# Courses Offered by the MNS Department for Different Disciplines of BRACU

a. Theoretical Courses:** APE 101 Mechanics, Properties of Matter, Waves & Oscillations 3 credits **

Vectors & scalars, unit vectors, scalar and vector products, scalar and vector fields, gradient, divergence and curl, curvilinear co-ordinates, 1 D & 2 D motions, work and energy, conservation laws, conservative force, projectile motion, uniform circular motion, rotation of rigid bodies, angular momentum, gravitation, gravitational field, potential.

Elasticity, Hooke's Law, adhesive and cohesive forces, molecular theory of surface tension, capillarity. Streamline & turbulent flow, Poiseulle's formula, streamline flow and turbulent flow, Reynold's Number, Equation of Continuity, Bernoulli's Theorem, Stokes' Law.

Principle of superposition, interference of waves, phase velocity and group velocity, simple harmonic motion, combination of SHM, Lissajous figures, damped SHM, forced oscillations, resonance, power and intensity of wave motion, waves in elastic media, vibration of strings, beats, Doppler Effect, velocity of sound, ultrasonics, and their applications.** APE 102 Thermal Physics, Radiation & Statistical Mechanics 3 credits **

Heat and temperature, thermal equilibrium, specific heat & calorimetry, Newton's Law of Cooling, Kinetic Theory of Gases, Boltzmann Distribution Law, Brownian motion, Law of equipartition of energy; Vander Waals' equation of state, heat transfer, conduction, convection and radiation, co-efficient of thermal conductivity and its measurement, First Law of thermodynamics, isothermal & adiabatic changes, reversible and irreversible processes, Carnot's cycle, Second Law of thermodynamics, entropy and disorder, absolute scale of temperature, Maxwell's relations, Clausius-Clapeyron Equation, Gibb's phase rule, Third Law of thermodynamics, Nernst heat theorem, radiation theory, black body radiation, Wien's Law, Stefan-Boltzman Law, Rayleigh Jeans Law, Planck's Law, variation of specific heat with temperature, Einstein's theory, Debye's theory, Joule-Thomson expansion, cryogenics, measurement of high temperature.

Statistical Mechanics: Phase space, concept of state and ensemble, microcanonical, canonical and grand canonical ensembles, Boltzmann probability distribution, Maxwell velocity distribution, derivation of Bose-Einstein and Fermi-Dirac statistics, ideal Fermi gas, degenerate Fermi system, equation of state of ideal gases, ideal Bose gas.** APE 103 Electrical Circuits I 3 credits**

Circuit variables and elements: Voltage, current, power, energy, independent and dependent sources, resistance. Basic laws: Ohm's law, Kirchhoff's current and voltage laws. Simple resistive circuits: Series and parallel circuits, voltage and current division, Wye-Delta transformation. Techniques of circuit analysis: Nodal and mesh analysis including supernode and super mesh. Network theorems: Source transformation, Thevenin's, Norton's and Superposition theorems with applications in circuits having independent and dependent sources, maximum power transfer condition and reciprocity theorem. Energy storage elements: Inductors and capacitors, series parallel combination of inductors and capacitors. Responses of RL and RC circuits: Natural and step responses. Magnetic quantities and variables: Flux, permeability and reluctance, magnetic field strength, magnetic potential, flux density, magnetization curve. Laws in magnetic circuits: Ohm's law and Ampere's circuital law. Magnetic circuits: series, parallel and series-parallel circuits.

Prerequisite PHY 115** APE 201 Solid State Physics and Materials Science 3 Credits**

Crystalline state, Bravais lattices, crystal symmetry, point group & space group, unit cells, Miller indices, x-ray diffraction, Bragg's Law, reciprocal lattice, structure factor, interatomic force and classification of solids, ionic, covalent, molecular, hydrogen bonded crystals, lattice energy of ionic crystals, Madelung constant, lattice vibration, phonons, normal modes in monatomic and diatomic linear chains, theory of specific heat, Einstein and Debye models, thermal expansion, defects in crystals, dislocations, consequences of defects on mechanical properties, elastic properties. Amorphous, composite, fibrous materials, polymers, plastics, binding forces, thermal and electrical conductivity of metals, dielectric properties of solids , modes of dielectric polarisation, ferro electricity, piezo electricity, optical properties of solids ,classical and semi classical theory, free carrier effects, lattice absorption, electronic absorption, magnetic properties of solid, dia and paramagnatism, ferro & ferrimagnetism, antiferromagnetism, ferrites, magnetic resonance, superconductivity, liquid crystals.** APE 202 Electrodynamics & Electromagnetic Waves & Fields 3 Credits **

Solution of Laplace's equation and Poisson's equation and applications to electrostatic problems, dielectrics, electrostatic energy, Maxwell's equations, electromagnetic waves, propagation of electromagnetic waves in conducting and non-conducting media, reflection and refraction, polarization, dispersion, scattering, waves in the presence of metallic boundaries, Waves between parallel planes, attenuation, wave impedance, waves in coaxial lines & modes, waves in strip and micro-strip lines, waveguides and resonators, solution of the inhomogeneous wave equations, simple radiating system, antennas, accelerated charge, Cerenkov radiation, elements of plasma physics. Prerequisite PHY 115.

APE 203 Electrical Circuits II 3 credits

Sinusoidal functions: Instantaneous current, voltage, power, effective current and voltage, average power, phasors and complex quantities, impedance, real and reactive power, power factor. Analysis of single phase ac circuits: Series and parallel RL, RC and RLC circuits, nodal and mesh analysis, application of network theorems in ac circuits, circuits simultaneously excited by sinusoidal sources of several frequencies, transient response of RL and RC circuits with sinusoidal excitation. Resonance in ac circuits: Series and parallel resonance. Magnetically coupled circuits. Analysis of three phase circuits: Three phase supply, balanced and unbalanced circuits, power calculation.

Prerequisite APE 103** APE 204 Digital Logic Design 3 credits**

An introduction to digital systems such as computer, communication and information systems. Topics covered include Boolean algebra, digital logic gates, combinational logic circuits, decoders, encoders, multiplexers. Asynchronous and synchronous counters. Registers, flip-flops, adders, Sequential circuit analysis and design. Simple computer architecture. ** APE 205 Electronic Devices and Circuits I 3 credits**

P-N junction as a circuit element: Intrinsic and extrinsic semiconductors, operational principle of p-n junction diode, contact potential, current-voltage characteristics of a diode, simplified dc and ac diode models, dynamic resistance and capacitance. Diode circuits: Half wave and full wave rectifiers, rectifiers with filter capacitor, characteristics of a zener diode, zener shunt regulator, clamping and clipping circuits. Bipolar junction transistor (BJT) as a circuit element: Basic structure. BJT characteristics and regions of operation, BJT as an amplifier, biasing the BJT for discrete circuits, small signal equivalent circuit models, BJT as a switch. Single stage BJT amplifier circuits and their configuarations: Voltage and current gain, input and output impedances. Metal-Oxide-Semiconductor Field-Effect-Transistor (MOSFET) as circuit element: structure and physical operation of MOSFETs, body effect, current- voltage characteristics of MOSFETs, biasing discrete and integrated MOS amplifier.

Prerequisite APE 103** APE 302 Electronic Devices and Circuits II 3 credits**

Frequency response of amplifiers: Poles, zeros and Bode plots, amplifier transfer function, techniques of determining 3 dB frequencies of amplifier circuits, frequency response of single-stage and cascade amplifiers, frequency response of differential amplifiers. Operational amplifiers (Op-Amp): Properties of ideal Op-Amps, non-inverting and inverting amplifiers, inverting integrators, differentiator, weighted summer and other applications of Op-Amp circuits, effects of finite open loop gain and bandwidth on circuit performance, logic signal operation of Op-Amp, dc imperfections. General purpose Op-Amp: DC analysis, small-signal analysis of different stages, gain and frequency response of 741 Op-Amp. Negative feedback: properties, basic topologies, feedback amplifiers with different topologies, stability, frequency compensation. Active filters: Different types of filters and specifications, transfer functions, realization of first and second order low, high and bandpass filters using Op-Amps. Signal generators: Basic principle of sinusoidal oscillation, Op-Amp RC oscillators, LC and crystal oscillators. Power Amplifiers: Classification of output stages, class A, B and AB output stages. Prerequisite APE 205** APE 400 Thesis / Project 3 credits**

A student is required to carry out thesis/project work in the 7th and 8th semester in a chosen field. There will be a supervisor who will either be a BRAC University faculty or any other suitable expert from universities and R/D organizations of the country to guide the thesis/ project work .On completion of study and research s/he will have to submit the dissertation/report and face a viva board for the defence.** APE 401 Measurements & Measuring Instruments 3 Credits**

Significance and methods of measurements, direct and indirect methods. Mechanical, electrical and electronic types of instruments, absolute and secondary instruments, analog and digital instruments, analog voltmeters and ammeters, AC transformer types, Flux gate magnetometer type. Accuracy and error of analog voltmeters and ammeters. Different types of Digital voltmeters, digital multimeters, Automation in multimeters.

Oscilloscopes, signal generators. Transducers. Absorption and detection of radiation, Nucleonic instruments. Analytical & medical instruments, Industrial instruments, Instrument systems.

Prerequisite APE 302** APE 402 Plasma Physics with Industrial Applications 3 credits**

General introduction to plasma physics, plasma as a fourth state of matter, definition, screening and Debye shielding, plasma frequency, ideal plasma, temperature and pressure of plasma, magnetic pressure and plasma drifts, plasma waves, Landau damping, collisions in plasmas, hydrodynamic description of plasma, one fluid model, two fluid model, Chew-Goldberg theory, low waves in maneto-hydrodynamics, description of plasma, dielectric tensor, longitudinal and transverse waves, plasma instabilities, transport in plasmas, plasma kinetic theory, Vlasov equation, linear waves, waves in magnetized plasma, electromagnetic waves, waves in hot plasmas, nonlinear waves, Landau damping, quasi linear theory, plasmas in fusion research, plasmas in industrial applications.** APE 403 Control Engineering 3 credits**

Introduction to control systems, electric circuits and components, transfer function and block diagram, mechanical translation systems, analogous circuits, mechanical rotational systems, rotating power amplifiers, DC & AC servomotors.

Inputs & responses, modeling of continuous systems; computer-aided solutions to systems problems; feedback control systems; stability, frequency response and transient response using root locus, frequency domain and state variable methods. Position control system, simulation diagrams, signal flow graphs, parallel state diagrams from transfer function. General frequency transfer function relationships, drawing the Bode plot, system type and gain as related to log magnitude curve, Nyquist's criterion and applications.

Prerequisites ECE 220 and MAT 203** APE 404 Microprocessors and Assembly Language Programming 3 credits**

Introduction to different types of microprocessors. Microprocessor architecture, instruction set, interfacing/O operation, interrupt structure, DMA. Microprocessor interface ICs. RAM, ROM, PROM, EPROM and EPROM's. Advanced microprocessor concept of microprocessor based system design. Microcomputer systems, representation of numbers and characters, introduction to IBMPC assembly language. Logic, shift, multiplication & division instructions, arrays and addressing modes, string instructions, text display and keyboard programming, memory management.

Prerequisite APE 204** APE 405 Computer Organization and Architecture 3 credits**

A systematic study of the various elements in computer design, including circuit design, storage mechanisms, addressing schemes, and various approaches to parallelism and distributed logic. Information representation and transfer; instruction and data access methods; CPU structure and functions processor and register organization, instruction cycles and pipe linings, the control unit; memory organisation. RISC and CISC machines. The course includes a compulsory 3 hour laboratory work each week.

Prerequisite APE 204. ** APE 406 Radar Engineering 3 credits**The course is oriented towards the understanding and design of radar systems. The contents will be radar principles & techniques, nature of radars, radar frequencies, radar target, radar equation, continuous & frequency modulated radars, detection & processing of radar signals, MTI radar, Pulse Doppler radar, tracking radar, radar indicators and displays, noise, ground & sea echoes & clutter, weather effect of radar, radar applications. Prerequisites: ECE 220 and MAT 203** APE 407 Renewable Energy Technology 3 credits**

Energy & development, energy consumption, world energy demand & future projection, energy units, earth's energy resource base, renewable and non-rewable sources of energy.

