# Course Description: B.Sc.(Hons.) in Mathematics

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### Description of Mathematics Core Courses: Bachelor of Science in Mathematics

#### MAT 111: Fundamentals of Mathematics

Credits: 3
Prerequisite: None

Course Content
Set theory, De Morgan’s laws; Relations; Mathematical logic: Propositions, Basic logical operations, Truth tables, Tautologies and Contradictions or fallacy, Logical implication and equivalence, Algebra of propositions, Conditional and Biconditional Statements, Arguments, Quantifiers, Deductive reasoning, Methods of proof, Method of induction; Trigonometry: De Moivre’s theorem, Complex arguments, Hyperbolic functions;  Inequalities: Weierstrass’s inequalities, Cauchy’s inequality, Tchebychef’s inequality, Cauchy-Schwartz’s inequality;  Summation of finite series: Method of difference;Theory of equations, Synthetic division, Transformation of equations; Matrices and Determinants: types of matrices, rules of matrix arithmetic, elementary matrices and solution using Cramer's rule.

#### MAT 112: Calculus I

Credits: 3
Prerequisite: None

Course Content
Functions of single variable and their graphs, Limit, Continuity, Differentiability; Tangent lines and rate of change, Techniques of differentiation, L’ Hopital’s Rule and Indeterminate forms; Increasing, Decreasing functions and Concavity, Relative Extrema, Absolute Maxima and Minima, Rolle’s Theorem, Mean Value Theorem; Successive Differentiation and Leibnitz Theorem, Maclaurin’s and Taylor’s theorem; The indefinite integral, Principles of integral Evaluation, Riemann sums and the definite integral, The Fundamental theorem of Calculus; Area between two curves, length of a plane curve, volumes, area of a surface of revolution; Improper Integrals, Gamma and Beta functions.

#### MAT 114: Calculus II

Credits: 3
Prerequisite: MAT 112

Course Content
Functions of multivariable's; limit, continuity, Partial derivatives and Euler’s theorem on homogeneous functions; Parametric equations, Tangent lines and Arc length for parametric curves; Tangent planes and Normal vectors, Maxima and Minima of functions of two variables, Lagrange Multipliers; Calculus of vector valued functions: unit tangent, normal and bi-normal vectors, curvature, motion along a curve; Double integrals and iterated integrals; Triple integrals and iterated integrals; General multiple integrals, Change of variables in multiple integrals, Jacobians, differentiation under the integral sign; Differentiation of vectors, Gradient, Divergence and Curl and their physical meanings.

#### MAT 115: Analytic and Vector Geometry

Credits: 3
Prerequisite: None

Course Content
Two-Dimensional Geometry: Change of axes, transformation of co-ordinates, Pair of straight lines, System of Circles: Orthogonal Circles. Conic Section:  Parabola, Ellipse & Hyperbola. The general equation of second degree, Identification of Conics; Three-Dimensional Geometry: Co-ordinate systems; Direction cosines & direction ratios, Plane, Straight line: The Shortest distance, Sphere, Orthogonal projection; Vector Analysis: Vectors and Scalars, Algebra of vectors, Vector differentiation and vector integration; Curvilinear Coordinates: Cartesian, Spherical, Polar and cylindrical systems, Green’s theorem, Divergence theorem, Stoke's theorem and their applications.

#### MAT 116: Differential Equations I

Credits: 3
Prerequisite: MAT 112

Course Content
Ordinary differential equations;Separation of variables, Homogeneous equations, Exact equation, Integrating factor, Linear and Bernoulli’s equations , Clairaut’s form and Lagrange’s form; Modeling with first order differential equations; Orthogonal and oblique trajectories;Linear second and higher order differential equations with constant coefficients; Singular solution; Homogeneous and non-homogeneous equations (variation of parameter, undetermined co-efficient, reduction of order), System of first and second order ordinary differential equations; Modeling with second-order equations; Series Solution, Initial value problem, Eigenvalue problems and Strum-Liouvile boundary value problems.

