Advanced Engineering Mathematics

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.