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Course Co-ordinated by IIT Kharagpur
 Coordinators IIT Kharagpur IIT Kharagpur IIT Kharagpur IIT Kharagpur

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This is a course suitable for B.Tech / M.Tech students of various discipline. It deals with some advanced topics in Engineering Mathematics usually covered in a degree course.

CONTENTS:- Linear Algebra:-  Review of Groups, Fields, and Matrices;  Vector Spaces, Subspaces,  Linearly dependent/independent, Basis, Dimensions;  Isomorphism, Linear transformations and their matrix representations; Rank, Inverse of Matrices, System of Equations; Inner-product spaces, Cauchy- Schwarz Inequality;
Orthogonality, Gram-Schmidt orthogonalisation process ; Eigenvalue, Eigenvectors, Eigenspace; Cayley-Hamilton Theorem;  Diagonalisation of matrices, Jordan canonical form; Spectral representation of real symmetric, hermitian and normal matrices, positive definite and negative definite matrices.

Theory of Complex variables:-A review of concept of limit, continuity, differentiability & analytic functions. Cauchy Riemann Equations, Line Integral in the complex plane, Cauchy Integral Theorem & Cauchy Integral Formula &  its consequences, Power series & Taylor Series(in brief ) ,Zeros & Singularity, Laurent’ Series, Residues,  Evaluation of Real Integrals

Transform Calculus:- Concept of Transforms, Laplace Transform(LT) and its existence, Properties of Laplace Transform, Evaluation of LT and inverse LT,
Evaluation of integral equations with kernels of convolution type and its
Properties, Complex form of Fourier Integral, Introduction to Fourier Transform,
Properties of general (complex) Fourier Transform, Concept and properties of Fourier
Sine Transform and Fourier Cosine Transform, Evaluation of Fourier Transform,
Solution of ordinary differential equation and one dim. Wave equation using
Transform techniques, Solution of heat conduction equation and Laplace equation in 2
dim. Using Transform techniques

Probability & Statistics :- A review of concepts of probability and random variables: Classical, relative frequency and axiomatic definitions of probability, addition rule, conditional probability, multiplication rule, Bayes’ Theorem.  Random Variables: Discrete and continuous random variables, probability mass, probability density and cumulative distribution functions, mathematical expectation, moments, moment generating function. Standard Distributions:  Uniform, Binomial, Geometric, Negative Binomial, Poisson, Exponential, Gamma, Normal.  Sampling Distributions: Chi-Square, t and F distributions. Estimation: The method of moments and the method of maximum likelihood estimation, confidence intervals for the mean(s) and variance(s) of normal populations. Testing of Hypotheses: Null and alternative hypotheses, the critical and acceptance regions, two types of error, power of the test, the most powerful test, tests of hypotheses on a single sample, two  samples.
 Module Lectures/Topics Module I Linear Algebra Lecture Review: Groups, Fields, & Matrices Vector Spaces, Subspaces, Linearly dependent / independent of vectors Basis, Dimension, Rank and Matrix Inverse Linear Transformation, Isomorphism & Matrix Representation System of Linear Equations, Eigenvalues and Eigenvectors Method to find Eigenvalues and Eigenvectors, Diagonalization of Matrices Jordan Canonical Form, Cayley Hamilton Theorem Innerproduct Spaces, Cauchy-Schwarz Inequality Orthogonality, Gram-Schmidt Orthogonalization Process Spectrum of Special Matrices, Positive / Negative Definite Matrices Module II Theory of Complex Variables Lecture  Concept of Domain, Limit, Continuity & Differentiability Analytic Functions, C-R Equations Harmonic Functions Line Integral in the complex Cauchy Integral Theorem Cauchy Integral Theorem (contd.) Cauchy Integral Formula Power & Taylor’s Series of Complex Numbers Power & Taylor’s Series of Complex Numbers (Contd.) Taylor’s Laurent Series of f(z) & Singularities Classification of Singularities , Residue and Residue Theorem Module III Transform Calculus Lecture  Laplace Transform and its Existence Properties of Laplace Transform Evaluation of Laplace and Inverse Laplace Transform Applications of Laplace Transform to Integral Equations and ODEs Applications of Laplace Transform to PDEs Fourier Series Fourier Series (Contd.) Fourier Integral Representation of a Function Introduction to Fourier Transform Applications of Fourier Transform t PDEs Module IV Probability & Statistics Lecture  Laws of Probability – 1 Laws of Probability – 2 Problems in Probability Random Variables Special Discrete Distributions Special Continuous Distributions Joint Distributions and Sampling Distributions Point Estimation Interval Estimation Basic Concepts of Testing of Hypothesis Tests for Normal Populations

Calculus , Real Analysis

Kreyszig, : Engineering Mathematics

R.S.L.Srivastava :- Engineering Mathematics , Tata Mc-Graw Hill

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