#### MAT 117: Programming Language I

Credits: 4
Prerequisite: None

Course Content
Programming in C: Concept of programming language and its classification; Programming logic and flow Chart; Structured programming using C-Constants, variables and data types, arithmetic and logical operation, loops and decision making, user-defined functions, character and strings, arrays, pointers, structures and unions, file management, graphics programming.

Programming in MATLAB: Function of several variables and graphs, multiple integrals, area and volume. Solution of ODEs of different types. Algebra of matrices, rank, inverse of matrices, matrix solution of linear system, eigenvalue, eigenvectors, diagonalization.

• The course includes lab work based on theory taught.

#### MAT 212: Differential Equations II

Credits: 3
Prerequisite: MAT 116

Course Content
First order equations: Classification of PDE, Complete integral, general solution; Method of characteristics for linear and quasi-linear Cauchy problems. Weak solutions, Charpit’s method, Jacobi’s method; Second order equations: Classification of general second order linear PDE, canonical forms; Hyperbolic Differential equations: Occurrence of the wave equation, IVPs and D’Alemberts solution of IBVPs; Method of separation of variables, method of eigen function, method of characteristics, method of Green’s functions, Fourier transformation method, Laplace transformation method. Periodic solution of 1D wave equation in cylindrical and spherical polar co-ordinates; Parabolic Differential equation; Elliptic Differential Equation.

#### MAT 213: Linear Algebra

Credits: 3
Prerequisite: MAT 112

Course Content
Methods for finding inverse of matrix, canonical forms of matrices, rank of a matrix; Cofactor expansion, Formation of adjoint matrix; System of Linear Equations: Gaussian elimination and Gauss-Jordan elimination method, Application of Matrices for solving system of linear equations, LU- decompositions; General vector space: Subspace, Linear combination, Linear independence and dependence, Basis and dimension, Row space, column space and null space; Inner Product spaces: Angle and orthogonality in inner product spaces, Orthonormal bases, Gram-Schmidt Process, QR-Decomposition, Best approximation, Least squares, Change of bases; Linear transformations: kernel and image of a linear transformation, rank and nullity, Matrix representation of linear transformations, Isomorphism; Diagonalization of matrices: Eigen values and Eigen vectors, the minimum polynomial of a matrix and the Cayley-Hamilton theorem.

#### MAT 214: Real Analysis

Credits: 3
Prerequisite: MAT 112

Course Content
Real Numbers, Supremum, Infimum, The completeness axiom and its consequences, Dedekind’s theorems; Neighbourhood, Limit point, Finite, Countable and Uncountable sets, Open set and closed set, Convex set and their properties; Bolzano-Weierstrass theorem; Open covering, compact sets, Heine-Borel theorem; Convergence of sequences, Monotone sequence, Cauchy convergence principle of sequence, bounded sequences, subsequences, Bolzano-Weierstrass theorem for sequence, Cluster point of sequence; Infinite series of real numbers: Convergence and absolute convergence, Gauss’s tests (simplified form), Alternating series (Leibnitz’s test), Product of infinite series; Continuity and Discontinuity, Extreme value theorem, Intermediate-value theorem and other properties of continuous functions; Uniform continuity; Lipschitz functions; The Riemann integral, Darboux’s sums, Darboux’s theorem,. Improper integrals: tests for convergence; Metric spaces.

#### MAT 215: Abstract Algebra

Credits: 3
Prerequisite: MAT 211

Course Content
Groups and Subgroups: Binary operations, Concept and examples of Groupoid, Semi-group, Monoid, Cyclic groups, Abelian group, Cosets and Lagrange’s theorem, Normal (invariant) subgroups, Factor (quotient) groups, Normalizer, centre of a group and Centralizer; Permutations, Symmetric group of permutations, cyclic permutation, Transposition, Even and odd permutations and alternating groups; Concept of Homomorphism, Isomorphism, Monomorphism, Epimorphism and Automorphism; Kernel and Image of a homomorphism, Homomorphism and Isomorphism theorems, Caley’s theorem, Direct Product of Groups, Classification of groups of small orders; Introduction to Rings and Fields.

