### NET JRF Syllabus And Reference Book

- CSIR NET-JRF Syllabus is distributed into 4 units, Unit-1: Real Analysis , Linear Algebra , Unit-2: Complex Analysis, Algebra, Topology, Unit-3: Ordinary Differential Equations (ODEs), Partial Differential Equations (PDEs), Numerical Analysis, Calculus of Variations , Linear Integral Equations and ,Classical Mechanics Unit-2: Descriptive Statistics, exploratory data analysis, Linear programming problem

- Elementary set theory, finite, countable and uncountable sets,
- Real number system as a complete ordered field, Archimedean property, supremum, infimum.
- Sequences and series, convergence, limsup, liminf. Bolzano Weierstrass theorem, Heine Borel theorem.
- Continuity, Monotonic functions, types of discontinuity,
- Uniform continuity,
- Differentiability, mean value theorem.
- Sequences and series of functions, Uniform convergence.
- Riemann sums and Riemann integral,
- Improper Integrals.
- Functions of bounded variation,
- Lebesgue measure, Lebesgue integral.
- Functions of several variables, directional derivative, partial derivative,
- Derivative as a linear transformation, inverse and implicit function theorems.
- Metric spaces, compactness, connectedness.
- Normed linear Spaces. Spaces of continuous functions as examples.
- Vector spaces, subspaces,
- Linear dependence,
- Basis, dimension,
- Algebra of linear transformations.
- Matrix representation of linear transformations. Change of basis,
- Algebra of matrices,
- Rank and determinant of matrices,
- Linear equations.
- Eigenvalues and eigenvectors, Cayley-Hamilton theorem.
- Canonical forms, Diagonal forms, triangular forms,
- Jordan forms.
- Inner product spaces, orthonormal basis.
- Quadratic forms, reduction and classification of quadratic forms

#### Analysis:

#### Reference Book

Book Name | Book By |

Introduction to Real Analysis | Sadhan Kumar Mapa |

Introduction to Real Analysis | Donald R. Sherbert and Robert G. Bartle |

Principles of Real Analysis | S.L. Gupta ,N.R.Gupta |

Mathematical Analysis | S.C. Malik and Savitha Arora |

#### Linear Algebra:

#### Reference Book

Book Name | Book By |

Linear Algebra | Vivek Sahai, Vikas Bist |

Linear Algebra | Arnold J. Insel, Lawrence E. Spence, and Stephen H. Friedberg |

Matrices and Linear Algebra | George Phillip Barker and Hans Schneider |

Linear Algebra | Seymour Lipschutz and, Marc Lipson |

#### Complex Analysis:

#### Reference Book

Book Name | Book By |

Complex Variables: | H.S. Kasana |

Foundations of Complex Analysis | S. Ponnusamy |

Complex Variable | Murray Spiegel, Seymour Lipschutz |

Complex analysis | Jane Gilman |

#### Algebra:

#### Reference Book

Book Name | Book By |

Contemporary Abstract Algebra | Joseph A. Gallian |

A Course in Abstract Algebra, | V.K. Khanna & S.K Bhamri |

Abstract Algebra | David S. Dummit, Richard M. Foote |

Fields and Galois Theory | John Mackintosh Howie |

#### Topology:

#### Reference Book

Book Name | Book By |

Basic topology | M. A. Armstrong |

Introduction to Topology | Bert Mendelson |

Topology | James Munkres |

#### Ordinary Differential Equations (ODEs):

- Existence and uniqueness of solutions of initial value problems for first order ordinary differential equations,
- singular solutions of first order ODEs,
- General theory of homogenous and non-homogeneous linear ODEs,
- system of first order ODEs.
- Variation of parameters,
- Sturm-Liouville boundary value problem,
- Green’s function.

#### Reference Book

Book Name | Book By |

Ordinary differential equations | Earl A. Coddington |

Differential Equations | Shepley L. Ross |

Essentials of ODE | Ravi Agarwal |

#### Partial Differential Equations (PDEs):

- Lagrange and Charpit methods for solving first order PDEs,
- Cauchy problem for first order PDEs.
- Classification of second order PDEs,
- General solution of higher order PDEs with constant coefficients,
- Method of separation of variables for Laplace, Heat and Wave equations.

#### Reference Book

Book Name | Book By |

Ordinary & Partial Diff.Equation | M. D. Raisinghania |

Partial Differential Equations | K. Sankara Rao |

Partial Differential Equations | T. Amaranath |

#### Numerical Analysis:

- Numerical solutions of algebraic equations,
- Method of iteration and Newton-Raphson method,
- Rate of convergence,
- Solution of systems of linear algebraic equations using Gauss elimination and Gauss-Seidel methods,
- Finite differences, Lagrange, Hermite and spline interpolation,
- Numerical differentiation and integration,
- Numerical solutions of ODEs using Picard, Euler, modified Euler and Runge-Kutta methods.