Non-renewable energy sources-fossil fuels, coal, natural gas, petroleum, etc. and non-fossil fuels like uranium (fission energy).

Renewable energy sources-solar energy, wind energy, tidal & wave & ocean energy, geothermal energy, biomass & hydropower, hydrogen energy. Advantages of renewable over conventional technologies, solar thermal conversion, radiation characteristics of materials, solar collectors, solar photovoltaic energy conversion, photovoltaic cells, design of PV systems, wind turbines, biopower, biofuels, integrated bioenergy systems, geothermal heat pumps, hydroelectricity & micro hydroelectric power, ocean thermal energy conversion. Energy, sustainability and environment, EIA.** BI0 101 Introduction to Biology 3 credits**

An introduction to the cellular aspects of modern biology including the chemical basis of life, cell theory, energetics, genetics, development, physiology, behaviour, homeostasis and diversity, and evolution and ecology. This course will explain the development of cell structure and function as a consequence of evolutionary process, and stress the dynamic property of living systems. ** CHE 101 Introduction to Chemistry 3 credits**

The course is designed to give an understanding of basics in chemistry. Topics include nature of atoms and molecules; valence and periodic tables, chemical bonds, aliphatic and aromatic hydrocarbons, optical isomerism, chemical reactions. ** ENV 101 Introduction to Environmental Science 3 credits**

Fundamental concepts and scope of environmental science, Earth's atmosphere, hydrosphere, lithosphere and biosphere, men and nature, technology and population, ecological concepts and ecosystems, environmental quality and management, agriculture, water resources, fisheries, forestry and wildlife, energy and mineral energy sources; renewable and non renewable resources, environmental degradation; pollution and waste management, environmental impact analysis, remote sensing & environmental monitoring.**ENV 103 Elements of Environmental Sciences 3 credits**

Fundamental concepts and scope of environmental science, Earth's atmosphere, hydrosphere, lithosphere and biosphere, men and nature, technology and population, ecological concepts and ecosystems, environmental quality and management, agriculture, water resources, fisheries, forestry and wildlife, energy and mineral energy sources; renewable and non renewable resources, environmental degradation; pollution and waste management, environmental impact analysis, remote sensing & environmental monitoring.

** GEO 101 Introduction to Economic Geography 3 credits**

Introduction: The field and environment of economic Geography; Bases of Economic Geography: Relief, Climate, Vegetation, Soils and Population; Extractive resources and human-environment relations; Primary Activities: types and brief descriptions; Secondary Activities: types and factors of localization, Stages in growth; Tertiary Activities: Trade, Transportation, Utilities, Technical and Professional services; Regional Economy: classification, Growth and Development; Economic Geography of Bangladesh: A brief account.

**MAT091: Basic Course in Mathematics (Non-Credit)**

The real number system, exponents, polynomial, factoring, rational expression, radicals, complex numbers, linear equation, quadratic equation, variation, inequalities, equation of a line, equation of a circle,coordinate system, functions, polynomial(linear, quadratic and cubic) functions,rational functions, exponential and logarithmic functions, system of equations, system of inequalities, properties of matrix, matrix solution of linear system, determinant, Cramer’s rule, Basic concepts of calculus, limit,rate of change, derivative.Mean (arithmetic, geometric, harmonic), mode, median, sample, variable, data, frequency distribution, graphical presentation, measures of central tendency and measures of dispersion, introduction to probability.

Reference Books:

1. Algebra and Trigonometry (6th Edition) - By Lial & Miller (LM).

2. College Algebra: Michael Sullivan

**MAT092: Remedial Course in Mathematics (Non-credit)**

The idea and methods of describing a set, finite and infinite sets, empty set, subset, universal set, union and intersection of sets, disjoint set, difference of sets, complement of set, power set, cartesian product of sets, real and complex numbers system, number line, interval notation, absolute value, polynomials, exponents and radicals, algebraic expressions; variable, constant, power, coefficient, factorization, highest common factor, least common multiple, quadratic and cubic equations, variation, relation, functions and their graphs, rational functions and their graphs, simple functions and their graphs, systems of linear equations, matrices and determinants with applications; binomial theorem, sequences, summation of series (arithmetic and geometric), permutations and combinations, elementary trigonometry; trigonometric, exponential and logarithmic functions; calculus (rate of change, derivative) Cartesian coordinate system; various form of linear equations, distance between two points, formulas of area and volume (rectangles, parallelogram, triangles, circles, cylinders, spheres), composition and resolution of forces, equilibrium of concurrent forces; dynamics, statics, speed and velocity, acceleration, equations of motion, Mean (arithmetic, geometry, harmonic), mode, median, sample, variable, data, frequency distribution, graphical presentation, measures of central tendency and measures of dispersion, introduction to probability.

Reference Books:

1. Sullivan, M. College Algebra, 9th ed. Prentice Hall, 2012.

2. Lial, M.L., Greenwell, R.N., Ritchey, N.P. Finite Mathematics and Calculus with Applications, 9th ed. Pearson, 2012.

**MAT 101 Fundamentals of Mathematics 3 credits** The real number system, exponents, polynomial, factoring, rational expression, radicals, complex number, linear equation, quadratic equation, variation, inequalities, coordinate system, functions, equations of line, equation of circle, exponential and logarithmic function, system of equations, system of inequalities properties of matrix, matrix solution of linear system, determinant, Cramer's rule, limit, rate of change, derivative.

MAT 102 Introduction to Mathematics 3 credits

MAT 102 Introduction to Mathematics 3 credits

Factorisation, Synthetic Division, Zeros (Roots) of Polynomials, Relation between Roots and Coefficients, Nature of Roots (Descarte's Rule of signs); Complex Number System, Graphical representation of Complex Numbers (Argand Diagram), Polar form of Complex Numbers; Conic Sections, Parabola, Circle, Ellipse, Hyperbola, Transformation of Coordinates and Applications; Exponential Growth & Decay. Applications; Mathematical Induction; Determinants, Fundamental Properties of Determinants, Minors and Cofactors, Application of Determinants to solve System of Linear Equations (Cramers, Rule); Introduction to Matrix Algebra, Matrix Multiplication, Augmented Matrix, Adjoint Matrix, Inverse Matrix, Application of Matrices-solution of System of Linear Equations (homogeneous & non-homogeneous), Consistency of System of Equations.

MAT 103 Basic Concepts in Mathematics 3 credits

The real numbers, Absolute value of real numbers, Exponents, Polynomials, Basic operation and Factoring of polynomials, Rational expressions, Radicals. Linear Equations, Solution, graphs and applications. Variation, Linear inequalities. Exponential and Logarithmic Functions, Exponential growth and decay, Ratios, proportions, percent, application of simple and compound interest. Trigonometric Functions, The Sine and Cosine Functions, Cartesian coordinate systems, Graphing, Relations. Equations of a straight line its slope, Equation of a circle, Systems of Linear Equations, Matrix. Population, Sample, Variable, Raw data, Frequency distribution table, Graphical presentation, Measures of central tendency and measures of dispersion.

**MAT 104 Mathematics 2 credits**

Calculus, definition of limit, continuity and differentiability, successive and partial differentiation, maxima and minima. Integration by parts, standard integrals, definite integrals. Solid geometry, system of coordinates. Distance between two points. Coordinate Transformation, Straight lines sphere and ellipsoid.

MAT 105 Calculus 3 credits

MAT 105 Calculus 3 credits

Differential Calculus: Limits, continuity and differentiability, differentiation, Taylor's, Maclaurine's & Euler's theorems, indeterminate forms, tangent and normal, sub tangent and subnormal, maxima and minima, radius of curvature & their applications, introduction to calculus of function of several variables, Taylor's theorem, maxima and minima for function of several variables. Transformation of coordinates & rotation of axes, conic sections.

Integral Calculus: Definition of integration, techniques of integration for definite & indefinite integrals, improper integrals, area, volume and surface integration, arc length and their applications, multiple integrals, Jacobian, line integrals, divergence theorem and Stokes' theorem, beta function and gamma function.

**MAT 110 MATH I: Differential Calculus and Co-ordinate Geometry 3 credits**

Differential Calculus: Limits, Continuity and differentiability. Differentiation. Taylor's Maclaurine's & Euler's theorem. Indeterminate forms. Partial differentiation. Tangent and normal. Subtangent and subnormal. Maximum and minimum, radius of curvature & their applications. Co-ordinate Geometry: Transformation of coordinates & rotation of axis. Pair of straight lines. General equation of second degree. System of circles. Conics section. Tangent and normal, asymptotes & their applications.

**MAT 111 Principles of Mathematics 3 credits**

Sets: Elementary idea of Set, Set notations, Set operations: union, intersection, complement, difference; Set operations and Venn diagrams. Set of Natural numbers, Integers, Rational numbers, Irrational numbers and Real numbers alongwith their geometrical representation, Idea of Open and Closed interval,. Idea of absolute value of real number, Variables and Constants, Product of two sets: Idea of product of sets, Product set of real numbers and their geometric representation, Axioms of real number system and their application in solving algebraic equations. Equation and Inequality, Laws of inequality, Solution of equations and inequalities. Variable and Functions: Variable of a set, Functions of single variable, Polynomial, Graph of Polynomial functions in single variable. Exponential, Logarithmic, Trigonometric functions and their graphs, Permutation and Combination. Binomial theorem.

**MAT 120 MATH II Integral Calculus and Differential Equations 3 credits**

Integral Calculus: Definitions of integration. Integration by the method of substitution. Integration by parts. Standard integrals. Integration by method of successive reduction. Definite integrals, its properties and use in summing series. Walli's formula. Improper integrals. Beta function and Gamma function. Area under a plane curve in Cartesian and polar coordinates. Area of the region enclosed by two curves in Cartesian and polar coordinates. Trapezoidal rule. Simpson's rule. Arc lengths of curves in Cartesian and polar coordinates, parametric and pedal equations. Intrinsic equations. Volumes of solids of revolution. Volume of hollow solids of revolutions by shell method. Area of surface of revolution. Ordinary Differential Equations: Degree of order of ordinary differential equations. Formation of differential equations. Solution of first order differential equations by various methods. Solutions of general linear equations of second and higher order with constant coefficients. Solution of homogeneous linear equations. Applications. Solution of differential equations of the higher order when the dependent and independent variables are absent. Solution of differential equations by the method based on the factorisation of the operators. [Students will be expected to attend a 3 hour tutorial class, once each week and submit tutorial worksheets.]

Prerequisite: MAT 110

MAT 121 Basic Algebra 3 credits

MAT 121 Basic Algebra 3 credits

Elements of logic: Mathematical statements, Logical connectives, Conditional and biconditional statements, Truth tables and tautologies, Quantifications, Logical implication and equivalence, Deductive reasoning. The concept of sets: Sets and subsets, Set operations, Family of Sets. Relations and functions: Cartesian product of two sets, Relations, Order relation, Equivalence relations, Functions, Images and inverse images of sets, Injective, surjective, and bijective functions, Inverse functions. Real number system: Field and order properties, Natural numbers, integers and rational numbers. Absolute value, Basic inequalities. (Including inequalities of means, powers, Weierstrass, Cauchy) Complex number system: Field of complex numbers, De Moivre's theorem and its applications. Elementary number theory: Divisibility, Fundamental theorem of arithmetic, Congruence. Summation of finite series: Arithmetic-geometric series, Method of difference, Successive differences, Use of mathematical induction. Theory of equations: Synthetic division, Number of roots of polynomial equations, Relations between roots and coefficients, Multiplicity of roots, Symmetric functions of roots, Transformation of equations.

**MAT 122 Analytic and Vector Geometry 3 credits**

Two dimensional geometry

Coordinates: Cartesian and polar Coordinates, Transformations of coordinates and its applications. Reduction of second degree equations to standard forms, Pairs of straight lines, Circles, Identification of conics, Equations of conics in polar Coordinates.

