This course covers topics in advanced engineering mathematics such as analytic functions, complex numbers, Laplace transform, Fourier series, probability, vector spaces, matrices, eigenvalues, and linear transformations. The course aims to teach students how to analyze and solve complex mathematical problems commonly encountered in engineering. The teaching method includes theoretical explanations, mathematical derivations, and problem-solving exercises. This course is intended for engineering students or professionals looking to deepen their understanding of advanced mathematical concepts relevant to engineering applications.
Overview
Syllabus
Analytic Functions, C-R Equations.
Harmonic Functions.
Line Integral in the Complex.
Cauchy Integral Theorem.
Cauchy Integral Theorem (Contd.).
Cauchy Integral Formula.
Power and Taylor Series of Complex Numbers.
Power and Taylor Series of Complex Numbers (Contd.).
Taylor's , Laurent Series of f(z) and Singularities.
Classification of Singularities, Residue and Residue Theorem.
Laplace Transform and its Existence.
Properties of Laplace Transform.
Evaluation of Laplace and Inverse Laplace Transform.
S30 2072.
Applications of Laplace Transform to PDEs.
Fourier Series (Contd.).
Fourier Integral Representation of a Function.
Introduction to Fourier Transform.
Applications of Fourier Transform to PDEs.
Laws of probability I.
Laws of probability II.
Problems in probability.
Random variables.
Special Discrete Distributions.
Special Continuous distributions.
Vector Spaces, Subspaces, Linearly Dependent / Independent of Vectors.
Review Groups, Fields and Matrices.
Basis, Dimension, Rank and Matrix Inverse.
Jordan Canonical Form,Cayley Hamilton Theorem.
Concept of Domain, Limit, Continuity and Differentiability.
Method to Find Eigenvalues and Eigenvectors, Diagonalization of Matrices.
Spectrum of special matrices,positive/negative definite matrices.
System of Linear Equations, Eigen values and Eigen vectors.
Linear Transformation, Isomorphism and Matrix Representation.
Orthogonality , Gram-Schmidt Orthogonalization Process.
Inner Product Spaces, Cauchy - Schwarz Inequality.
Taught by
Ch 30 NIOS: Gyanamrit