#### MAT 216: Data Structures and Algorithms

Credits: 4
Prerequisite: MAT 117

Course Content
Elementary Data Organization, Data Structure Operations, Control Structures, Algorithms, String Processing and Pattern Matching Algorithms;Arrays, Records and Pointers: Linear Arrays, Inserting and Deleting, Multidimensional Arrays, Pointer Arrays, Record Structures, Parallel Arrays And Matrices;Queues And Lists, Linked Lists, Two-Way Lists, Header Linked Lists, Deques, Priority Queues. Stacks: Array Representation of Stacks, Quicksort. Recursion, Towers of Hanoi, Implementation of Recursive Procedures. Trees: Rooted Trees, Binary Trees, Header Nodes, Threads, Huffman’s Algorithm, Game Trees, and General Trees. Graphs: Sequential Representation Of Graph, Adjacency Matrix, Path Matrix, Warshall’s Algorithm, Linked Representation Of Graphs; Sorting, Searching and Data Modification.

The course includes lab work based on theory taught.

#### MAT 312: Programming Language II

Credits: 4
Prerequisite: MAT117

Course Content
Introduction to Python programming languages: generation of computer programming language, print, input/output. Lexical matters: operators, blocks and indentation, names and tokens.  The concept of data types, variables, assignments, immutable variables, numerical types, comments in the program, understanding error messages. Conditions, Boolean logic, logical operators, ranges. Conditional control statements: if, if-else, chained if-else, nested if-else, match-case statement. Loop Statement: while, for, break, continue, pass, pattern and nested loop related problems. Strings- various types of string manipulation such as subscript operator, indexing, slicing a string, concatenation, and more built-in string operations. Strings and number system: converting strings to numbers and vice versa. Binary, octal, hexadecimal numbers. Lists, tuples, set and dictionaries: basic list operators, replacing, inserting, removing an element, searching and sorting lists, list comprehension, dictionary literals, adding and removing keys, accessing and replacing values; traversing dictionaries, mutable and immutable property. Functions-with/out parameter, with/out return, recursion, keyword arguments, default arguments, variable-length arguments, solving complex problems in modular fashion using user defined function. Working with anonymous function – Lambda, Map, Filter, Reduce. Files I\O and operations: manipulating files and directories, os and sys modules, text files- reading and writing text and numbers from or to a file, creating and reading a formatted file (csv or tab-separated) Object oriented basics: creating class in Python, private identifier, constructor, inheritance, polymorphism. The necessity of import. Math library: gcd, lcm, factorial, comb, ceil, trigonometric functions, angular conversion, hyperbolic functions. NumPy: array, multidimensional array, different types of matrix operations, and several NumPy operations and functions. Matplotlib: basic sinusoidal plotting, scatter plot, pie chart etc. Differentiation and integration with NumPy and Matplotlib. Solving and visualizing first order equations with NumPy and Matplotlib. SciPy: differentiation, integration, linear algebra operation.

#### MAT 313: Mathematical Methods

Credits: 3
Prerequisite: MAT 212

Course Content
Laplace Transforms of some Elementary Functions, Transforms of Derivatives, Relations involving Integrals, Laplace Transforms of Periodic Functions, Inverse Laplace Transforms, Convolution Theorem, Simple Initial Value Problems, Solution of Differential Equations by Laplace Transforms; Fourier Series and Fourier Transform: Fourier Series, Half Range Fourier Sine and Cosine Series, Complex Form of Fourier Series, Parseval's Identity for Fourier Series, Finite Fourier Transforms, The Fourier Integral, Complex Form of Fourier Integrals,  Fourier Sine and Cosine Transforms, The Convolution Theorem, Parseval's Identity for Fourier Integrals, Relationship of Fourier and Laplace Transforms; Legendre Functions, Bessel Functions, Hermite Polynomials, Laguerre Polynomials; Hyper Geometric Functions.

