Review of Fourier series and Applications, Review of Laplace Transforms and Applications. Classification of second order linear partial differential equations, Canonical forms for hyperbolic, parabolic and elliptic equations, Homogeneous and Non Homogeneous equations with constant coefficients. Applications
Vector Functions, General rules for differentiation, Velocity and Acceleration, Gradient of a scalar field, Directional Derivative, Properties of Gradient, Divergence of vector point function, Curl of a vector point function, Properties of Divergence and Curl. Applications Integration of vector functions, Line integral, Circulation, Work done by a force, Surface integrals, Volume integrals, Divergence Theorem of Gauss, Green’s Theorem in the plane, Stoke’s Theorem, problems on all the three theorems and Applications.
Review of Complex analysis, Complex analysis applied to potential theory, Electrostatic fields, conformal mapping, Heat problems, Fluid flow, General properties of Harmonic functions, Complex Integration, Cauchy’s Theorem, Cauchy’s Integral Formula, Cauchy’s Integral Formula for Derivatives, Taylor’s and Laurent’s series. Applications. Singular point, Residue, Method of finding Resides, Residue Theorem,Contour Integration, Integration round the unit circle, Rectangular contour. Applications.
Numerical Solutions algebraic and transcendental equations: False position method, Newton – Raphson method, Iteration method, Aitken’s method, Solution of linear simultaneous equations. Gauss elimination method, Inverse of a matrix , Gauss-Seidal method, Crout’s method. Solution of Ordinary Differential Equations: Taylor’s Series method, Picard’s method, Euler’s method, Euler’s Modified method, Runge-Kutta 4thorder method. Predictor and corrector method (Milen’s and Adams-Bashfourth) Applications.
Finite differences, Interpolation, Newton’s Forward & Backward Interpolation formulae, Lagrange’s formula, Newton’s Divided difference, Central difference formulae (all formulae with proof). Numerical Differentiation, Numerical Integration (all rules with proof). Applications.