Three dimensional geometry

Coordinates in three dimensions, Direction cosines, and Direction ratios. Planes, straight lines, shortest distance, sphere, orthogonal projection and conicoids.

Vector geometry

Vectors in plane and space, Algebra of vectors, scalar and vector products, Triple scalar products, its applications to Geometry.

**MAT 123 Calculus I 3 credits**

Differential Calculus: Real number system and its geometrical representation, real variable, function of single (real) variable, parametric equations, limit, continuity and differentiability, derivatives of different types of functions, geometrical significance of derivative, Rolle's Theorem, Mean Value Theorem, Taylor's Theorem; maxima, minima, point of inflexion, concavity and convexity, sketching of curves using concepts of calculus; Indeterminate Form, L'Hospital Rule, Successive Differentiation, Leibnitz's Theorem, tangent, normal and related formulas, curvature.

Integral Calculus: Indefinite integrals of different types of functions, various methods of integrations, definite integrals, Fundamental Theorems of Definite Integrals, properties of definite integrals, Reduction formulas, Arc Length, Area under the curves, Surface area and Volume of a 3-D objects. Improper Integrals and applications.

Prerequisite: MAT122

**MAT 124 FORTRAN Programming 3 credits**

Problem-solving techniques using computers: Flowcharts, Algorithms, Pseudocode. Programming in FORTRAN: Syntax and semantics, data types and structures, input/output, loops, decision statements, arrays, user-defined functions, subroutines and recursion. Computing using FORTRAN: Construction and implementation of FORTRAN programs for solving problems in mathematics and sciences.

MAT 203 Matrices, Linear Algebra & Differential Equations 3 credits

Matrices: Types of matrices, algebraic operation on matrices, determinants, adjoint & inverse matrix, orthogonality & diagonalization of matrix.

Linear Algebra: System of linear equations, vector space; 2D- space, 3D- space, Euclidean nD- space, sub space, linear dependence, basis and dimension, row space, column space, rank and nullity, linear transformation, eigen value and eigen vector, matrix diagonalization and similarity, application of linear algebra.

Ordinary Differential Equations: Introduction to differential equations, first-order differential equations and applications, higher order differential equations and applications, series solutions of linear equations, systems of linear first-order differential equations.

Prerequisite MAT 105

**MAT 204 Complex Variables & Fourier Analysis 3 credits**

Complex Variables: Complex number systems, general functions of a complex variable, limits and continuity of a function of complex variables and related theorems, complex differentiation and Cauchy-Riemann equations, mapping by elementary functions, line integral of a complex function. Cauchy's integral theorem, Cauchy's integral formula, Liouville's theorem, Taylor's and Laurent's theorem, singular points, residue, Cauchy's residue theorem, evaluation of residues, contour integration and conformal mapping. Fourier analysis: Real and complex form, finite Fourier transform, Fourier integrals, Fourier transforms and their use in solving boundary value problems.

Prerequisite MAT 105

**MAT 205 Introduction to Numerical Methods 3 credits**

Computer arithmetic: floating point representation of numbers, arithmetic operations with normalized floating point numbers; iterative methods, different iterative methods for finding the roots of an equation f (x) = 0 and their computer implementation; solution of simultaneous algebraic equations by various methods, solution of tri-diagonal system of equations, interpolation for equispaced and non-equispaced nodes, least square approximation of functions, curve fitting, Taylor series representation, Chebyshev series, numerical differentiation and integration and numerical solution of ordinary differential equations & partial differential equations.

Prerequisite MAT 203

**MAT 211 Calculus II 3 credits**

Functions of several variables, concept of surface, sketching of , contour sketch for surface, limit and continuity, partial derivative and its geometrical significance, chain rule of partial differentiation, concept of gradient, divergence and curl, directional derivative and tangent plane, concept of differential and perfect differential, linear approximation and increment estimation, maxima, minima and saddle point, Lagrange multiplier, higher order derivatives, Taylor's theorem of function of several variables. Multiple integrals: Double integrals, Double integrals in Polar coordinates, Triple integrals, Triple integrals in Cylindrical and Spherical coordinates, Change of variables in Multiple integrals, Jacobian, Line integrals, Green's theorem, Surface integrals, Applications of Surface integrals, Divergence theorem, Stoke's theorem.

Prerequisite: MAT123.

**MAT 212 Linear Algebra 3 credits**

Introduction to matrix, different types of matrices, equivalent matrices, determinants, properties of determinants, minors, cofactors, evaluation of determinants, adjoint matrix, inverse matrix, method for finding inverse matrix, elementary row operations and echelon form of matrix, system of linear equations (homogeneous and non-homogeneous equations) and their solutions; Vector, vector spaces and subspaces, linear independence and dependence, basis and dimension, change of bases, rank and nullity, linear transformation, kernel and images of a linear transformation and their properties, eigenvalues and eigenvectors, diagonalization, Cayley Hamilton theorem,

**MAT 215 MATH III Complex Variables and Laplace Transformations 3 credits**

Complex Variables: Complex number systems. General functions of a complex variable. Limits and continuity of a function of complex variables and related theorems. Complex differentiation and Couchy-Riemann equations. Mapping by elementary functions. Line integral of a complex function. Cauchy's integral theorem. Cauchy's integral formula. Liouville's theorem. Taylor's and Laurent's theorem. Singular points. Residue. Cauchy's residue theorem. Evaluation of residues. Contour integration. And conformal mapping Laplace Transforms: Definition. Laplace transforms of some elementary functions. Sufficient conditions for existence of Laplace transforms. Inverse Laplace transforms. Laplace transforms of derivatives. The unit step function. Periodic function. Some special theorems on Laplace transforms. Solutions of differential equations by Laplace transformations. Evaluation of improper integrals.

Prerequisite: MAT120

MAT 216 MATH IV Linear Algebra and Fourier Analysis 3 Credits

MAT 216 MATH IV Linear Algebra and Fourier Analysis 3 Credits

Linear Algebra: Basic subject on matrix theory and linear algebra, emphasizing topics useful in other discipline, including systems of equations, vector spaces, determinants, Eigenvalues, similarity, and positive definite matrices, Applications to Gauss elimination with pivoting. Fourier Analysis: Real and complex form. Finite transform. Fourier integral. Fourier transforms and their uses in solving boundary value problems. Multiple integrals; surface and volume integrals, divergence and Stoke's theorem.

Prerequisite: MAT 120

**MAT 221 Real Analysis I 3 credits**

Real number system: Completeness of real numbers, supremum and infimum principles and their consequences, Dedekind's theorems, Bolzano-Weierstrass theorem. Sequences of Real Numbers: Infinite sequence, Convergent sequences, Monotone sequences, subsequences, Cauchy sequence, Cauchy criteria for convergence of sequences. Infinite Series: Concept of sum and convergence, series of positive terms, alternating series, absolute and conditional convergence, test for convergence, Convergence of sequences and series of functions. Continuity and Limits: Properties of continuous functions, Extreme Value Theorem and Intermediate Value Theorem, Uniform continuity concepts, Limits, Heine-Borel theorem. Integration: Necessary and sufficient conditions for integrability, Darboux Sums and Riemann Sums, Improper integral and their tests for convergence.

**MAT 222 Differential Equations I 3 credits**

Ordinary differential equations and their solutions: Classification of differential equations, Solutions, Implicit solutions, Singular solutions, Initial Value Problems, Boundary Value Problems, Basic existence and uniqueness theorems (statement and illustration only), Solution of first order equations: Separable equations, Linear equations, Exact equations, Special integrating factors, Substitutions and transformations, Modeling with first order differential equations, Model solutions and interpretation of results, Orthogonal and oblique trajectories, Solution of higher order linear equations: Linear differential operators, Basic theory of linear differential equations, Solution space of homogeneous linear equations, Fundamental solutions of homogeneous solutions, Reduction of orders, Homogeneous linear equations with constant coefficients, Non homogeneous equation, Method of undetermined coefficients, Variation of parameters, Euler Cauchy differential equation, Modeling with second order equations, Initial Value Problems and Boundary Value problems, reduction of order, Euler equation, generating functions, eigenvalue problems. Inhomogeneous linear difference equations (variation of parameters, reduction of order, Series solutions of second order linear equations: Taylor series solutions about an ordinary point. Frobenious series solutions about regular singular points. Series solutions of Legendre, Bessel, Laguerre and Hermite equations. Systems of linear first order differential equations: Elimination method. Matrix method for homogeneous linear systems with constant coefficients. Variation of parameters. Matrix exponential.

Prerequisite: MAT211

**MAT 223 Numerical Analysis I 3 credits**

Solution of equation of single variable: Fixed point iteration, Bisection algorithm, Method of False position, Newton-Raphson's method, Error Analysis for Iterative methods, Accelerating limit of convergence. Interpolation: Interpolating polynomials for equispaced and nonequispaced nodes, Lagrange's polynomial, Newton-Gregory's Interpolating polynomials, curve fitting with Least Square method, Iterated interpolation, Extrapolation. Differentiation and Integration: Numerical differentiation, Single point and (n+1)point formulae of differentiation, Richardson's extrapolation, Numerical Integration, Gaussian quadrature formula, Trapezoidal, Simpson's, Weddle's Rules.

Prerequisite: MAT124

**MAT 301 Group Theory 3 Credits**

Definition and various examples of groups, subgroups, cosets, normal subgroups, quotient (factor) groups, permutation groups, cyclic groups, generator of a cyclic group, centre of a group, Abelian group, normalizer and centralizer of an element/ subset of a group and its application to physics, group homomorphism, isomorphism and automorphism and related theorems, symmetry groups, SU (3), SU (6), application of group theory in solid state physics & elementary particles.** MAT 303 Tensor Analysis 3 Credits**

Definition of tensor, tensor density, affine tensor and geometrical object, properties of tensor symmetry, criteria of tensor properties, metric tensor, Kronecker symbol and LeviCivita's symbol, determinant of metric tensor, connection between metric tensor and Dirac's matrices in the Sommerfeld representation, evolution of square root from four dimensional interval in matrix sense, transformation properties of vector partial derivatives by coordinates, connection coefficients and covariant derivatives, Christoffel's symbols, geodetic lines (geodetics) as a generalization of notion of straight line, variation principle for geodetics, parallel transport, connection between geodetics and covariant differentiation, transport along closed line curvature tensor of the 4th rank, curvature tensor of the 2 D rank, scalar curvature, equations of geodetic deviation, curvature expression in terms of Dirac's matrices, Bianchi's identity, Einstein's conservative tensor, integral operations and corresponding theorems.** MAT 311 Abstract Algebra 3 credits**

Equivalence relation and residue classes modulo n. Groups and subgroups, Cyclic groups, Symmetric groups. Cosets and Lagrange's Theorem: Normal subgroups, Quotient groups, Permutation groups, Homomorphism, Isomorphism and Automorphism of groups with related theorems and problems, Cayley's Theorem, centralizer and normalizer of an element/ subset in a group. Rings, Ideals, and Quotient Rings, Prime and Maximal Ideals. Integral Domain, Field of fractions. Principal Ideal Domain, Euclidian Domain, Unique Factorization Domain.

Polynomial Rings, Primitive polynomials, Gauss Theorem, Eisenstein's criterion for irreducibility. Prime Fields, characteristic of Fields.

Prerequisite: MAT121** MAT 312 Numerical Analysis II 3 credits**

**Part A: Theory**

Solutions of linear system of equations: Gaussian Elimination method with pivoting, Matrix inversion, Direct factorization of matrices, Iterative Techniques for solving linear system of equations: Jacobi's and Gauss-Seidel Method. Solution of tridiagonal system, Eigen values and Eigen vectors (Power Method). Numerical solution of Nonlinear system: Fixed point for functions of several variables, Newton's method, Quasi-Newton's method. Initial Value Problem for ODE: Euler's method, Higher order Taylor's method, Runge-Kutta methods, Multistep methods, Variable Stepsize Multistep methods. Boundary Value Problem: Linear Shooting method, Shooting method for non linear BVP. Boundary Value problem involving elliptic, parabolic and hyperbolic equations, explicit and implicit Finite Difference method.