#### MAT 314: Mechanics

Credits: 3
Prerequisite: MAT 115

Course Content
Reduction of System of Forces, General Conditions of Equilibrium, Principle of Virtual Work, Stable and Unstable Equilibrium, Centre of Gravity; Dynamics of a Particle: Simple Harmonic Motion, Motion of Two Dimensions, Motion of a Particle Under a Central Force, Motion of a Particle in Space, Kepler’s Laws, Motion in Three Dimensions, Angular Momentum; Dynamics of Rigid Bodies: Moments and Products of Inertia, D’Alembert Principles, Motion about a Fixed Axis, Motion in Two Dimensions, Conservation of Momentum and Energy, Euler’s and Lagrange’s Theorem.

#### MAT 315: Complex Analysis

Credits: 3
Prerequisite: MAT 112

Course Content
Complex Numbers: Introduction to Complex Numbers and their Properties, Geometry of the Complex Plane, Polar Form and Spherical Representation of Complex Numbers, Stereographic Projection;Functions of a Complex Variable, Limits and Continuity of Complex Functions, Differentiability of a Complex Function, Multiple-Valued Functions, Branch Points and Branch Lines; Analytic Functions, Harmonic Functions, Cauchy Riemann Equations, Necessary and Sufficient Conditions, Singularities, Classification of Singularities, Elementary Functions; Complex Integration: Complex Line Integrals, Cauchy's Theorem, Simply and Multiply Connected Regions, Cauchy Integral Formula, Liouville’s Theorem and the Fundamental Theorem of Algebra, Maximum Modulus Principle; Convergence of Sequence and Series, Laurent and Taylor Series, Residues, Cauchy's Residue Theorem and its Applications, Evaluation of Definite Integrals by Contour Integration; Conformal Mappings and its Physical Applications.

#### MAT 316: Discrete Mathematics

Credits: 3
Prerequisite: MAT 111

Course Content
Logic: Propositional Logic, Applications of Propositional Logic, Proof Methods; Sequences and Summations, Cardinality of Sets, Algorithms; Induction and Recursion, Recursive Algorithms, Counting principles, The Pigeonhole Principle; Recurrence Relations, Inclusion–Exclusion with Applications; Relation and their properties, Closure of Relations, Equivalence Relations; Graphs and Graph Models, Euler and Hamilton Paths, Shortest Path Algorithm, Planar Graph, Graph Coloring; Introduction to Trees, Spanning Tree Problems; Boolean Algebra: Functions and Logic Gates, Minimization of Circuits.

#### MAT 317: Numerical Analysis I

Credits: 4
Prerequisite: MAT 112

Course Content
Mathematical Preliminaries: Review of Calculus, Round-Off Errors And Computer Arithmetic, Algorithms And Convergence; Solution Of Equation In One Variable: Bisection Method, Method of False Position, Fixed Point Iteration, Secant Method, Newton-Raphson Method, Error Analysis for Iterative Method, Accelerating Limit of Convergence; Interpolation and Polynomial Approximation: Concept of Interpolation and Extrapolation, Interpolation and Lagrange Polynomial, Divided Differences, Hermite Interpolation, Cubic Spline Interpolation; Differentiation and Integration: Numerical Differentiation, Richardson’s Extrapolation, Elements Of Numerical Integration, Adaptive Quadrature Method, Romberg’s Integration, Gaussian Quadrature; Solution of Linear Systems (Simultaneous Equations): Gauss Elimination Method, Gauss-Jordan Elimination Method. Gauss-Jacobi and Gauss-Seidel Iterative Methods.

The course includes lab work based on theory taught.

