This is an introductory mechanics course designed to consolidate the understanding of fundamental concepts in mechanics such as force, energy, momentum etc. more rigorously as needed for further studies in physics, engineering and technology. Topcs covered include kinematics and dynamics of point masses, Newton’s laws, momentum, energy, angular momentum and torque, conservation laws, motion under gravity, central force problem, Virial theorem, Kepler’s laws, Rutherford problem, coupled oscillations, dynamics of rigid bodies, moment of inertia tensor, Euler’s equations, orthogonal transformation and Euler’s angle, Cayley Klein parameters, symmetric top, Lagrangian dynamics, generalized coordinates and forces, Lagrange’s equation, Hamilton’s principle, and variational methods.
This course serves as an introduction to the field of operations research. It will quip students with scientific approaches to decision-making and mathematical modelling techniques required to design, improve and operate complex systems in the best possible way. Topics covered include linear programming, the simplex method, duality and sensitivity analysis, integer programming , nonlinear programming, dynamic programming and network models.
This course is intended to introduce the student to the basic concepts and theorems of functional analysis and its applications. Topics covered include linear spaces, topological spaces, normed linear spaces & Banach Spaces, inner product spaces, Hilbert spaces, linear functional and the Hahn-Banach theorem.
This course is designed to equip students with the basic techniques for the efficient numerical solution of problems in science and engineering. Topics covered include round off errors and floating-point arithmetic, solution of non-linear equations, bracketing, fixed point methods, secant method, Newton's method, zeros of polynomials, Polynomial interpolation, orthogonal polynomial, least squares approximations, approximation by rational function, numerical differentiation, numerical integration, and adaptive quadrature.
This course introduces students to the theory of boundary value and initial value problems for partial differential equations with emphasis on linear equations. Topics covered include first and second order partial differential equations, classification of second order linear partial differential equations, derivation of standard equation, methods of solution of initial and boundary value problems, separation of variables, Fourier series and their applications to boundary value problems in partial differential equation of engineering and physics, internal transform methods; Fourier and Laplace transforms and their application to boundary value problems.
The construction of mathematical models to address real-world problems has been one of the most important aspects of each of the branches of science. It is often the case that these mathematical models are formulated in terms of equations involving functions as well as their derivatives. Such equations are called differential equations. If only one independent variable is involved, often time, the equations are called ordinary differential equations. The course will demonstrate the usefulness of ordinary differential equations for modelling physical and other phenomena. Complementary mathematical approaches for their solution will be presented. Topics covered include linear differential equation of order n with coefficients continuous on some interval J, existence-uniqueness theorem for linear equations of order n, determination of a particular solution of non-homogeneous equations by the method of variation of parameters, Wronskian matrix of n independent solutions of a homogeneous linear equation, ordinary and singular points for linear equations of the second order, solution near a singular point, method of Frobenius, singularities at infinity, simple examples of Boundary value problems for ordinary linear equation of the second order, Green’s functions, eigenvalues, eigenfunctions, Sturm-Liouville systems, properties of the gamma and beta functions, definition of the gamma function for negative values of the argument; Legendre, Bessel, Chebyshev, Hypergeometic functions and orthogonality properties.
This course provides an introduction to complex analysis which is the theory of complex functions of a complex variable. The topics to be covered in the course are: complex numbers, sequences and series of complex numbers, limits and continuity of functions of complex variables, elementary functions of a complex variable, Cauchy-Riemann criterion for differentiability, analytic functions, complex integrals, Taylor’s and Laurent’s series, calculus of residues, contour integration and conformal mapping.
This course is designed to offer a basic introduction to measure theory and Lebesgue’s integral. The topics to be covered are: countable and uncountable sets, countability of the rationals, uncountability of the reals, measurable sets and functions, the Lebesgue’s integral where E is a measurable subset of the real line and f is measurable on E, the spaces as metric spaces, Cauchy sequences in spaces, completeness of spaces, the Riesz-Fischer theorem and Mean convergence in the space .
This course is designed as a basic introductory course in the analysis of metric spaces. It is aimed at providing the abstract analysis components for the degree course of a student majoring in mathematics. This course affords students an opportunity to gain some familiarity with the axiomatic method in analysis. The topics to be covered are: metric spaces, open spheres, open sets, limit points, closed sets, interior, closure, boundary of a set, sequences in metric spaces, subsequences, upper and lower limits of real sequences, continuous functions on metric spaces, uniform continuity, isometry, homomorphism, complete metric spaces, compact sets in a metric space, Heine-Borel theorem, connected set, and the inter-mediate value theorem.
Laser sources, application formula, optical system design, He-Ne laser, spectroscopy, mode selections, stabilization methods, gas lasers, measuring techniques.