Part B: Numerical Analysis Lab

Construction and implementation of FORTRAN / C, C++ programs of techniques in Numerical Analysis. There will be at least 15 lab assignments.

Prerequisite: MAT 223** MAT 313 Differential Geometry 3 credits**

Curves in space: Vector functions of one variable, space curves, unit tangent to a space curve, equation of a tangent line to a curve, Osculating plane (or plane of curvature), vector function of two variables, tangent and normal plane for the surface , Principal normal, binormal and fundamental planes, curvature and torsion, Serret Frenet's formulae, theorems on curvature and torsion, Helices and its properties, Circular helix. Spherical indicatrik, Curvature and torsion. Curvature and torsion for spherical indicatrices. Involute and Evolute of a given curve, Bertrand curves. Surface: Curvilinear coordinates, parametric curves, Metric (first fundamental form), geometrical interpretation of metric, relation between coefficients E, F, G. properties of metric, angle between parametric curves, elements of area, second fundamental form. Derivatives of surface normal M (Weingarten equations), Third fundamental form, Principal sections, Principal sections, direction and curvature, first curvature, mean curvature, Gaussian curvature, normal curvature, lines of curvature, centre of curvature, Rodrigues's formula, condition for parametric curves to be line of curvature, Euler's Theorem, Elliptic, hyperbolic and parabolic points, Dupin Indicatrix. ** MAT 314 Complex Analysis 3 credits**

Introduction to complex numbers and their properties, complex functions, limits and continuity of complex functions, Analytic functions, Cauchy Riemann equations, harmonic functions, Rational functions, Exponential functions, Trigonometric functions, Logarithmic functions, Hyperbolic functions. Contour integration: Cauchy's Theorem, Simply and Multiply connected domain, Cauchy integral formula, Morera's theorem, Liouville's theorem. Convergent series of analytic functions, Laurent and Taylor series, Zeroes , Singularities and Poles, residues, Cauchy's Residue theorem and its applications, Conformal Mapping.

Prerequisite: MAT123** MAT 315 History of Mathematics 3 credits**

A Survey of the development of mathematics beginning with the history of numeration and continuing through the development of the calculus. The study of selected topics from each field is extended to the 20th century. Biographical and historical aspects will be reinforced with studies of procedures and techniques of earlier mathematical cultures. ** MAT 316 Operations Research I 3 credits**

Convex sets and related theorems, Introduction to linear programming, Formulation of linear programming problems, Graphical solutions, Simplex method, Duality of linear programming and related theorems, Sensitivity. Unconstrained optimization: Newton's method, Trust region algorithms, Least Squares and zero finding. Constrained optimization: linear/nonlinear

equality/inequality constraints, Duality, working set methods. Linear programming: Simplex method, primal dual interior point methods.** MAT 321 Real Analysis II 3 credits**

Metric spaces: definition and some examples, open sets, closed sets, Convergence, Completeness, Baire's theorem. Connected set: Compact sets, locally compact sets and related theorems, connected sets, locally connected sets, continuity and compactness. Sequence in metric space: Convergent and Cauchy sequence, Completeness, Banach Fixed Point theorem with applications, sequence and series of functions, pointwise and uniform convergence, differentiation and integration of series. Continuous function on metric space: Boundedness, Intermediate Value Theorem, uniform continuity. Differentiation in Ñ : Jacobian , implicit and inverse function theorems. Integration in Ñ : contents and integrals, Fubini's theorem, change of variables.

Prerequisite: MAT221** MAT 322 Differential Equations II 3 credits**

ODE: Existence and uniqueness theory: Fundamental existence and uniqueness theorem. Dependence of solutions on initial conditions and equation parameters. Existence and uniqueness theorems for systems of equations and higher order equations. Eigen value problems and Strum-Liouville boundary value problems: Regular Strum-Liouville boundary value problems. Solution by eigenfunction expansion. Green's functions. Singular Strum-Liouville boundary value problems. Oscillation and comparison theory. Nonlinear differential equations: Phase plane, paths and critical points. Critical points and paths of linear systems.

PDE: First order equations: complete integral, General solution. Cauchy problems. Method of characteristics for linear and quasilinear equations. Charpit's method for finding complete integrals. Methods for finding general solutions. Second order equations: Classifications, Reduction to canonical forms. Boundary value problems related to linear equations. Applications of Fourier methods (Coordinates systems and separability. Homogeneous equations.) Boundary value problems involving special functions. Transformation methods for boundary value problems, Applications of the Laplace transform. Application of Fourier sine and cosine transforms.

Prerequisite: MAT222.** MAT 323 Vector Mechanics 3 credits**

Statics: Fundamental concept and principle of Mechanics. Statics of Particles: Review of vectors, vector addition of forces, resultant of several concurrent forces, resolution of forces into components, equilibrium of particles in a plane and in space. Rigid bodies: momentum of a force and a couple, Varignon's theorem, equivalent system of forces and vectors, reduction of system of forces. Equilibrium of rigid bodies: reactions at supports and connections of rigid bodies in two dimensions. Centroid and center of gravity: CG of two and three dimensional bodies, centroids of areas, lines and volumes. Moment of inertia, moments and products of inertia, radius of gyration, parallel axis theorem, principal axis and principal moments of inertia.

Dynamics: Kinematics of particles: rectilinear and curvilinear motion of particles. Kinematics of particles: Newton's second law of motion, linear and angular momentum of a particle, conservation of energy and momentum, principle of work and energy and their applications, motion under a central force and conservative central force, impulsive motion. System of particles: Newton's law, effective forces, linear and angular momentum, conservation of momentum and energy, work energy principle. Kinematics of rigid bodies: translation, rotation, and plane motion relative to rotating frame, Coriolis force. Plane motion of a rigid body: motion in two dimensions, Euler's equation of motion about a fixed point.

Prerequisite: MAT122.** MAT 324 Discrete Mathematics 3 credits**

Number System: Numbers with different bases, their conversion and arithmetic operations, normalized scientific notations. Logic: Introduction to logic, logical operations, application of logic to sets. Mathematical Reasoning: Methods of proof, Mathematical induction, recurrence relations, generating functions. Boolean Algebra: Ordered sets, lattices, Boolean algebra and operations, Boolean expressions, logic gates, minimization of Boolean expressions, Karnaugh maps, Karnaugh map algorithm. Graphs: introduction and definitions, representing graphs, graph isomorphism, connected graph, planar graph, path and circuit, shortest path algorithm, Eulerian path, Euler's theorem, Seven Bridges of Problem, graph coloring. Application of graph: tree, tree reversal, trees and sorting, spanning trees, minimum spanning trees: related algorithms. Search trees: binary search tree, leaves on a rooted tree, spanning trees and the MST problems, network and flows, the max flow and the min-cut theorem. Binary trees. ** MAT 325 Mathematical Methods 3 credits**

Series solution: singularity of a rational function, series solution of linear differential equations at nonsingular and regular singular points. Fourier series: Introduction to orthogonal functions and Integral transform, Fourier integral, Fourier transform and their applications. Laplace transformation method: Definition, existence and properties of Laplace transform, Inverse Laplace transform, Transforms of discontinuous and periodic functions. Convolution. Impulses and Dirac delta function. Solving initial value problems. Solving linear systems, Harmonic functions: Laplace equation in different coordinates and its applications. Special functions: Legendre functions of first and second kinds, Hermite polynomials, generating function, Hypergeometric functions, Laguerre functions, Bessel function and their properties.

Prerequisite: MAT 322** MAT 326 Hydrodynamics 3 credits**

Preliminaries: Concept of viscosity; Inviscid fluid; stream line, path line and streak lines; steady and unsteady motion. Equation of motion: Equation of continuity; Euler's equation of motion, conservative forces, Bernoulli's equation; circulation and Kelvin's circulation theorem; vorticity, irrotational and rotational motion, velocity potential; energy equation, Kelvin's minimum energy theorem. Two dimensional motion: vorticity, stream function and velocity potential function, streaming motion, complex potential and complex velocity, stagnation points, motion past a circular cylinder, circle theorem, motion past a cylinder, Joukowaski transformation, Blasius theorem; two dimensional source, sink and doublets, source and sink in a stream, the method of image. Vortex motion: vortex line, tube and filament, rectilinear and circular vortices, kinetic energy of system of vortices, vortex sheet, Karman's vortex street. ** MAT 400 Project /Thesis 3 Credits**

A student is required to carry out project / thesis work in the last two semesters in her/his chosen field. There will be a supervisor who will either be a BRAC University faculty or any other suitable expert from universities and R/D organizations of the country to guide the project / thesis work .On completion of study and research s/he will have to submit the dissertation report and to face a viva board for the defence of the dissertation.** MAT 411 Topology 3 credits**

Metric Spaces: Definition and some examples. Open sets. Closed sets. Convergence. Completeness. Baire's theorem. Continuous mappings. Spaces of continuous functions. Euclidean and unitary spaces. Topological Spaces: Definition and some examples. Elementary concepts. Open bases and open subbases. Weak topologies. Function algebras. Compactness: Compact spaces. Product spaces. Tychonoff's Theorem. Locally compact spaces. Compactness for metric spaces. Separation: T1-spaces and Hausdorff spaces. Completely regular spaces and normal spaces. Urysohn's lemma. Connectedness: Connected spaces. Locally connected spaces. Pathwise connectedness. Banach Spaces: Definition and some examples. Continuous linear transformations. Hahn-Banach theorem. Natural embedding. Open mapping theorem. Conjugate of an operator. Hilbert Spaces: Definition and some simple properties. Orthogonal complements. Orthogonal sets. Conjugate spaces. Adjoint and self-adjoint operators. Fixed point theory: Banach contraction principle. Schauder Principle. Applications.

Prerequisite: MAT 221** MAT 415 Finite Element Methods 3 credits**

Basic concept of finite element method, approximate solution of BVP, direct approach to Finite Element Methods. Galerkin's weighted residual method for one-dimensional BVP, the modified Galerkin's technique. Shape functions for one-dimensional elements. Division of a region into elements, linear and quadratic elements, numerical integration over elements. Finite element solution of one dimensional BVPs. Finite Element approximations of line and double integrals: line integral using quadratic elements, double integrals using triangular and quadrilateral elements, double integrals using curved elements. Finite Element solution of two-dimensional BVP: Galerkin formulation, matrix formulation for 2–D finite elements. Three-nodded triangular elements. Variational formulation of BVP: construction of variational functions, the Ritz method and finite elements, matrix formulation of the Ritz procedure, solution of two-dimensional problems. Pre-processing and solution assembly: mesh generation in one and two dimensions, techniques of assembly and solutions.

Prerequisite: MAT312.** MAT 416 Tensor Calculus 3 credits**

Tensor: Coordinates, Vectors and tensors: Curvilinear coordinates, Kronecker delta, summation convention, space of N dimensions, Euclidean and Riemannian space, coordinate transformation, Contravariant and covariant vectors, the tensor concept, symmetric and skewsymmetric tensor. Riemannian metric and metric tensors: Basis and reciprocal basis vectors, Euclidean metric in three dimensions, reciprocal or conjugate tensors, Conjugate metric tensor, associated vectors and tensor's length and angle between two vector's, The Christoffel symbols. Covariant Differentiation of Tensors and applications: Covariant derivatives and its higher rank tensor and covariant curvature tensor.

