Matrices : Linear dependence of vectors and rank of matrices. Elementary transformations, Gauss - Jordan method to find inverse of a matrix, reduction to normal form, Consistency and solution of algebraic equations, Linear transformations, Orthogonal transformations, Eigen values, Eigen vectors, Cayley Hamilton theorem, Reduction to diagonal form, Bilinear and quadratic form, Orthogonal, Unitary, Hermitian and similar matrices.
Ordinary Differential Equations : Exact differential equations, Equations reducible to exact form by integrating factors; Equations of the first order and higher degree. Clairaut’s equation.
Linear Differential Equations : Leibnitz’s linear and Bernoulli’s equation, Methods of finding complementary functions and particular integrals. Special methods for finding particular integrals : (i) Method of variation of parameters
(ii) Method of underdetermined coefficients. Cauchy’s homogeneous and Legendre’s linear equation. Simultaneous linear equations with constant coefficients.
Applications of Differential Equations : Applications to electric / electronic
L-R-C circuits. Deflection of beams, Simple harmonic motion, Oscillation of a spring.
Vector Calculus : Scalar and vector fields, Differentiation of vectors, Velocity and acceleration. Vector differential operators Del, Gradient, Divergence and curl, Their physical interpretation. Formulae involving Del applied to point functions and their products. Line, Surface and Volume integrals.
Application of Vector Calculus : Flux, Solenoidal and irrotational vectors. Gauss Divergence theorem. Green’s theorem in plane. Stoke’s theorem. Applications to electromagnetics and fluid mechanics.
Statistics : Recapitulation of statistics and probability. Discrete and continuous probability distributions. Binomial, Poisson and Normal distribution, Applications. Curve fitting.
Sampling and Testing of Hypothesis : Sampling methods. Student’s t-test, Chi-square test, F-test and Fisher’s z-test.