#### Reference Book

Book Name | Book By |

Numerical Methods | S. R. K.Iyengar,R. K. Jain, M. Kumar Jain |

Methods of Numerical Analysis | S. S. Sastry |

Numerical analysis | Kendall Atkinson |

#### Calculus of Variations:

- Variation of a functional, Euler-Lagrange equation,
- Necessary and sufficient conditions for extrema.
- Variational methods for boundary value problems in ordinary and partial differential equations.

#### Reference Book

Book Name | Book By |

Calculus of Variations with Applications | Gupta A.S |

Calculus of variations | Israel Gelfand |

#### Linear Integral Equations:

- Linear integral equation of the first and second kind of Fredholm and Volterra type,
- Solutions with separable kernels.
- Characteristic numbers and eigenfunctions,
- Resolvent kernel.

#### Reference Book

Book Name | Book By |

Integral Equations | Shanti Swarup |

Linear Integral Equations | Ram P. Kanwal |

linear integral equations | M.D Raisinghania |

#### Classical Mechanics:

- Generalized coordinates, Lagrange’s equations,
- Hamilton’s canonical equations,
- Hamilton’s principle and principle of least action,
- Two-dimensional motion of rigid bodies,
- Euler’s dynamical equations for the motion of a rigid body about an axis,
- theory of small oscillations.

#### Reference Book

Book Name | Book By |

Classical Mechanics | Binoy Bhattacharyya |

Classical Mechanics | Tom W. B. Kibble and Frank H. Berkshire |

#### Descriptive Statistics, exploratory data analysis

- Sample space, discrete probability, independent events,
- Bayes theorem. Random variables and distribution functions (univariate and multivariate);
- expectation and moments.
- Independent random variables, marginal and conditional distributions.
- Characteristic functions. Probability inequalities (Tchebyshef, Markov, Jensen).
- Modes of convergence, weak and strong laws of large numbers,
- Central Limit theorems (i.i.d. case).
- Markov chains with finite and countable state space, classification of states,
- limiting behaviour of n-step transition probabilities,
- stationary distribution,
- Poisson and birth-and-death processes.
- Standard discrete and continuous univariate distributions.
- sampling distributions,
- standard errors and asymptotic distributions,
- distribution of order statistics and range.
- Methods of estimation, properties of estimators,
- confidence intervals. Tests of hypotheses:
- most powerful and uniformly most powerful tests, likelihood ratio tests.
- Analysis of discrete data and chi-square test of goodness of fit.
- Large sample tests. Simple nonparametric tests for one and two sample problems,
- rank correlation and test for independence.
- Elementary Bayesian inference. Gauss-Markov models,
- estimability of parameters, best linear unbiased estimators, confidence intervals,
- tests for linear hypotheses. Analysis of variance and covariance.
- Fixed, random and mixed effects models.
- Simple and multiple linear regression.
- Elementary regression diagnostics. Logistic regression.
- Multivariate normal distribution, Wishart distribution and their properties.
- Distribution of quadratic forms.
- Inference for parameters, partial and multiple correlation coefficients and related tests.
- Data reduction techniques: Principle component analysis, Discriminant analysis, Cluster analysis,
- Canonical correlation. Simple random sampling,
- stratified sampling and systematic sampling.
- Probability proportional to size sampling.
- Ratio and regression methods. Completely randomized designs,
- randomized block designs and Latin-square designs.
- Connectedness and orthogonality of block designs, BIBD. 2K factorial experiments:
- Confounding and construction.
- Hazard function and failure rates,
- Censoring and life testing, series and parallel systems.

#### Reference Book

Book Name | Book By |

Fundamentals of Descriptive Statistics | Zealure C Holcomb |

Fundamentals of Mathematical Statistics | Gupta S C |

Miller & Freund'S Probability And Statistics | Richard A. Johnson |

#### Linear programming problem:

- Simplex methods, duality.
- Elementary queuing and inventory models.
- Steady-state solutions of Markovian
- queuing models: M/M/1, M/M/1 with limited waiting space,
- M/M/C, M/M/C with limited waiting space, M/G/1.

#### Reference Book

Reference Books | |

Book Name | Book By |

Introduction to Management Science OR | Man Mohan P. K. Gupta Kanti Swarup |

Opeartions Research And Introduction | Hamdy A. Taha |