# CSIR-UGC JRF Engineering Science Mathematics Syllabus 2013

CSIR-UGC
Engineering Science Question paper shall contain Engineering Mathematics and
Engineering Aptitude question which comprises of 70 marks out of 200 total
allotted marks.

Generally Engineering
Mathematics and Engineering Aptitude shall have the following major divisions
as Linear Algebra, Calculus, Complex variables, Vector Calculus, Ordinary
Differential, Equations
and Probability.

The detailed
Syllabus under the major topics mentioned above are discussed below which will
be very essential while preparing for your CSIR-UGC JRF Engineering Sciences
Examination.

**Linear Algebra:**The Linear algebra consists of Inverse, Rank, Algebra of matrices, and system of linear equations, symmetric & skew-symmetric as well orthogonal matrices. Hermitian & Skew-Hermitian as well unitary matrices. Eigen values & eigenvectors, diagonalisation of matrices alone.

**Calculus:**limit, Functions of single variable, continuity and differentiability, Indeterminate forms and L'Hospital rule, Mean value theorems, Maxima and minima, Taylor's series. Fundamental and mean value-theorems of integral calculus. Newton’s method for finding roots of polynomials. Numerical integration by trapezoidal and Simpson’s rule. Evaluation of definite and improper integrals, Beta & Gamma functions, Functions of two variables, continuity, limit, partial derivatives, total derivatives, Euler's theorem for homogeneous functions, maxima and minima, Lagrange method of multipliers, double integrals & their applications, series and sequence, tests for convergence, Fourier Series, power series, Half range sine as well cosine series.

**Complex variables**: Cauchy-Riemann equations, Analytic functions, Line integral, Cauchy's integral theorem and integral formula Laurent' series and Taylor’s, Residue theorem & its applications.

**Equations:**Higher order linear differential equations with constant coefficients, parameters method, Cauchy-Euler's equations, Legendre polynomials and Bessel's functions of the first kind and their properties, power series solutions. Numerical solutions of first order ordinary differential equations by Runge-Kutta and Euler’s methods.

**Probability:**Definitions of probability and simple theorems, Bayes Theorem and conditional probability.

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