit 1 ā Algebra
Groups: subgroups, cyclic & permutation groups, homomorphisms, Lagrange, Cauchy & Sylow theorems, classification of finite abelian groups.
Rings & fields: ideals, quotient rings, ring homomorphisms, Euclidean & polynomial rings, UFDs, field of fractions; extension fields & basics of Galois theory; finite fields.
Linear algebra: vector spaces, basis & dimension, linear maps & rankānullity, eigenvalues/eigenvectors; diagonal/triangular/Jordan forms; nilpotent maps; inner-product spaces; quadratic forms; Hermitian/unitary/normal operators.
š Unit 2 ā Real Analysis
Sets & completeness of ā, supremum/infimum, sequences & series (lim sup/lim inf), BolzanoāWeierstrass, HeineāBorel; continuity, uniform continuity, differentiability & mean value theorems; uniform convergence of sequences/series of functions; RiemannāStieltjes integral & its properties; power series & Fourier series; multivariable calculus: directional/partial derivatives, derivative as linear map, inverse & implicit function theorems.
š Unit 3 ā Topology
Topological spaces, bases & subspaces; product & order topologies; closed sets & limit points; continuous maps; connectedness & components; compactness (HeineāBorel, local & limit-point compactness); countability axioms; separation axioms (T1āT4), normal spaces; Urysohn lemma, Tietze extension theorem, metrization ideas.
š Unit 4 ā Complex Analysis
Analytic functions; power-series expansions (Maclaurin), uniform convergence & Abelās theorem; conformal mappings, Mƶbius transforms & cross-ratio; complex integration & Cauchyās theorems; Cauchy integral formula; Taylor series & zeros; isolated singularities & residues; maximum modulus principle & applications.
š Unit 5 ā Functional Analysis
Banach spaces: Hƶlder & Minkowski, continuous linear maps, HahnāBanach, natural embedding, open mapping & closed graph theorems; dual operators.
Hilbert spaces: orthonormal sets/bases, adjoint & projections, basic spectral facts in finite dimension.
Banach algebras: spectrum, spectral radius, radical & semisimplicity.
š Unit 6 ā Differential Geometry
Space curves: SerretāFrenet formulas, curvature/torsion, intrinsic equations, helices, spherical indicatrix.
Surfaces: first/second fundamental forms, Gaussian/mean curvature, surfaces of revolution & helicoids, isometries; Meusnier & Euler theorems; lines of curvature & asymptotic lines; Dupinās indicatrix; developables; geodesics & conjugate points.
š Unit 7 ā Differential Equations
ODEs: linear equations (constant/variable coefficients), Wronskian, nonhomogeneous equations; regular singular points; special equationsāLegendre, Bessel, Hermite; existence & uniqueness (Lipschitz).
PDEs: first-order (Lagrange & Charpit); classification of second-order PDEs; higher-order with constant coefficients; separation of variables for Laplace/Heat/Wave (up to 2D).
š Unit 8 ā Classical Mechanics & Numerical Analysis
Mechanics: generalized coordinates; Lagrangeās equations; Hamiltonās equations & principle of least action; canonical transformations; generating functions; Poisson & Lagrange brackets.
Numerical: roots (iteration, NewtonāRaphson); linear systems (Gauss, GaussāSeidel); finite differences; interpolation (Lagrange, Hermite, splines); numerical differentiation/integration; ODE solvers (Picard, Euler, modified Euler, RungeāKutta).
š Unit 9 ā Operations Research
Linear programming & simplex (primal/dual/dual-simplex/revised); integer programming; dynamic & nonlinear programming; network analysis, max-flow/min-cut; queueing models (M/M/1, M/M/c, limited space, M/G/1); inventory models (deterministic, with/without shortages, price break).
š Unit 10 ā Probability & Statistics
Probability: independence, Bayes; random variables & distributions (binomial, Poisson, uniform, normal, exponential, gamma, beta, Cauchy); expectation, moments, MGFs/characteristic functions; inequalities (Markov, Chebyshev, Jensen); convergence; LLN & CLT.
Statistics: standard errors; sampling distributions (t, F, ĻĀ²) & applications; hypothesis testing (large-sample tests), ANOVA.