**MA8251 Engineering Mathematics 2 **

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**Padeepz – MA8251 Engineering Mathematics 2 **

Unit 1 MATRICES

Eigenvalues and Eigenvectors of a real matrix – Characteristic equation – Properties of

Eigenvalues and Eigenvectors – Cayley-Hamilton theorem – Diagonalization of matrices – Reduction of a quadratic form to canonical form by orthogonal transformation – Nature of quadraticforms.

UNIT II VECTOR CALCULUS

Gradient and directional derivative – Divergence and curl – Vector identities – Irrotational and

Solenoidal vector fields – Line integral over a plane curve – Surface integral – Area of a curved

surface – Volume integral – Green’s, Gauss divergence and Stoke’s theorems – Verification and

application in evaluating line, surface and volume integrals.

UNIT III ANALYTIC FUNCTIONS

Analytic functions – Necessary and sufficient conditions for analyticity in Cartesian and polar

coordinates – Properties – Harmonic conjugates – Construction of analytic function – Conformal

mapping – Mapping by functions, – Bilinear transformation.

UNIT IV COMPLEX INTEGRATION

Line integral – Cauchy’s integral theorem – Cauchy’s integral formula – Taylor’s and Laurent’s

series – Singularities – Residues – Residue theorem – Application of residue theorem for

evaluation of real integrals – Use of circular contour and semicircular contour.

UNIT V LAPLACE TRANSFORMS

Existence conditions – Transforms of elementary functions – Transform of unit step function and

unit impulse function – Basic properties – Shifting theorems -Transforms of derivatives and

integrals – Initial and final value theorems – Inverse transforms – Convolution theorem –

Transform of periodic functions – Application to solution of linear second order ordinary differential

equations with constant coefficients

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