#### MAT 318: Theory of Numbers

Credits: 3
Prerequisite: MAT 111

Course Content
Divisibility and Factorization: Definition, Properties, Division Algorithm, Greatest Integer Function, Primes: Definition, Euclid's Theorem, Prime Number Theorem (Statement only), Goldbach and Twin Primes Conjectures, Fermat Primes, Mersenne Primes, the Greatest Common Divisor: Definition, Properties, Euclid's Algorithm, Linear Combinations and the Gcd, the Least Common Multiple: Definition and Properties, the Fundamental Theorem of Arithmetic: Euclid's Lemma, Canonical Prime Factorization, Divisibility, Gcd, and Lcm in Terms of Prime Factorizations, Primes in Arithmetic Progressions: Dirichlet's Theorem on Primes in Arithmetic Progressions (Statement Only); Congruence’s :Definitions And Basic Properties, Residue Classes, Complete Residue Systems, Reduced Residue Systems, Linear Congruences in One Variable, Euclid's Algorithm, Simultaneous Linear Congruence’s, Chinese Remainder Theorem, Wilson’s Theorem, Fermat's Theorem, Pseudo primes and Carmichael Numbers, Euler's Theorem; Arithmetic Functions: Arithmetic Function, Multiplicative Functions: Definitions and Basic Examples, The moebius Function, Moebius Inversion Formula, the Euler Phi Function, Carmichael Conjecture, the Number-Of-Divisors and Sum-of-Divisors Functions, Perfect Numbers, Characterization of Even Perfect Numbers; Quadratic Residues: Quadratic Residues And Nonresidues, the Legendre Symbol: Definition and Basic Properties, Euler's Criterion, Gauss' Lemma, and the Law of Quadratic Reciprocity; Primitive Roots; Continued Fractions And Rational Approximations, Sums of Squares, Pythagorean Triples, Pell's Equation, Partitions, Recurrences, Applications to Primality Testing, Application to Cryptography.

#### MAT 319: Operations Research

Credits: 3
Prerequisite: MAT 211

Course Content
Convex Set: Line Segment, Hyperplane, Halfspaces, Hypersphere, Convex Combination of a Set of Points, Extreme Point; Linear Programming: Two Variable Linear Model Formulation, Basic Properties of Linear Programming Problems and Related Theorems; Solution Procedure of Linear Programming Problems: Graphically Solution, Simplex Method, Big-M Simplex Method, Two Phase Simplex Method, Degeneracy, Unboundedness, Sensitivity Analysis; Duality Theory: Definition of The Duality Problem, Primal Dual Relationship, Dual Simplex Method; Non-Linear Programming: Constrained And Unconstrained Optimization, Separable Programming, Quadratic Programming; Theory of Games: Description of Games, Characteristics of Game Theory, Maxmini And Minimax Principles, Solution of Two Person Game Problems.

#### MAT 411: Numerical Analysis II

Credits: 4
Prerequisite: MAT 317

Course Content
Iterative Techniques in Matrix Algebra: Linear Systems of Equations, Eigenvalues and Eigenvectors, the Power Method, Householder’s Method, Q-R Method; Nonlinear System of Equations: Fixed Point for Functions of Several Variables, Newton’s Method, Quasi-Newton’s Method, Steepest Descent Techniques; Initial Value Problems for ODE : Euler’s And Modified Euler’s Method, Higher Order Taylor’s Method, Single-Step Methods (Runge-Kutta Methods, Extrapolation Methods-Higher Order Differential Equations and Systems of Differential Equations), Multi-Step Methods (Adams-Bashforth, Adams-Moulton, Predictor-Corrector and Hybrid Methods), Variable Step-Size Multi-Step Methods, Error and Stability Analysis; Boundary Value Problem for ODE: Shooting Method for Linear And Nonlinear Problems, Finite Difference Methods for Linear and Nonlinear Problems.

The course includes lab work based on theory taught.

#### MAT 412: Tensor Analysis

Credits: 3
Prerequisite: MAT 115

Course Content
Coordinates, Vectors and Tensors: Curvilinear Coordinates, Kronecker Delta, Summation Convention, Space of Dimensions, Euclidean and Riemannian Space, Coordinate Transformation, Contravariant and Covariant Vectors,Mixed and Invariant Tensors, Addition, Subtraction and multiplication of Tensors, Contraction, Symmetric and skew-Symmetric tensors, Quotient law; 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, Christoffel Symbols and their transformation laws; Covariant Differentiation of  Tensors: Covariant Derivative of A Tensor and Higher Rank Tensors, Tensor Forms of Gradient, Divergence, Curl and Laplacian, Riemann Christoffel Tensor, Curvature Tensor, Ricci Tensor, Zero Tensor, Intrinsic Derivative, Bianchi Identity, Flat Space and Einstein space; Applications Of Tensor Analysis.