Prerequisite: MAT313** MAT 421 Fluid Mechanics 3 credits**

Preliminaries: Real and ideal fluids, Viscosity, Reynolds number, laminar and turbulent flows, boundary layers. Stress and rate of strain, General stress state of deformable bodies, General state of deformation of flowing fluid, Relation between stress and rate of deformation in general orthogonal coordinates. Equations of motion: Thermodynamic equation of state, Equation of continuity, Navier-Stokes equations, Energy equation, Equations of motion in different coordinate system. Exact solution of Navier-Stokes equations, Steady plane flow, Couette-Poiseuille flow, Plane stagnation-point flow, flow past parabolic body and circular cylinder, Steady axisymmetric flow, Circular pipe flow (Hagen-Poiseuille flow), Flow between two concentric rotating cylinders, Flow at a rotating disc, Unsteady plane flow, First Stokes problem, Second Stokes problem, Startup of Couette flow, Unsteady plane stagnation-point flow. Similarity analysis: Reynolds law of similarity, Dimensional analysis and theorem, Important non-dimensional quantities. Very slow motion: Equations of slow motion, Motion of a sphere in a viscous fluid, Theory of lubrication. Laminar boundary layer: Introduction to boundary layer, boundary layer equations in two dimensions, Dimensional representation of boundary layer equations, Displacement thickness, Friction drag, Flat plate boundary layer, Momentum thickness, Energy thickness, Similar solutions of boundary layer equations: Derivation of ODE, Wedge flow, Flow in a convergent channel, Integral relations of the boundary layer: Momentum-Integral equation, Energy-Integral equation.

Prerequisite: MAT 326

**MAT 422 Theory of Numbers 3 credits**

Arithmetic in Ù. Euclidean algorithm. Continued fractions. The ring Ù and its group of units. Chinese Remainder Theorem. Linear Diophantine equations. Arithmetical functions. Dirichlet convolution. Multiplicative function. Representation by sum of two and four squares. Arithmetic of quadratic fields. Euclidean quadratic Fields.

**MAT 423 Mathematical Modelling 3 credits**

**Modelling in Biology**

Continuous population models for single species: Continuous growth models, Delay models, Periodic fluctuations, Harvesting models. Discrete population models for single species: Simple models, Cobwebbing, Discrete logistic models, Stability, Periodic fluctuations and Bifurcations, Discrete Delay models, Continuous models for interacting populations: Predator-prey models, Lotka-Volterra systems, Complexity and stability, Periodic behaviour, Competition Models, Mutualism. Discrete growth models for interacting populations: Predator-prey models. Epidemic models and dynamics of infectious diseases: Simple epidemic models and practical applications.

**Modelling in Economics**

Theory of the household: Preference and indifference relations, Utility function, Order conditions of optimization, Stutsky equation, Demand functions, Revealed Preference hypothesis, Von Neumann-Morgenstern utility. Theory of the firm: Production function, Laws of production and scale, Optimizing behaviour, Cost curves and cost functions, Constrained output maximization. Theory of factor demand: Optimal input mix, Factor demand and supply curves, Elasticity of derived demand. Market structures and equilibrium: Market Economy and equilibrium, Stability of equilibrium, Dynamic stability.

Prerequisite: MAT 322.

**MAT 424 Operations Research II 3 credits**

Transportation and Assignment Problem: Introduction and formulation; relationship with linear programming. Network models: shortest route problems, minimal spanning, maximal-flow problem. Sequencing problem: Minimax-maximin strategies, mixed strategies, expected pay-off, solution of and games, games by linear programming and Brown's algorithm. Dynamic programming: Investment problem, Production Scheduling problem, Stagecoach problem, Equipment replacement problem. Non-linear programming: Introduction, unconstrained problem, Lagrange method for equality constraint problem, Kuhn-Tucker method for inequality constraint problem. Quadratic programming.

Prerequisite: MAT 316

**MAT 425 Advanced Numerical Methods 3 credits**

Non-iterative (Newton's, steepest descent) methods for solution of a system of equation(linear and non-linear). Approximation theory: discrete least square applications, Chebyshev polynomial applications, rational function approximation, trigonometric polynomial approximation, Fast Fourier Transform. Approximating Eigenvalues: Honseholder's method, QR algorithm. BVP involving ODE: shooting method for linear and nonlinear problems, finite difference method for linear and nonlinear problems. PDE: Finite difference methods for elliptic, parabolic and hyperbolic problems.

Prerequisite: MAT 312

**PHY 101 Introduction to Physics 3 Credits**

Vectors and scalars, Newton's Laws of motion, inertia, force, momentum, conservation of linear momentum, work, energy, conservation of energy, power, gravitation, escape velocity, projectile motion, simple harmonic motion, uniform circular motion. Structural properties of matter, elasticity, Hooke's Law, viscosity, surface tension. Heat and temperature, different scales of temperature, thermal expansion, specific heat, gas laws, heat transfer. Waves and oscillations, longitudinal and transverse waves, sound waves, velocity of sound, ultrasonic waves & their applications. Reflection and refraction of light, mirrors and lenses, total internal reflection, interference, diffraction. Coulomb's Law, ohm's law; resistance, potential difference, capacitance. Magnetic force on a moving charge, electromagnetic spectrum, velocity of light. Atoms and nuclei, mass number and atomic number, isotopes, isobars & isotones, atomic theory, Planck's Law, Photo-electric effect, wave-particle duality, special theory of relativity, radioactive decay, nuclear fission & nuclear fusion, nuclear energy, fossil fuels & other sources of energy. Structure & vastness of the universe, big bang theory, light year, solar system, Kepler's Laws of planetary motion, cosmological principle, Hubble's Law, red shift, stellar energy, neutron stars, quasars, supernovae, pulsars, black holes.

**PHY 102 Fundamentals of Physics 2 Credits**

Vectors and scalars, Newton's Laws of motion, principles of conservation of linear momentum and energy, gravitation, projectile motion, simple harmonic motion, rotation of rigid bodies. Elasticity, Hooke's Law, viscosity, Stokes' Law , surface tension. Heat & temperature, specific heat, gas laws, Newton's Law of cooling, First and Second Laws of thermodynamics, kinetic theory of gases, heat transfer. Wave motion, stationary waves, sound waves, Doppler Effect, beats, acoustics, ultrasonic & applications. Huygens' principle, electromagnetic waves, reflection, refraction, interference, diffraction.

**PHY 110 Mechanics and Properties of Matter 3 credits**

Mechanics: Vectors & scalars, vector addition and subtraction, unit vectors, scalar and vector products, scalar & triple vector product, scalar and vector fields, gradient, divergence and curl, curvilinear coordinates, motion in one dimension, motion in a plane, work and energy, conservation laws, conservative force, projectile motion, uniform circular motion, simple harmonic motion, rotational motion, moment of inertia, radius of gyration, angular momentum, Kater's pendulum, Newton's Law of gravitation, gravitational field, potential, escape velocity.

Properties of Matter: Hooke's Law, elastic modulii, adhesive and cohesive forces, molecular theory of surface tension, capillarity, variation of surface tension with temperature. Streamline flow, Poiseulle's formula, streamline flow and turbulent flow, Reynold's Number, Equation of Continuity, Bernoulli's Theorem, Stokes' Law.

**PHY 111 Principles of Physics I 3 Credits**

Vectors and scalars, unit vector, scalar and vector products, static equilibrium, Newton's Laws of motion, principles of conservation of linear momentum and energy, friction, elastic and inelastic collisions, projectile motion, uniform circular motion, centripetal force, simple harmonic motion, rotation of rigid bodies, angular momentum, torque, moment of inertia and examples, Newton's Law of gravitation, gravitational field, potential and potential energy. Structure of matter, stresses and strains, Modulii of elasticity Poisson's ratio, relations between elastic constants, work done in deforming a body, bending of beams, fluid motion and viscosity, Bernoulli's Theorem, Stokes' Law, surface tension and surface energy, pressure across a liquid surface, capillarity. Temperature and Zeroth Law of thermodynamics, temperature scales, isotherms, heat capacity and specific heat, Newton's Law of cooling, thermal expansion, First Law of thermodynamics, change of state, Second Law of thermodynamics, Carnot cycle, efficiency, kinetic theory of gases, heat transfer. Waves & their propagation, differential equation of wave motion, stationary waves, vibration in strings & columns, sound wave & its velocity, Doppler effect, beats, intensity & loudness, ultrasonics and its practical applications. Huygens' principle, electromagnetic waves, velocity of light, reflection, refraction, lenses, interference, diffraction, polarization.

**PHY 112 Principles of Physics II 3 Credits**

Electric charge, Coulomb's Law, electric field & flux density, Gauss's Law, electric potential, capacitors, steady current, Ohm's law, Kirchhoff's Laws. Magnetic field, Biot-Savart Law, Ampere's Law, electromagnetic induction, Faraday's Law, Lenz's Law, self inductance and mutual inductance, alternating current, magnetic properties of matter, diamagnetism, paramagnetism and ferromagnetism. Maxwell's equations of electromagnetic waves, transmission along wave- guides. Special theory of relativity, length contraction and time dilation, mass-energy relation. Quantum theory, Photoelectric effect, x-rays, Compton effect, dual nature of matter and radiation, Heisenberg uncertainty principle. Atomic model, Bohr's postulates, electron orbits and electron energy, Rutherford nuclear model, isotopes, isobars and isotones, radioactive decay, half-life, alpha, beta and gamma rays, nuclear binding energy, fission and fusion.Fundamentals of solid state physics, lasers, holography.

**PHY 113 Waves, Oscillation &Acoustics 3 Credits**

Principle of superposition, interference of waves, phase velocity and group velocity, simple harmonic motion, combination of SHM, Lissajous figures, damped SHM, forced oscillation, resonance, power and intensity of wave motion, waves in elastic media, vibration of strings, beats, Doppler Effect, acoustics, stroboscopy, velocity of sound, ultrasonics, and their applications.

**PHY 114 Thermal Physics & Radiation 3 Credits**

Heat and temperature, thermal equilibrium, Zeroth Law of thermodynamics, specific heat & calorimetry, Newton's Law of cooling, Kinetic Theory of Gases, idea of pressure due to collisions of molecules, mean free path, Boltzmann Distribution Law, Brownian motion, Law of equipartition of energy; Vander Waals' equation of state, heat transfer, conduction, convection and radiation, conduction of heat in solids, coefficient of thermal conductivity and its measurement, First Law of thermodynamics, isothermal & adiabatic changes, reversible and irreversible processes, Carnot's cycle, efficiency of heat engines, Second Law of thermodynamics, entropy and disorder, absolute scale of temperature, thermodynamic functions, Maxwell's relations, Clausius-Clapeyron Equation, Gibb's phase rule, Third Law of thermodynamics, Nernst heat theorem, radiation theory, black body radiation, Wien's Law, Stefan-Boltzman Law, Rayleigh Jeans Law, Planck's Law, variation of specific heat with temperature, Einstein's theory, Debye's theory, conduction of heat in solids, measurement of conductivity, JouleThomson expansion, refrigeration, heat engines, Rankine cycles, cryogenics, measurement of high temperature.

**PHY 115 Electricity and Magnetism 3 Credits**

Charge, quantization of charge, Coulomb's Law, electric field and potential. Gauss's Law, electric dipole, dielectrics, capacitance, energy of charged systems, electrical images, magnetic dipole, energy in a magnetic field. Direct current and electromotive force, Ohm's Law, Kirchhoff's Laws, Wheatstone Bridge, Lorentz force, magnetic field of a current and Ampere's Law, Biot-Savart Law, electromagnetic induction, Faraday's Law, selfinduction, mutual induction, alternating current, RMS value, power factor, CR, LR and LCR circuits, resonance.

**PHY 201 Solid State Physics 3 Credits**

Crystalline state, Bravais lattices, crystal symmetry, point group & space group, unit cells, Miller indices, xray diffraction, Bragg's Law, reciprocal lattice, structure factor, interatomic force and classification of solids, ionic, covalent, molecular, hydrogen bonded crystals, lattice energy of ionic crystals, Madelung constant, lattice vibration, phonons, normal modes in monatomic and diatomic linear chains, theory of specific heat, Einstein and Debye models, thermal expansion, defects in crystals, dislocations, consequences of defects on mechanical properties.

