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Note for Advanced Engineering Mathematics - AEM By SHIVA PRASAD DAS

  • Advanced Engineering Mathematics - AEM
  • Note
  • Silicon Institute of Technology - SIT
  • 9 Topics
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Shiva Prasad Das
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GATE 2017 ENGINEERING MATHEMATICS Handwritten Notes Common for AE/AG/BT/CE/CH/EC/EE/IN/ME/MN/MT/PE/PI Prepared by K. Manikantta ReddyM.Tech E-mail: reddy78922@gmail.com Mobile: +91 96666 78922

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This book is intended to provide basic knowledge on Engineering Mathematics to the GATE aspirants. Even though the syllabus is same, the questions appearing in different papers follows different patterns. So the GATE aspirants are advised to go through their respective paper syllabus (available in official GATE website) and previous question papers to understand the depth of the subject required to prepare for GATE exam. Most of the solved problems in this material are the questions appeared in EC/EE/ME/CE/IN/PI papers. So the remaining branches students are advised to solve the previous problems from their respective papers. EE branch students are advised to prepare transform theory, which is not available in this book. This book is still under preparation. If you find any mistakes in this, please inform me through an e-mail. Don’t share this book without proper citation and credits. Thank You All The Best

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SYLLABUS – GATE 2016 1. MECHANICAL ENGINEERING – ME 2. METALLURGICAL ENGINEERING - MT 3. PRODUCTION AND INDUSTRIAL ENGINEERING - PI Linear Algebra: Matrix algebra, systems of linear equations, eigenvalues and eigenvectors. Calculus: Functions of single variable, limit, continuity and differentiability, mean value theorems, indeterminate forms; evaluation of definite and improper integrals; double and triple integrals; partial derivatives, total derivative, Taylor series (in one and two variables), maxima and minima, Fourier series; gradient, divergence and curl, vector identities, directional derivatives, line, surface and volume integrals, applications of Gauss, Stokes and Green’s theorems. Differential equations: First order equations (linear and nonlinear); higher order linear differential equations with constant coefficients; Euler-Cauchy equation; initial and boundary value problems; Laplace transforms; solutions of heat, wave and Laplace's equations. Complex variables: Analytic functions; Cauchy-Riemann equations; Cauchy’s integral theorem and integral formula; Taylor and Laurent series. (Except for MT paper) Probability and Statistics: Definitions of probability, sampling theorems, conditional probability; mean, median, mode and standard deviation; random variables, binomial, Poisson and normal distributions. Numerical Methods: Numerical solutions of linear and non-linear algebraic equations; integration by trapezoidal and Simpson’s rules; single and multi-step methods for differential equations. CIVIL ENGINEERING - CE Linear Algebra: Matrix algebra; Systems of linear equations; Eigen values and Eigen vectors. Calculus: Functions of single variable; Limit, continuity and differentiability; Mean value theorems, local maxima and minima, Taylor and Maclaurin series; Evaluation of definite and indefinite integrals, application of definite integral to obtain area and volume; Partial derivatives; Total derivative; Gradient, Divergence and Curl, Vector identities, Directional derivatives, Line, Surface and Volume integrals, Stokes, Gauss and Green’s theorems. Ordinary Differential Equation (ODE): First order (linear and non-linear) equations; higher order linear equations with constant coefficients; Euler-Cauchy equations; Laplace transform and its application in solving linear ODEs; initial and boundary value problems. Partial Differential Equation (PDE): Fourier series; separation of variables; solutions of onedimensional diffusion equation; first and second order one-dimensional wave equation and twodimensional Laplace equation. Probability and Statistics: Definitions of probability and sampling theorems; Conditional probability; Discrete Random variables: Poisson and Binomial distributions; Continuous random variables: normal and exponential distributions; Descriptive statistics - Mean, median, mode and standard deviation; Hypothesis testing. Numerical Methods: Accuracy and precision; error analysis. Numerical solutions of linear and nonlinear algebraic equations; Least square approximation, Newton’s and Lagrange polynomials, numerical differentiation, Integration by trapezoidal and Simpson’s rule, single and multi-step methods for first order differential equations.

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ELECTRONICS AND COMMUNICATION ENGINEERING – EC Linear Algebra: Vector space, basis, linear dependence and independence, matrix algebra, eigen values and eigen vectors, rank, solution of linear equations – existence and uniqueness. Calculus: Mean value theorems, theorems of integral calculus, evaluation of definite and improper integrals, partial derivatives, maxima and minima, multiple integrals, line, surface and volume integrals, Taylor series. Differential equations: First order equations (linear and nonlinear), higher order linear differential equations, Cauchy's and Euler's equations, methods of solution using variation of parameters, complementary function and particular integral, partial differential equations, variable separable method, initial and boundary value problems. Vector Analysis: Vectors in plane and space, vector operations, gradient, divergence and curl, Gauss's, Green's and Stoke's theorems. Complex Analysis: Analytic functions, Cauchy's integral theorem, Cauchy's integral formula; Taylor's and Laurent's series, residue theorem. Probability and Statistics: Mean, median, mode and standard deviation; combinatorial probability, probability distribution functions - binomial, Poisson, exponential and normal; Joint and conditional probability; Correlation and regression analysis. Numerical Methods: Solutions of non-linear algebraic equations, single and multi-step methods for differential equations, convergence criteria. ELECTRICAL ENGINEERING - EE Linear Algebra: Matrix Algebra, Systems of linear equations, Eigenvalues, Eigenvectors. Calculus: Mean value theorems, Theorems of integral calculus, Evaluation of definite and improper integrals, Partial Derivatives, Maxima and minima, Multiple integrals, Fourier series, Vector identities, Directional derivatives, Line integral, Surface integral, Volume integral, Stokes’s theorem, Gauss’s theorem, Green’s theorem. Differential equations: First order equations (linear and nonlinear), Higher order linear differential equations with constant coefficients, Method of variation of parameters, Cauchy’s equation, Euler’s equation, Initial and boundary value problems, Partial Differential Equations, Method of separation of variables. Complex variables: Analytic functions, Cauchy’s integral theorem, Cauchy’s integral formula, Taylor series, Laurent series, Residue theorem, Solution integrals. Probability and Statistics: Sampling theorems, Conditional probability, Mean, Median, Mode, Standard Deviation, Random variables, Discrete and Continuous distributions, Poisson distribution, Normal distribution, Binomial distribution, Correlation analysis, Regression analysis. Numerical Methods: Solutions of nonlinear algebraic equations, Single and Multi‐step methods for differential equations. Transform Theory: Fourier Transform, Laplace Transform, z‐Transform.

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