#### MAT 414: Differential Geometry

Credits: 3
Prerequisite: MAT 412

Course Content
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 Functions of Two Variables. Tangent and Normal Plane to the Surface F (X ,Y ,Z )=0, Principal Normal, Binormal and Fundamental Planes, Curvature and Torsion, Serret-Frenet’s Formulae, Theorems on Curvature and Torsion, Helices and Their Properties, Circular Helix. Spherical Indicatrices, Curvature and Torsion of Spherical Indicatrices, Involutes and Evolutes in Space Curves, Bertrand Curves; Surfaces: Parametric Curves, Monge’s Form of the Surface, First Fundamental Form or Metric, Geometrical Representation of Metric, Relation Between Coefficients E, F, G; Properties of Metric, Angle Between Any Two Directions and Parametric Curves, Condition of Orthogonality of Parametric Curves, Elements of Area, Second Fundamental Form, Weingarten Equations (or Derivatives of Unit Surface Normal), Normal Curvature, Meusnier’s Theorem, Curvature Directions, Condition of Orthogonality of Curvature Direction, Principal Curvatures, Lines of Curvature, Mean Curvature (First Curvature), Gaussian Curvature (Second Curvature), Centre of Curvature, Rodrigues’s Formula, Euler’s Theorem, Elliptic, Hyperbolic and Parabolic Points, Dupin indicatrix, Asymptotic Lines, Third Fundamental Form, Theorem of Beltrami-Ennerper.

#### MAT 415: Fluid Mechanics

Credits: 3
Prerequisite: MAT 212

Course Content
Real and Ideal Fluids, Viscosity, Laminar and Turbulent Flows, Steady and Unsteady Flows, The Velocity Potential, The Vorticity Vector; Density and Specific Gravity of Liquid And Mixture; Pressure at a Point. Pressure on a Surface. Centre of Pressure. Archimedes Principle; Basic Principles, Stream Lines. Path Lines. Velocity Potential, Bernoulli’s Theorem; Equations of Continuity and Motion, Navier-Stokes Equations, Energy Equation; Two Dimensional Motion, Complex Potential and Streaming Motion, Stream Functions, Kinetic Energy of Irrotational Motion, Circulations; Sources Sinks Doublets in Two Dimensions, Images; Circle Theorem, Blasius Theorem; Stokes Stream Function and Three Dimensional Motions. Rectilinear Vortices and Vortex Motion, Gravity Waves of Small Amplitudes; Laminar Boundary Layer: Introduction to Boundary Layer, Boundary Layer Equations in Two Dimensions, Dimensional Representation of Boundary Layer Equations.

#### MAT 416: Industrial Mathematics

Credits: 3
Prerequisite: STA204

Course Content
Statistical Reasoning: Random Variables, Uniform Distributions, Gaussian Distributions, The Binomial Distribution, The Poisson Distribution, Taguchi Quality Control; Monte Carlo Methods: Computing Integrals, Mean Time Between Failure, Servicing Requests, The Newsboy Problem; Data Acquisition and Manipulation: The z-Transform, Linear Recursions, Filters, Stability, Polar and Bode Plots, Aliasing, Closing the Loop, Decibels; The Discrete Fourier Transform: Real Time Processing, Properties of the DFT, Filter Design, The First Fourier Transform, Image Processing, Applications; Cost Benefit Analysis: Present Value, Life Cycle Costing; Microeconomics: Supply and Demand, Revenue, Cost, and Profit, Elasticity of Demand, Duopolistic Competition, Theory of Production, Leontiev Input/Output; Frequency Domain Methods: The Frequency Domain, Generalized Signals, Plants in Cascade, Surge Impedance, Stability, Filters, Feedback and Root-Locus, Nyquist Analysis, Control.

#### MAT 499: Research Project

Credits: 3

Prerequisite: Completed at least 95 credits