**PHY 202 Optics 3 Credits**

Laws of reflection and refraction, total internal reflection, Huygens' Principle, velocity of light, Young's experiment, Fresnel's biprism, Newton's rings, Michelson's interferometer, multiple reflections, FabryPerot interferometer, diffraction of light, Fresnel and Fraunhoffer diffraction, single, double and multipleslit diffraction, diffraction grating, spectrometer, resolving power of a grating, polarization of light, production of polarized light, plane, circular and elliptically polarized light, optical activity, double refraction, optic axis, halfwave and quarterwave plate, nicol prism, dispersion of light, scattering of light, Thomson scattering.

**PHY 204 Classical Mechanics and Special Theory of Relativity 3 Credits**

Classical Mechanics: Newtonian equations of motion, conservation laws of a system of particles, variable mass, generalized coordinates, generalized force, D' Alembert's Principle, variational method, EulerLagrange equations of motion, Hamilton's principles, twobody central force problem, elliptic orbit, scattering in a central field, Rutherford formula, kinematics of rigid body motion, Euler angles, rotating coordinates, Coriolis force, wind motion, principal axis transformation, top motion, principle of least action, Hamiltonian equations of motion, small oscillations, normal coordinates, normal modes.

Special Theory of Relativity: Galilean relativity, Michelson-Morley experiment, postulates of special theory of relativity, Lorentz transformation, length contraction, time dilation, twin paradox, variation of mass, relativistic kinematics, mass energy relation.

**PHY 205 Statistical Mechanics 3 Credits**

Statistical Mechanics: Phase space, concept of state and ensemble, microcanonical, canonical and grand canonical ensembles, Boltzmann probability distribution, Maxwell velocity distribution, derivation of Bose-Einstein and FermiDirac statistics, ideal Fermi gas, degenerate Fermi system, equation of state of ideal gases, ideal Bose gas, application of Statistical mechanics in various fields in physical, biological, social sciences, economics, finance and in engineering & ICT.

**PHY 210 Quantum Physics of Atoms, Solids and Nuclei 3 Credits**

Special Theory of Relativity: Michelson-Morley Experiment, Special Theory of Relativity, Lorentz Transformations, Time Dilation, Length Contraction, Mass-Energy Relation. Quantum Phenomena: Blackbody Radiation, Planck's Law, Photoelectric Effect, Bohr Atomic Model, Energy Levels & Atomic Spectra, Correspondence Principle, Dual Nature of Matter & Waves. Introductory Quantum Mechanics: Wave Function, Operators, Expectation Values, Schrödinger's Wave Equation, Particle in Box, Schrödinger Equation for Hydrogen Atom, Energy Levels, Magnetic & Orbital Angular Momentum, Concept of Quantum Numbers. Solid State Physics: Crystal Structure, Crystal Diffraction, Bragg Law, Lattice Vibrations & Phonons, Free Electron Model, Energy Levels & Density of States, Fermi-Dirac distribution function, Free Electron gas in Three dimension, Electrical conductivity & Thermal Conductivity, Hall Effect, Band Theory of Solids, Band Diagrams of Insulator, Semiconductor & Metals, Superconductivity, Lasers & Holography. Nuclear Physics: Rutherford Nuclear Model, Radioactivity, Half life & Mean life, Nuclear Binding Energy, Fission & Fusion, Particle Accelerator, Elementary Particles & Nuclear Interactions, Quarks, Lepton & Hadrons, Big Bang & Origin of the Universe.

**PHY 301 Classical Electrodynamics 3 Credits**

Solution of Laplace's equation and Poisson's equation and applications to electrostatic problems, dielectrics, electrostatic energy, Maxwell's equations, electromagnetic waves, propagation of electromagnetic waves in conducting and non-conducting media, reflection and refraction, polarization, dispersion, scattering, waves in the presence of metallic boundaries, waveguides and resonators, solution of the inhomogeneous wave equations, simple radiating system, antennas, accelerated charge, Cerenkov radiation, elements of plasma physics.

Prerequisite PHY 115.

**PHY 302 Fluid Mechanics 3 Credits**

Fluid properties, fluid statics, manometry, force on submerged planes and curved surface, buoyancy and floatation, one dimensional flow of fluid, equation of continuity, Euler's equation, flow of fluid in pipes, Bernoulli's equation, flow through orifice, mouthpiece, venturimeter, fundamental relations of compressible flow, frictional losses in pipes and fittings, types of fluid machinery, impulse and reaction turbines, centrifugal and axial flow pumps, deep well turbine pumps, specific speed, unit power, unit speed, unit discharge, performance and characteristics of turbines and pumps, design of pumps, reciprocating pumps.

Prerequisite PHY 110

**PHY 303 Quantum Mechanics 3 Credits**

Breakdown of classical physics, quantum nature of radiation, Planck's Law, photoelectric effect, Einstein's photon concept and explanation of photoelectric effect, de Broglie wave, wave particle duality, electron diffraction, Davisson-Germer experiment, emergence of quantum mechanics, Schrodinger equation, basic postulates of quantum mechanics, physical interpretation of wave function, wave packets, Heisenberg's uncertainty principle, linear operators, Hermitian operators, eigenvalue equation, one-dimensional potential problem, harmonic oscillator, orbital angular momentum, rotation operator, spherical harmonics, spin angular momentum, addition of angular momenta, solution of the Schrodinger equation for hydrogen atom, matrix formulation of quantum mechanics.

**PHY 304 Atomic & Molecular Physics 3 Credits**

Rutherford scattering experiment, Discovery of the nucleus, Bohr quantization rules, hydrogen atom spectra, Franck-Hertz experiment, Sommerfeld-Wilson quantization rules, electron spin, Stern – Gerlach experiment, Pauli exclusion principle, electronic configuration of atoms, vector atom model, coupling schemes, Hund's rule, multiplet structure, fine structure in hydrogen spectral lines, Zeeman effect, Paschen-Beck effect, production of X-rays, measurement of X-ray wavelength, X-ray scattering, Compton Effect, Mosely's Law, molecular spectra, rotational and vibrational levels, Raman Effect and its applications, lasers.

**PHY 305 Nuclear Physics I 3 Credits**

Basic properties of nuclei, constituents of nuclei, nuclear mass, charge, size and density, nuclear force, spin, angular momentum, electric and magnetic moments, binding energy, separation energy, semi-empirical mass formula, radioactive decay law, transformation laws of successive changes, measurement of decay constant, artificial radioactivity, radioisotopes, theory of alpha decay, gamma radiation, energy measurement, pair spectrometer, classical treatment of gamma emission, internal conversion, Mossbauer Effect, beta decay, energy measurement, conservation of energy and momentum in beta decay, neutrino hypothesis, orbital electron capture, positron emission, interaction of radiation in matter, ionisation, multiple scattering, range determination, bremsstrahlung, pair production, annihilation. Discovery of neutrons, production and properties of neutrons, nuclear reactions, elastic and inelastic scattering, Q-value of a reaction and its measurements, nuclear cross-section, compound nucleus theory, direct reaction and kinematics.

Prerequisite PHY 304

**PHY 306 Basic Electronics 3 Credits**

Network theorems, filters, transmission line, basic semiconductor concepts, energy bands, electrons and holes, semiconductor diode, rectification, regulators, Zener diode, diode circuits, unijunction transistor, FET and its characteristics, transistor amplifier, FET amplifier; amplifier circuits, voltage amplifiers, RC coupled amplifiers and tuned amplifiers, frequency response, bandwidth, power amplifier, push-pull amplifier, feedback and amplifier stability, operational amplifier and its characteristics, oscillators, modulation and demodulation, digital electronics, digital logic, logic gates, Boolean algebra, logic circuits, information registers, flipflop circuit.

**PHY 308 Methods of Experimental Physics & Instrumentation 3 Credits**

Optical and spectroscopic instruments, defects of images and their remedies, optical blooming, phase contrast and polarizing microscope, spectrophotometers, optical transmittance, reflectance and absorption, application of interferometry, production and measurement of high and ultrahigh vacuum. Rotary pump, diffusion pump, ion pump and turbo pump, pirani, penning and ionisation gauges, measurement of current and voltages, potentiometer, VTVM, oscilloscope, D.C. amplifier, lock-in amplifier, frequency meter and counter, four point probe, flux meter and Hall probed transducers , piezoelectric, thermistor, photo-transducers, voltage regulator, SCR type temperature controllers. Prerequisites PHY 202 and PHY 306

**PHY 309 Introduction to Materials Science 3 Credits**

Crystalline solids, amorphous, composite, fibrous materials, polymers, plastics, binding forces, elastic properties, dislocations, defects etc, specific heat, thermal expansion, thermal conductivity and electrical conductivity of metals, dielectric properties of solids , modes of dielectric polarisation, ferro electricity, piezo electricity, optical properties of solids ,classical and semi classical theory, free carrier effects, lattice absorption, electronic absorption, magnetic properties of solid, atomic magnetic moments, dia and paramagnatism, ferro & ferrimagnetism, antiferromagnetism, ferrites, magnetic resonance, superconductivity, type-1, type –2 superconductors, liquid crystals. Prerequisite PHY 201

**PHY 310 Advanced Solid State Physics 3 Credits**

Free electron theory, transport properties, Sommerfeld theory, Hall Effect, box quantization, density of states, Fermi surface, Fermi energy, electrical conductivity, Wiedmann Franz law, band theory of solids, electron in a periodic potential, Schrödinger equation, Bloch function, LCAO and OPW methods, dielectric properties of insulators, Clausius-Mosotti relations, dielectric loss, relaxation time, polarization mechanism, direct & indirect band gap semiconductors, extrinsic semiconductors, charge carrier concentration, recombination process of p-n junction, superconductivity, Meissner Effect, London equation, BCS theory, introduction to high temperature superconductivity, magnetic materials, quantum theory of diamagnetism and paramagentism, theory of ferromagnetic, ferrimagnetic and antiferromagnetic orders, magnetic resonance. Prerequisite PHY 201

**PHY 311 X-Rays 3 Credits**

Continuous and Characteristic X-rays, Bremsstrahlung, Properties of X-rays, X-ray technique, Weissenberg and precession methods, identification of crystal structure from powder photograph and diffraction traces, Laue photograph for single crystal, geometrical and physical factors affecting X-ray intensities, analysis of amorphous solids and fibre textured crystal.

Prerequisite PHY 201

**PHY 312 Nuclear Physics II 3 Credits**

Determination of nuclear size by scattering methods and electromagnetic methods, mirror nuclei, electron scattering, nuclear shapes, electric and magnetic multiple moments, isotopic spin formalism, two-nucleon problem, nuclear forces, exchange force, meson theory of nuclear forces. Shell model, refinement of extreme single particle model, collective model, nuclear reactions, compound nucleus model, concept of optical potential, energy averaged cross section and the optical model at low energies, phenomenological optical model, direct reactions, parity violation in beta decay, nuclear fission and nuclear reactor, nuclear fusion, nuclear liquid drop model & shell model, magic numbers (qualitative) accelerators, Van de Graaff generator, linear accelerator, cyclotron, synchrotron, detection of charged particles, photons and neutrons, nuclear pulse counting systems, elementary particles.

Prerequisite PHY 305

**PHY 313 Physics for Development 3 Credits**

Twenty first century development issues, physics and break through technologies, ICT, fibre optics, quantum information theory, physics in genetics engineering and molecular biology, physics and health issues, bio and medical physics, materials science and physics, high temperature superconducting materials, space physics, microgravity experiments, econo-physics, physics principles applied in sociology.

**PHY 400 Thesis / Project 4.5 Credits**

A student is required to carry out thesis/project work in the 7th and 8th semester in a chosen field. There will be a supervisor who will either be a BRAC University faculty or any other suitable expert from universities and R/D organizations of the country to guide the thesis/ project work .On completion of study and research s/he will have to submit the dissertation paper and to face a viva board for the defence.

**PHY 401 Reactor Physics 3 Credits**

Interactions of neutrons with matter, cross – sections for neutron reactions, thermal neutron cross-sections, nuclear fission, energy release in fission, neutron multiplication, nuclear chain reaction, steady state reactor theory, criticality condition, homogeneous and heterogeneous reactor systems, neutron moderation, neutron diffusion, control of nuclear reactions, coolant, types of nuclear reactors: power reactor, research reactor, fast reactor, breeder reactor, reactor shielding, waste disposal.

