# Advanced Engineering Mathematics

IIT Kharagpur Course , Prof. P. Panigrahi

### Lecture 1: Mod-01 Lec-01 Review Groups, Fields and Matrices

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##### Lecture Details :

Advanced Engineering Mathematics by Prof. P.D. Srivastava,Dr. P. Panigrahi,Prof. Somesh Kumar,Prof. J. Kumar, Department of Mathematics, IIT Kharagpur. For more details on NPTEL visit http://nptel.iitm.ac.in

##### Course Description :

Contents:
Linear Algebra : 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.

Theory of Complex Variables : Concept of Domain, Limit, Continuity & Differentiability - Analytic Functions, C-R Equations - Harmonic Functions - Line Integral in the complex - Cauchy Integral Theorem - Cauchy Integral Formula - Power & Taylors Series of Complex Numbers - Power & Taylors Series of Complex Numbers (Contd.) - Taylors Laurent Series of f(z) & Singularities - Classification of Singularities , Residue and Residue Theorem.

Transform Calculus : 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 Integral Representation of a Function - Introduction to Fourier Transform - Applications of Fourier Transform t PDEs.

Probability & Statistics : Laws of Probability - 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.

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