Sunday 26 April 2020

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Multiple Integrals Online Tutor, Double Integration In Cartesian And Polar Coordinates – Change Of Order Of Integration – Area As A Double Integral – Triple Integration In Cartesian Coordinates – Conversion From Cartesian To Polar – Volume As A Triple Integral. Vector Calculus Online Tutor, Gradient, Divergence, Curl – Solenoidal And Irrotational Fields – Vector Identities (Without Proof) – Directional Derivatives – Line, Surface And Volume Integrals –Green's, Gauss Divergence And Stoke's Theorems (Without Proof) – Verification And Applications To Cubes And Parallelepipeds Only. Laplace Transforms Online Tutor, Transforms Of Simple Functions – Basic Operational Properties – Transforms Of Derivatives And Integrals – Initial And Final Value Theorems – Inverse Transforms – Convolution Theorem – Periodic Functions – Applications Of Laplace Transforms For Solving Linear Ordinary Differential Equations Up To Second Order With Constant Coefficients Only. Analytic Functions Online Tutor, Definition Of Analytic Function – Cauchy Riemann Equations – Properties Of Analytic Functions - Determination Of Harmonic Conjugate – Milne-Thomson's Method – Conformal Mappings: 1/Z, Az, Az+B And Bilinear Transformation. Complex Integration Online Tutor, Line Integral – Cauchy's Integral Theorem (Without Proof) – Cauchy's Integral Formulae And Its Applications – Taylor's And Laurent's Expansions (Statements Only) – Singularities – Poles And Residues – Cauchy's Residue Theorem – Contour Integration – Unit Circle And Semi Circular Contour.

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Characteristic equation – Eigen values and Eigen vectors of a real matrix – Properties of Eigen values – Cayley – Hamilton theorem orthogonal reduction of a symmetric matrix to diagonal form – Orthogonal matrices – Reduction of quadratic form to canonical form by orthogonal transformations.  Function of two variables – Partial derivatives – Total differential – Taylor's expansion – Maxima and Minima – Constrained Maxima and Minima by Lagrangian Multiplier method – Jacobians – Euler's theorem for homogeneous function.

 

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Periodic functions: Properties, Even & Odd functions: Properties, Special wave forms: Square wave, Half wave Rectifier, 
Full wave Rectifier, Saw-toothed wave, Triangular wave.
Euler's Formulae for Fourier Series, Fourier Series for functions of period 2π, Fourier Series for functions of period 2l, Dirichlet's conditions, Sum of  Fourier series. 
Theorem for the convergence of Fourier Series (statement only). Fourier Series of a function with its periodic  extension. Half Range Fourier Series: Construction of Half range Sine Series, Construction of Half range Cosine Series. Parseval's identity (statement only). Complex functions, Concept of Limit, Continuity and Differentiability. Analytic functions, Cauchy-Riemann Equations (statement only). Sufficient condition for a function to be analytic. Harmonic function and Conjugate Harmonic function, related problems. 
Construction of Analytic functions: Milne Thomson method, related problems.