### NET JRF Syllabus And Reference Book

- CSIR NET-JRF Syllabus is distributed into 11 Subjects Linear Algebra ,Complex Analysis ,Real Analysis ,Ordinary Differential Equations ,Algebra ,Functional Analysis ,Numerical Analysis ,Partial Differential Equations ,Topology ,Probability and Statistics ,Linear programming

#### Linear Algebra

- Finite dimensional vector spaces;
- Linear transformations and their matrix representations, rank;
- Systems of linear equations,
- Eigenvalues and eigenvectors, minimal polynomial, Cayley-Hamilton Theorem,
- Diagonalization, Jordan-canonical form,
- Hermitian, Skew- Hermitian and unitary matrices;
- Finite dimensional inner product spaces, Gram-Schmidt orthonormalization process, self-adjoint operators,
- Definite forms.

#### Complex Analysis

- Analytic functions,
- Complex integration: Cauchy’s integral theorem and formula;
- Liouville’s theorem, maximum modulus principle; Zeros and singularities;
- Taylor and Laurent’s series; residue theorem and applications for evaluating real integrals.
- Conformal mappings, bilinear transformations;

#### Real Analysis

- Riemann integration,
- Sequences and series of functions, uniform convergence,
- Power series, Fourier series,
- Functions of several variables, maxima, minima;
- Multiple integrals, line, surface and volume integrals,
- Theorems of Green, Stokes and Gauss;
- Metric spaces, compactness, completeness,
- Weierstrass approximation theorem;
- Lebesgue measure, measurable functions; Lebesgue integral,
- Fatou’s lemma, dominated convergence theorem.

#### Ordinary Differential Equations

- First order ordinary differential equations,
- Existence and uniqueness theorems for initial value problems,
- Systems of linear first order ordinary differential equations,
- Linear ordinary differential equations of higher order with constant coefficients;
- Linear second order ordinary differential equations with variable coefficients;
- Method of Laplace transforms for solving ordinary differential equations,
- Series solutions (power series, Frobenius method);
- Legendre and Bessel functions and their orthogonal properties.

#### Algebra

- Groups, subgroups, cyclic groups and permutation groups,
- normal subgroups, Quotient groups and
- Homomorphism theorems, automorphisms;
- Sylow’s theorems and their applications;
- Rings, ideals, prime and maximal ideals, quotient rings,
- Unique factorization domains, Principle ideal domains, Euclidean domains, polynomial
- Rings and irreducibility criteria;
- Fields, finite fields, field extensions.

#### Functional Analysis

- Normed linear spaces,
- Banach spaces, Hahn-Banach extension theorem,
- Open mapping and closed graph theorems,
- Principle of uniform boundedness;
- Inner-product spaces, Hilbert spaces, orthonormal bases,
- Riesz representation theorem, bounded linear operators.

#### Numerical Analysis

- Bisection, secant method,
- Newton-Raphson method,
- Fixed point iteration;

**Numerical solution of algebraic and transcendental equations:**

**Interpolation:**

- Error of polynomial interpolation,
- Lagrange, Newton interpolations;
- Numerical differentiation;
- Numerical integration:
- Trapezoidal and Simpson rules;

**Numerical solution of systems of linear equations:**

- Direct methods (Gauss elimination, LU decomposition);
- Iterative methods (Jacobi and Gauss-Seidel);

**Numerical solution of ordinary differential equations:**

- initial value problems:
- Euler’s method,
- Runge-Kutta methods of order 2.

#### Partial Differential Equations

- Linear and quasilinear first order partial differential equations,
- Method of characteristics;
- Second order linear equations in two variables and their classification;
- Cauchy, Dirichlet and Neumann problems;
- Solutions of Laplace, wave in two dimensional Cartesian coordinates,
- Interior and exterior Dirichlet problems in polar coordinates;
- Separation of variables method for solving wave and diffusion equations in one space variable;
- Fourier series and Fourier transform and
- Laplace transforms methods of solutions for the above equations.

#### Topology

- Basic concepts of topology,
- Bases, subbases, subspace topology,
- Order topology, product topology,
- Connectedness, compactness,
- Countability and separation axioms,
- Urysohn’s Lemma.

#### Probability and Statistics

- Probability space, conditional probability, Bayes theorem,
- Independence, Random variables, joint and conditional distributions,
- Standard probability distributions and their properties,Discrete Uniform, Binomial, Poisson, Geometric, Negative binomial,Normal, Exponential, Gamma
- Continuous probability distributions ,uniform, Bivariate , Normal, Multinomial
- Expectation, conditional expectation, moments; Weak and strong law of large numbers, Central limit theorem; Sampling distributions, UMVU estimators, Maximum likelihood estimators; Interval estimation ,Testing of hypotheses, standard parametric tests based on normal distributions; Simple linear regression
- Linear programming problem and its formulation,
- Convex sets and their properties,
- Graphical method,
- Basic feasible solution, Infeasible and unbounded LPP’s, alternate optima
- Simplex method, big-M and two phase methods;
- Dual problem and duality theorems,
- Dual simplex method
- Application in post optimality analysis;
- Balanced and unbalanced transportation problems,
- Vogel’s approximation method for solving transportation problems;
- Hungarian method for solving assignment problems.

#### Linear programming

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