MA 2161 Engineering Mathematics
Unit I Ordinary differential equations
Higher order linear differential equation with constant coeffecients – Methods of variation of parameters – Cauchy’s and legendre’s linear equations – Simultaneous first order linear equations with constant coefficients.
Unit II Vector calculus
Gradient divergence and curl – Directional derivative – Irrotational and solenoidal vector fields – Vector integration – Green’s theorem and stoke’s theorem (excluding proofs) – Simple applications involving cubes and rectangular parallelopipeds.
Unit III Analytic functions
Functions of a Complex variable – Analytic functions - Necessary conditions, Cauchy – Reimann equation and sufficient conditions (excluding proofs) – Harmonic and orthogonal properties of analytic function – Harmonic conjugate – Construction of analytic functions – conformal mapping: w=z+c, cz, 1/z and bilinear transformation.
Unit IV Complex integration
Complex integration – Statement and applications of cauchy’s integral theorem and cauchy’s integral formula – Taylor and Laurent expansions – singular points – Residues - Residues theorem – application of residue theorem to evaluate real integrals – Unit circle and semicircular contour (excluding poles and boundaries)
Unit V Laplace transform
Laplace transform – conditions for existence – Transform of elementary functions – Basic properties – Transform of derivatives and integrals – Transform of unit step function and impulse functions – Transform of periodic functions.
Definition of inverse laplace transform as contour integral – Convolution theorem (excluding proof) – Initial and final values theorems – Solution of linear ODE of second order with coefficients using laplace transformation techniques.