MA8251 Engineering Mathematics 2
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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