### Degree Type:

Bachelor of Science

### Department:

Department of Mathematics

### Programme Duration:

4 years (Standard Entry)

### Modes of Study:

Regular

By studying this degree programme you will be equipped with the skills and knowledge required for jobs in fields such as education, engineering, business, insurance, finance and accounting

### Entry Requirements:

Applicants must obtain passes in Elective Mathematics and any two (2) of the following elective subjects: Physics, Chemistry, Economics, Biology and Technical Drawing.

### Career Opportunities:

This programme will give you a good understanding of pure and applied mathematics and enhance your career prospects in an array of fields. You will cover a wide range of topics, from the abstract to how mathematics is used in the real world, and develop a secure understanding of mathematical concepts and approaches. In a broad sense, Mathematics goes beyond the study of numbers, counting and measuring to the study of number patterns, relationships and communicating concepts. The divisions within mathematics include arithmetic which studies numbers, algebra which studies structures, geometry which studies space, analysis which studies infinite processes [such as Calculus] and probability theory & statistics which study random processes.

## Programme Structure

### First Semester

MAT 201: Introduction to Abstract Algebra
3 Credit(s)
Pre-requisite: MAT 102

This course aims to provide a first approach to the subject of algebra, which is one of the basic pillars of modern mathematics. The focus of the course will be the study of certain structures called groups, rings, fields and some related structures. Abstract algebra gives to student a good mathematical maturity and enable learners to build mathematical thinking and skill. The topics to be covered are injective, subjective and objective mappings.  Product of mappings, inverse of a mapping. Binary operations on a set. Properties of binary operations (commutative, associative and distributive properties).  Identity element of a set and inverse of an element with respect to a binary operation. Relations on a set. Equivalence relations, equivalence classes. Partition of set induced by an equivalence relation on the set.  Partial and total order relations on a set. Well-ordered sets. Natural numbers; mathematical induction. Sum of the powers of natural numbers and allied series. Integers; divisors, primes, greatest common divisor, relatively prime integers, the division algorithm, congruencies, the algebra of residue classes.  Rational and irrational numbers. Least upper bound and greatest lower bound of a bounded set of real numbers. Algebraic structures with one or two binary operations. Definition, examples and simple properties of groups, rings, integral domains and fields.

MAT 203: Further Calculus
3 Credit(s)
Pre-requisite: MAT 102

This course is designed to develop advanced topics of differential and integral calculus. Emphasis is placed on the applications of definite integrals, techniques of integration, indeterminate forms, improper integrals and functions of several variables. The topics to be covered are differentiation of inverse, circular, exponential, logarithmic, hyperbolic and inverse hyperbolic functions.  Leibnitz’s theorem. Application of differentiation to stationary points, asymptotes, graph sketching, differentials, L’Hospital rule.  Integration by substitution, by parts and by use of partial fractions. Reduction formulae. Applications of integration to plane areas, volumes and surfaces of revolution, arc length and moments of inertia.  Functions of several variables, partial derivatives.

### Second Semester

MAT 202: Vector Algebra and Differential Equations
3 Credit(s)
Pre-requisite: MAT 102

The construction of mathematical models to address real-world problems has been one of the most important aspects of each of the branches of science. It is often the case that these mathematical models are formulated in terms of equations involving functions as well as their derivatives. Such equations are called differential equations. If only one independent variable is involved, often time, the equations are called ordinary differential equations. The course will demonstrate the usefulness of ordinary differential equations for modeling physical and other phenomena. Complementary mathematical approaches for their solution will be presented. The topics to be covered are vector algebra with applications to three-dimensional geometry. First order differential equations; applications to integral curves and orthogonal trajectories.  Ordinary linear differential equations with constant coefficients and equation reducible to this type. Simultaneous linear differential equations. Introduction to partial differential equations.

MAT 206: Complex Numbers and Matrix Algebra
3 Credit(s)
Pre-requisite: MAT 101

This course is designed to give an introduction to complex numbers and matrix algebra, which are very important in science and technology, as well as mathematics. The topics to be covered are complex numbers and algebra of complex numbers. Argand diagram, modulus-argument form of a complex number. Trigonometric and exponential forms of a complex number.  De Moivre’s theorem, roots of unity, roots of a general complex number, nth roots of a complex number. Complex conjugate roots of a polynomial equation with real coefficients.  Geometrical applications, loci in the complex plane. Transformation from the z-plane to the w-plane. Matrices and algebra of matrices and determinants, Operations on matrices up to . inverse of a matrix and its applications in solving  systems of equation. Gauss-Jordan method of solving systems of equations. Determinants and their use in solving systems of linear equations. Linear transformations and matrix representation of linear transformations.

### First Semester

MAT 303: Introductory Analysis
3 Credit(s)
Pre-requisite: MAT 201 and MAT 203

This course covers the fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, and the interchange of limit operations. It shows the utility of abstract concepts and teaches an understanding and construction of proofs. The topics to be covered include

limit of a sequence of real numbers, standard theorems on limits, bounded and monotonic sequences of real numbers, infinite series of real numbers, tests for convergence, power series, limit, continuity and differentiability of functions of one variable,  Rolle’s theorem, mean value theorems, Taylor’s theorem, definition and simple properties of the Riemann integral.

MAT 305: Linear Algebra I
3 Credit(s)
Pre-requisite: MAT 101

This course introduces more algebraic methods needed to understand real world questions. It develops fundamental algebraic tools involving matrices and vectors to study linear systems of equations and Gaussian elimination, linear transformations, orthogonal projection, least squares, determinants, eigenvalues and eigenvectors and their applications.  The topics to be covered are axioms for vector spaces over the field of real and complex numbers. Subspaces, linear independence, bases and dimension. Row space, Column space, Null space, Rank and Nullity.  Inner Products Spaces. Inner products, Angle and Orthogonality in Inner Product Spaces, Orthogonal Bases, Gram-Schmidt orthogonalization process. Best Approximation. