  ### Course Description

This is a comprehensive mathematics course for engineering students, covering topics including linear algebra, complex variables, laplace and fourier transforms to solve ordinary and partial differential equations, and probability and statistics. It 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.

TOPICS COVERED:
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. The Fourier Transform is one of the many mathematical tools applied to engineering which will be studied in this course.
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### Video Lectures & Study Materials

Visit the official course website for more study materials: http://nptel.ac.in/courses/111105035/

# Lecture Play Lecture
I. Linear Algebra
1 Review of Groups, Fields and Matrices (58:15) Play Video
2 Vector Spaces, Subspaces, Linearly Dependent/Independent of Vectors (1:03:30) Play Video
3 Basis, Dimension, Rank and Matrix Inverse (1:02:09) Play Video
4 Linear Transformation, Isomorphism and Matrix Representation (51:33) Play Video
5 System of Linear Equations, Eigenvalues and Eigenvectors (58:34) Play Video
6 Method to Find Eigenvalues and Eigenvectors, Diagonalization of Matrices (56:16) Play Video
7 Jordan Canonical Form, Cayley Hamilton Theorem (1:00:01) Play Video
8 Inner Product Spaces, Cauchy-Schwarz Inequality (56:16) Play Video
9 Orthogonality, Gram-Schmidt Orthogonalization Process (59:22) Play Video
10 Spectrum of special matrices,positive/negative definite matrices (53:48) Play Video
II. Theory of Complex Variables
11 Concept of Domain, Limit, Continuity and Differentiability (53:10) Play Video
12 Analytic Functions, CR Equations (54:08) Play Video
13 Harmonic Functions (55:19) Play Video
14 Line Integral in the Complex Plane (54:24) Play Video
15 Cauchy Integral Theorem (52:54) Play Video
16 Cauchy Integral Theorem (Contd.) (52:48) Play Video
17 Cauchy Integral Formula (54:03) Play Video
18 Power and Taylor's Series of Complex Numbers (54:11) Play Video
19 Power and Taylor's Series of Complex Numbers (Contd.) (55:01) Play Video
20 Taylor's, Laurent Series of f(z) and Singularities (55:12) Play Video
21 Classification of Singularities, Residue and Residue Theorem (55:56) Play Video
III. Laplace and Fourier Transforms
22 Laplace Transform and its Existence (59:05) Play Video
23 Properties of Laplace Transform (57:40) Play Video
24 Evaluation of Laplace and Inverse Laplace Transform (58:02) Play Video
25 Applications of Laplace Transform to Integral Equations and ODEs (57:43) Play Video
26 Applications of Laplace Transform to PDEs (57:26) Play Video
27 Fourier Series (57:24) Play Video
28 Fourier Series (Contd.) (58:00) Play Video
29 Fourier Integral Representation of a Function (57:56) Play Video
30 Introduction to Fourier Transform (57:57) Play Video
31 Applications of Fourier Transform to PDEs (57:53) Play Video
IV. Probability and Statistics
32 Laws of Probability I (57:10) Play Video
33 Laws of Probability II (57:20) Play Video
34 Problems in Probability (59:25) Play Video
35 Random Variables (59:26) Play Video
36 Special Discrete Distributions (58:01) Play Video
37 Special Continuous Distributions (58:06) Play Video
38 Joint Distributions and Sampling Distributions (58:29) Play Video
39 Point Estimation (55:38) Play Video
40 Interval Estimation (57:22) Play Video
41 Basic Concepts of Testing of Hypothesis (54:48) Play Video
42 Tests for Normal Populations (59:50) Play Video