Prerequisite PHY 305

**PHY 402 Atmospheric Physics 3 Credits**

Structure of the atmosphere, elementary ideas about the sun and the laws of radiation, definitions and units of solar radiation, depletion of solar radiation in the atmosphere, terrestrial radiation, radiation transfer, heat balance in the atmosphere, heat budget, vertical temperature profile, radiation charts and their uses, composition of the atmosphere, mean molecular weight, humidity, mixing ratio, density and saturation vapour pressure. Fundamental equations of atmospheric motion, approximations of the equation, circulation and vorticity and their equations. Introduction to atmospheric thermodynamics.

**PHY 403 Plasma & Astrophysics 3 Credits**

General introduction to plasma physics, plasma as a fourth state of matter, definition, screening, and Debye shielding, plasma frequency, ideal plasma, temperature and pressure of plasma, magnetic pressure and plasma drifts, plasma waves, Landau damping, collisions in plasmas, hydrodynamic description of plasma, one fluid model, two fluid model,Chew-Goldberg theory, low waves in magneto-hydrodynamics, description of plasma, dielectric tensor, longitudinal and transverse waves, plasma instabilities, transport in plasmas, plasma kinetic theory, Vlasov equation, linear waves, waves in magnetized plasma, electromagnetic waves, waves in hot plasmas, nonlinear waves, Landau damping, quasi linear theory, plasmas in fusion research, astrophysical plasmas.

Introduction to astrophysics, formation of stars and galaxies, evolution of stars, the notion of cosmology, Cosmological Principle, various cosmological models of the universe, expansion of universe, Hubble's Law, problem of singularity in time, solutions of Friedmann, de Sitter and others, density of matter in the universe, cosmological term, self screening effect for matter.

Prerequisites PHY 301 and PHY 304

**PHY 404 Electronic Devices and Circuits 3 Credits**

Modelling and application of Semiconductor devices and integrated circuits, advanced transistor amplifier analysis, including feedback effects. Design for power amplifiers, operational amplifiers (OPAMP), analog filters, oscillators, A/D and D/A converters and power converters. Introduction to transistor level design of CMOS digital circuits.

Prerequisite PHY 306

**PHY 405 Mathematical Physics 3 Credits**

Series solution of 2nd order ordinary differential equations about ordinary and singular points, orthogonal set functions, Sturm-Liouville boundary value problem (SLP), eigen values and eigen functions of different SLP, series of orthogonal set of function. Laplace transforms: definition, Laplace transformations of some elementary functions, inverse Laplace transformations, Laplace transformations of derivatives, Dirac delta function, some special theorem on Laplace transformations, solution of differential equations by Laplace transformations, evaluation of improper integrals; finite Fourier series, Fourier transforms, Fourier integrals, Fourier transform and application to solution BVP, beta and gamma functions, Legendre functions, Bessel functions, solution of boundary value problem by method of separation of variables, solution PDE of mathematical physics: Helmholtz equation, wave equation: vibrating string, vibrating membrane, diffusion equation, Laplace equation, Hermite polynomials, Laguerre polynomials, hyper-geometric functions.

Prerequisite MAT 203

**PHY 406 Medical Physics & Instrumentation 3 Credits**

Ultrasound imaging, A-scan, B-scan, M-scan, clinical applications, rectilinear scanner, gamma camera, CAT scanner, MRI, clinical applications, audiology, hearing aids, vascular measurements, blood pressure, blood flow, blood velocity, cardiac measurements; ECG, ECG planes, elementary ideas on heart disorders, defibrillators, pacemakers, neuromuscular measurements; EEG, EMG, stimulation of neural tissue, nerve conduction measurements, bio-electric amplifiers, patient safety, radiopharmaceuticals, radiotherapy, radiation protection, radiation dosimetry.

**PHY 407 Mathematical Modelling in Physics 3 Credits**

Basic concept of mathematical modelling, formulation and solution, overview of computational methods of classical and quantum physics, numerical procedure for special functions, Random numbers generator, Brownian motion simulation, linear system of equations, sparse linear system, eigen value problems, BVP involving ODE, Sturm-Liouville problems, BVP involving PDE: elliptic, parabolic and hyperbolic problems using finite difference and other methods, Monte Carlo integration and simulation, mathematical modelling of problems of physics using above techniques.

Prerequisite MAT 205

**PHY 408 Advanced Quantum Mechanics 3 Credits**

Heisenberg and Dirac or interaction pictures, time-independent perturbation theory, degenerate perturbation theory, variation method, hydrogen atom and helium atom, WB approximation method, Sommerfeld-Wilson quantisation condition, time-dependent perturbation theory, Fermi's golden rule, applications, identical particles, parity, Pauli principle, applications, non- relativistic scattering theory, partial wave expansion, optical theorem, S matrix, solution of the wave equation by the method of Green's function, Lippmann Schwinger equation, Neumann series, Born approximation, applications, Klein-Gordon and Dirac equations, existence of electron spin, magnetic moment, plane wave solution of the Dirac equation, hole theory; prediction of the positron.

Prerequisite PHY 303

**PHY 409 Physics of Radiology 3 Credits**

The production and properties of X-rays, diagnostic and therapeutic X-ray tubes, X-ray circuit with rectification, electron interaction, characteristic radiation, bremsstrahlung, angular distribution of X-rays, quality of X-rays, beam restricting devices, the grid, radiographic film, radiographic quality, factors affecting the image, image modification, image intensification, contrast media, modulation transfer function, exposure in diagnostic radiology, fluoroscopy, computed tomography, ultrasound, magnetic resonance imaging (MRI).

**PHY 410 Laser Physics 3 Credits**

Spontaneous and stimulated emission, absorption, pumping schemes, characteristic properties of laser beam, laser speckle, grain size calculation for freespace propagation, semi classical treatment of absorption and stimulated emission, spontaneous emission, results of QED treatment, electric dipole, allowed and forbidden transitions, Einstein's A and B coefficient, radiation trapping, superfluorescence, superradiane and amplified spontaneous emission, nonradiative decay, homogeneous and inhomogeneous broadening, linewidth calculations for naturally, collisionally and Doppler broadened line, two level and four, level saturation, saturation of absorption & inhomogeneously broadened line, passive optical resonators, continuous wave and transient laser behaviour, laser beam transformation, types of lasers, their construction and use, applications of lasers, optical communications, laser in fusion research, holography.

Prerequisite PHY 304

**PHY 411 Geophysics 3 Credits**

Solar system, the planets, meteorites, cosmic ray exposures of meteorites, Poynting-Robertson effect, compositions of the terrestrial planet, pre-radioactivity age problem, radioactive elements and the principle of radiometric dating, growth of constituents and of atmospheric argon, age of the earth and of meteorites, dating the nuclear synthesis, figure of the earth, precession of the equinoxes, the Chandler- wobble, tidal friction and the history of the earth moon system, fluctuation in rotation and excitation of the wobble, seismology of the earth, elastic wave and seismic rays, travel time and velocity depth curves for body waves, shockwave, internal pressure of earth core, internal density and composition, free oscillation, earthquake prediction problem, terrestrial magnetism, earth magnetic field, geophysical prospecting; seismic, gravitational, magnetic, electrical and nuclear methods.

**PHY 412 Dynamical & Tropical Meteorology 3 Credits**

Geophysical fluid dynamics, Navier-Stokes' equation, rotating and stratified flow, scale analysis, hydrostatic approximation, Coriolis force, geopotential etc., gradient and thermal wind, vorticity and circulation theorems, Proudmen-Taylor theorem, atmospheric wave, atmospheric turbulence, barotroic and baroclinic instabilities, numerical weather forecasting, quasi-geotropic approximation, barotropic vorticity equation, primitive equation, multilayered models, tropical cyclones, norwesters and tornadoes, the monsoons, dynamical climatology, physics of upper atmosphere: geomagnetism, neutral atmosphere, ionosphere and magnetosphere.

Prerequisite PHY 402

**PHY 413 General Theory of Relativity 3 Credits**

Gravitation, Lagrangian Einstein equations, approximation of weak field and Hilbert's auxiliary conditions, comparison of corresponding relations with those of Newton's theory of gravitation, source of gravitation field, Schwarzschild's solution in isotropic and other coordinate systems, analogy between gravitation and electromagnetism, motion of test mass and geodetic lines, motion in Schwarszchild's field, equations of motion in general relativistic mechanics as a consequence of Einstein's equation of gravitational field, gravitational waves in weak field approximation, problem of energy transfer, exact wave solutions in the case of gravitational field, waves of matrices or wave of curvature, locally plane gravitational waves, Weber's and Braginski's experiments, prospects of future gravitational experiments.

**PHY 414 Field Theory 3 Credits**

Equation of motion, quantization, conservation laws, construction of Hilbert space, Lagrangian, equation of motion, quantization of neutral and charged Klein-Gordon fields, Dirac equation, spinors, quantization of Dirac field, Maxwell fields, Gupta-Bleuler formalism, theory of gauge fields, invariant functions propagators for Klein-Gordon field, Dirac fields and electromagnetic fields, symmetries of interactions, interaction picture; U and S matrices, Feynman diagrams, Wick's theorem, Feynman rules, lowest order, amplitude and cross section for Compton scattering, GSW model of electroweak interactions, elements of QCD, path integral in field theory, introduction to string theory.

**PHY 415 Neutron Scattering 3 Credits**

Neutron sources, continuous and pulsed sources, monochromatization, collimation and moderation of neutrons, neutron detectors, scattering of neutrons and its advantages, elastic scattering of neutrons, magnetic scattering and determination of magnetic structure, inelastic scattering, thermal vibration of crystal lattices, lattice dynamics and phonons, neutron polarization, polarized neutron applications, scattering by liquids and molecules, Van Hove correlation formalism, some experimental results of scattering by liquids and molecules, small angle neutron scattering and its applications in the study of biological molecules and defects, experimental techniques of scattering measurements, time-of-flight method, crystal diffraction techniques, neutron diffractometer and triple-axis spectrometer, constant Q-method.

Prerequisite PHY 305

PHY 416 Radiation Biophysics 3 Credits

PHY 416 Radiation Biophysics 3 Credits

Nucleus, ionizing radiations, radiation doses, interaction of radiation with matter, cell structure, radiation effects on independent cell systems, oxygen effect, hyperthermia, LET and RBE, lethal, potentially lethal and sub-lethal radiation damage, dose-rate effect, acute effects of radiation, somatic effects, late effects, non-specific life shortening and carcinogenesis, genetic changes, nominal standard dose (NSD), time dose fractionation (TDF), Standquist curve.

Prerequisite PHY 305

**STA 101 Introduction to Statistics 3 credits**

Introduction to statistics, graphical displays, frequency distribution, mean, median, mode and other measures of central tendency, standard deviation and other measures of dispersion, measure of skewness and Whisker-Box plot, correlation and regression analysis, elementary probability theory, conditional and joint probability, sample survey, simple random sampling, stratified random sampling, systematic sampling, index number, time series and quality control.

**STA 201 Elements of Statistics and Probability 3 credits**

Frequency distribution, mean, median, mode and other measures of central tendency, standard deviation and other measures of dispersion, measure of skewness and Whisker-Box plot, correlation and regression analysis, elementary probability theory, conditional and joint probability, Bayes' theorem, discrete probability distributions, binomial, hypergeometric, Poisson, geometric and negative binomial distributions, continuous probability distributions, normal and exponential distributions, sampling distributions for relevant statistics (Normal, t, chi-square, and F distributions), central limit theorem, confidence intervals and hypothesis testing for parameter (mean and proportion).

