MA8251 – NOTES & QP
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MA8251 – SYLLABUS
UNIT I 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 quadratic forms.
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.