PGTRB Mathematics Syllabus 2025 – Complete Breakdown
Shiksha Class – Admission Going On
📞 Contact: 7448878882 / 7448878884
Unit I: Algebra
- Groups: Cyclic Groups, Permutation Groups, Lagrange’s theorem, Normal subgroups, Homomorphism, Cayley’s theorem, Cauchy’s theorem, Sylow’s theorems, Finite Abelian Groups
- Rings: Integral Domain, Field, Ring Homomorphism, Ideals and Quotient Rings, Field of Quotients, Euclidean Rings, Polynomial Rings, Unique Factorization Domain
- Fields: Extension fields, Galois theory, Finite fields
- Vector Spaces: Bases, Dual spaces, Inner product spaces, Linear transformations, Rank, Characteristic roots, Canonical forms, Jordan form, Quadratic forms, Hermitian, Unitary, Normal transformations
Unit II: Real Analysis
- Set theory, Countable and uncountable sets, Real number system, Archimedean Property
- Sequences and Series: Convergence, Supremum/Infimum, Limit theorems, Continuity, Uniform Continuity, Differentiability, Mean Value Theorem
- Series of functions: Uniform convergence, Continuity, Integration, Differentiation
- Riemann–Stieltjes integral: Definition, Properties, Integration of vector functions
- Power Series and Fourier Series
- Functions of several variables: Directional & Partial derivatives, Inverse & Implicit function theorems
Unit III: Topology
- Topological spaces: Basis, Order, Product, Subspace Topologies
- Closed sets, Limit points, Continuous functions, Box and Product Topologies
- Connectedness, Compactness, Local Compactness
- Countability and Separation Axioms
- Normal spaces: Urysohn Lemma, Urysohn Metrization Theorem, Tietze Extension Theorem
Unit IV: Complex Analysis
- Analytic Functions: Limits, Continuity, Power Series, Conformality, Mapping, Linear transformations
- Complex Integration: Line integrals, Cauchy’s Theorem and Integral Formula, Index of a point, Taylor’s theorem, Zeros and Poles, Maximum Modulus Principle
Unit V: Functional Analysis
- Banach Spaces: Definitions, Inequalities, Linear transformations, Hahn-Banach Theorem, Open Mapping, Closed Graph Theorem
- Hilbert Spaces: Orthonormal bases, Adjoint of operators, Spectral Theorem
- Matrices and Operators: Determinants, Spectral radius, Regular and Singular elements, Banach Algebra
Unit VI: Differential Geometry
- Curves and Surfaces: Frenet formulas, Curvature, Helices, Surfaces of revolution
- Geometric Concepts: Gaussian curvature, Fundamental forms, Isometry, Geodesics, Indicatrix, Dupin’s Theorem
Unit VII: Differential Equations
Ordinary Differential Equations:
- Linear equations (constant & variable coefficients), Wronskian, Non-homogeneous equations, Initial value problems
- Special equations: Legendre, Bessel, Hermite
- Existence and Uniqueness, Exact equations, Lipschitz condition
Partial Differential Equations:
- Lagrange and Charpit Methods
- Classification and Solution of Second Order PDEs
- Separation of Variables: Laplace, Heat, Wave equations (2D)
Unit VIII: Classical Mechanics and Numerical Analysis
Classical Mechanics:
- Generalized Coordinates, Lagrange’s Equations, Hamilton’s Equations, Canonical transformations, Poisson brackets
Numerical Analysis:
- Numerical solutions of equations: Iteration, Newton-Raphson
- Linear equations: Gauss Elimination, Gauss-Seidel
- Interpolation: Lagrange, Hermite, Spline
- Numerical differentiation & integration
- ODE Solvers: Picard, Euler, Modified Euler, Runge-Kutta
Unit IX: Operations Research
- Linear Programming: Simplex, Duality, Revised Simplex
- Integer & Dynamic Programming, Non-linear Programming
- Network Analysis: Max Flow-Min Cut
- Queuing Theory: M/M/1, M/M/C, M/G/1 models
- Inventory Models: Deterministic, Single Price Break
Unit X: Probability Theory
- Probability Basics: Sample space, Bayes’ Theorem, Random Variables, Distribution Functions
- Expectations, Moments, Characteristic Functions, Inequalities (Chebyshev, Markov)
- Convergence, Law of Large Numbers, Central Limit Theorem (i.i.d)
- Distributions: Binomial, Poisson, Normal, Exponential, Gamma, Beta, Cauchy
- Sampling Distributions: t, F, Chi-square, ANOVA, Large Sample Tests
Shiksha Class – PGTRB Mathematics Coaching 2025
📞 Contact: 7448878882 / 7448878884
0 Comments