**STA 301 Modern Probability Theory & Stochastic Processes 3 Credits**

Stochastic Processes: Definition of different types of stochastic processes, recurrent events, renewal equation, delayed recurrent events, number of occurrence of a recurrent event. Markov Chain: Transition matrix, higher transition probabilities, classification of sets and chains, ergodic properties. Finite Markov Chain: General theory of random walk with reflecting barriers, transient states, absorption probabilities, application of recurrence time, gambler's ruin problem. Homogeneous Markov Processes: Poisson process, simple birth process, simple death process, simple birth death process, general birth process, effect of immigration, nonhomogeneous birth death process, Queuing theory. Modern Probability Theory: Probability of a set function, Borel field and extension of probability measure, probability measure notion of random variables, probability space, distribution functions, expectation and moments.

Prerequisite: STA201

**b. Practical Courses:**

APE 104 APE Lab I 1.5 Credits

List of Experiments:

EXP 1: Determination of the Modulus of Rigidity of a Wire by the Method of Oscillations

EXP 2: Determination of Surface Tension of Mercury and the Angle of Contact by Quincke's Method

EXP 3: Determination of the Specific Heat of a Liquid by the Method of Cooling

EXP 4: Determination of the Thermal Conductivity of a Bad Conductor by Lee's Method

EXP 5: Determination of the Specific Resistance of a Wire using a Meter Bridge

EXP 6: Determination of the High Resistance of a Suspended Coil Galvanometer by the Method of Deflection

EXP 7: Determination of the Temperature Co-efficient of Resistance of the Material of a Wire

EXP 8: Determination of the Line Frequency by Lissajous Figure using an Oscilloscope and a Function Generator and Verification of the Calibration of Time/Div Knob at a Particular Position for Different Frequencies

EXP 9: Charging and Discharging of Capacitors and Study of Their Various Characteristics.

EXP 10: Verification of Thevenin's and Norton's Theorem.

EXP 11: Verification of Maximum Power Transfer Theorem.

EXP 12: Verification of Current Division Rule (CDR), KVL and KCL

EXP 13: Conversion of Galvanometer into Voltmeter.

EXP 14: Conversion of Galvanometer into Ohmmeter.

EXP 15: Determination of the e/m of Electron Using Helmholtz Coil.

EXP 16: Determination of the Threshold Frequency for Photoelectric Effect of a Photo-Cathode and the Value of Planck's Constant by Using a Photoelectric Cell.

APE 206 APE Lab II 1.5 Credits

List of Experiments:

EXP 1: Determination of the Refractive Index of the Material of a Prism by using a Spectrometer.

EXP 2: Determination of the Radius of Curvature of a Lens by Newton's Rings Method

EXP 3: Determination of the Wavelengths of Various Spectral Lines by Spectrometer by using Plane Diffraction Grating

EXP 4: Study of the Frequency Responses of Series and Parallel LRC Series Circuit and the Variation of Q-factor with Resistance.

EXP 5: Study of the Variation of Electrical Conductivity of a Semiconductor and Determine of its Energy Gap.

EXP 6: Study of the Characteristics of a PN Junction and Zener Diode.

EXP 7: Study of the Characteristics of a NPN Bipolar Junction Transistor (BJT) in Common Base configuration.

EXP 8: Study of the Characteristics of Junction Field Effect Transistor (JFET) in Common source configuration.

EXP 9: Design and construction of a 4-diode Full Wave Rectifier power supply and study the effect of Shunt Capacitor filter.

EXP 10: Implementation of AND, OR, NOT logic gates.

EXP 11: Design of S-R flip-flop

EXP 12: To design Code converters (Decimal-to-BCD, BCD-to-Decimal)

EXP 13: To design Ripple, Ring and Decade Counters using JK-FFs.

EXP 14: To study the characteristics of IC MUX, to realization of combinational circuits and generation of complex wavesforms.

EXP 15: Use of IC 74138 decoder as DEMUX, realization of 1-to-16 line DEMUX using 74138.

APE 301 APE Lab III 1.5 Credits

List of Experiments:

EXP 1: Study of the characteristics of a Uni-junction Transistor.

EXP 2: Study of the characteristics of a Silicon Controlled Rectifier Transistor.

EXP 3: To draw and study the I-V characteristics of a solar cell.

EXP 4: Design and construction of a BJT CE single-stage amplifier using potential divider biasing.

EXP 5: Study the frequency response characteristics of a two stage RC coupled BJT amplifier.

EXP 6: Study of the characteristics of 741 Operational Amplifier.

EXP 7: Design of Inverting and Non-inverting amplifiers.

EXP 8: Design and construction of active low pass and high pass filters using Op-Amps.

EXP 9: Design and construction of active Butterworth Band pass filter.

EXP 10: Design and construction of a Summing Amplifier using 741 Op-Amp.

EXP 11: Study of the percentage distortion and power output of a complimentary symmetry push-pull power amplifier.

EXP 12: Design and Construction of a Colpitts Oscillator.

EXP 13: Design and Construction of Astable and Monostable Multividrators using BJTs

EXP 14: Expt with minority carrier.

EXP 15: Design and Construction of a Crystal Oscillator.

APE 303 APE Lab IV 1.5 Credits

List of Experiments:

EXP 1: Design and Construction of an Amplitude modulator and a demodulator.

EXP 2: Design and Construction of a Frequency modulator and a demodulator.

EXP 3: Design and Construction of a Phase-Shift-Keying and its detection.

EXP 4: Design and Construction of a Pulse Amplitude modulation and its detection.

EXP 5: Design and Construction of a Pulse Width modulation and its detection.

EXP 6: Design and Construction of a Pulse Code modulation and its detection.

EXP 7: To study Time Division Multiplex System

EXP 8: Expt. with DSP (using DSP trainer)

EXP 9: To measure microwave standing wave ratio.

EXP 10: To measure microwave Frequency and Wavelength.

EXP 11: Expt. with microwave antenna.

EXP 12: Expt. with PLL.

EXP 13: Expt. with Fiber-Optic Communication.

EXP 14: To Study the TV Composite Video signal.

EXP 15: Design, construction and testing of an Astable, Monostable and Voltage Controlled Oscillator using 555 Timer.

EXP 16: Expt. with microprocessor (8086).

MAT 250 Mathematics Lab I 2 Credits

Introduction to the computer algebra package MATHEMATICA/Matlab. Evaluation and graphical representation of function. Solution of linear and nonlinear equations by using False- Position, Bisection, Newton Raphson methods. Solution of system of linear equations by using Gaussian Elimination method. Interpolation and extrapolation. Numerical differentiations. Curve fitting. Trapezoidal and Simpson's rules for numerical integration. Problem solving in concurrent courses (e.g. Calculus, Linear Algebra and Geometry), using FORTRAN, MATHEMATICA and Matlab. Lab Assignments: There shall be at least 15 lab assignments

MAT 350 Mathematics Lab II 2 Credits

Solution of initial value or boundary value problems for Ordinary differential equations. Solution of initial value or boundary value problems for partial differential equations. Problem solving in concurrent courses (e.g; Calculus, Advanced linear algebra problems, Differential Equations and Numerical Analysis, Complex Analysis, Linear Programming, Numerical Analysis and Applied Mathematics) using FORTRAN and MATHEMATICA/Matlab. Lab Assignments: There shall be at least 15 lab assignments.

PHY 116 Physics Lab I 1.5 Credits

List of Experiments:

EXP 1: Determination of the Young's Modulus of a Short Wire by Searle's Dynamic Method

EXP 2: Determination of the Modulus of Rigidity of a Wire by the Method of Oscillations

EXP 3: Determination of g by means of a Compound Pendulum

EXP 4: Determination of the Moment of Inertia of a Flywheel about its Axis of Rotation

EXP 5: Determination of the Spring Constant and Effective Mass of a given Spiral Spring

EXP 6: Determination of Surface Tension of Water by Capillary Tube Method

EXP 7: Determination of Surface Tension of Mercury and the Angle of Contact by Quincke's Method

EXP 8: Determination of the Viscosity of Glycerine by Applying Stokes' Law.

EXP 9: Determination of the Specific Heat of a Liquid by the Method of Mixture

EXP 10: Determination of the Specific Heat of a Liquid by the Method of Cooling

EXP 11: Determination of the Thermal Conductivity of a Bad Conductor by Lee's Method

EXP 12: Determination of the Pressure Co-efficient of a Gas at Constant Volume by Constant Volume Air Thermometer

EXP 13: Determination of the Stefan's Constant

EXP 14: Study of Variation of the Frequency of a Tuning Fork with the Length of a Sonometer (n-l curve) under given Tension and Hence to Determine the Unknown Frequency

EXP 15: Determination of the Frequency of a Tuning Fork by Melde's Experiment

EXP 16: Determination of Velocity of Sound by Kundt's Tube.

PHY 203 Physics Lab II 1.5 Credits

List of Experiments:

EXP 1: Determination of the Focal Length and Hence the Power of a Convex Lens by Displacement Method with the Help of an Optical Bench

EXP 2: Determination of the Refractive Index of a Liquid by Plane Mirror and Pin Method using a Convex Lens

EXP 3: Determination of the Radius of Curvature of a Lens by Newton's Rings Method

EXP 4: Determination of the Refractive Index of the Material of a Prism by using a Spectrometer

EXP 5: Determination of the Wavelengths of Various Spectral Lines by Spectrometer by using Plane Diffraction Grating

EXP 6: Determination of the Value of an Unknown Resistance and Verification of the Laws of Series and Parallel Resistances by Means of a Post Office Box

EXP 7: Determination of the Internal Resistance of a Cell by a Potentiometer

EXP 8: Determination of the Specific Resistance of a Wire using a Meter Bridge

EXP 9: Determination of the Resistance of a Galvanometer by the Half-Deflection Method

EXP 10: Determination of the High Resistance of a Suspended Coil Galvanometer by the Method of Deflection

EXP 11: Comparison of the EMF of Two Cells with a Potentiometer

EXP 12: Determination of the Resistance per Unit Length of a Meter Bridge

EXP 13: Determination of the Temperature Co-efficient of Resistance of the Material of a Wire

EXP 14: Determination of the Value of J by Electrical Method

EXP 15: Determination of the Line Frequency by Lissajous Figure using an Oscilloscope and a Function Generator and Verification of the Calibration of Time/Div Knob at a Particular Position for Different Frequencies

EXP 16: Determination of the Self-Inductance of a Coil by Anderson's Method.

EXP 17: Charging and Discharging of Capacitors and Study of Their Various Characteristics.

PHY 307 Physics Lab III 1.5 Credits

List of Experiments:

EXP 1: Determination of the Excitation and Ionization Potentials (of mercury) by Frank-Hertz Experiment.

EXP 2: Determination of the e/m of Electron Using Helmholtz Coil.

EXP 3: Determination of the Threshold Frequency for Photoelectric Effect of a Photo-Cathode and the Value of Planck's Constant by Using a Photoelectric Cell.

EXP 4: Determination of the Plateau of a Geiger-Muller Counter and Hence to Find its Operating voltage.

EXP 5: Study of the Variation of Electrical Conductivity of a Semiconductor and Determine of its Energy Gap.

EXP 6: Study of the Characteristics of a PN Junction and Zener Diode.

EXP 7: Study of the Characteristics of PNP and NPN Transistors.

EXP 8: Study of the Frequency Response Characteristics of an RC Low pass, RC High pass, a Band pass and a Parallel T Filter.

EXP 9: Study of the Frequency Response in LRC Series Circuit and the Variation of Q-factor with Resistance.

EXP 10: Determination of the Frequency Response in LRC Parallel Circuit and Determination of Q-factor.

EXP 11: Study of Variation of Reactance due to L and C with Frequency.

EXP 12: Designing and Construction a Summing Amplifier Using 741 Operation Amplifier (OPAMP).

EXP 13: Construction of Full Wave Bridge Rectifier Using Semiconducting Diodes and Study of the Effect of Filters.

EXP 14: Determination of Transistor Characteristics in Common Emitter Configuration and Determination of Hybrid Parameter.

EXP 15: Determination of the Coefficient of Mutual Inductance Between Two Coils and Hence to Show its Variation with the Separation Between the Coils.

EXP 16: Determination of the Absorption Coefficients of Different Materials for the Radiation Emitted by a Radioactive Source by Using a Geiger-Mueller Counter.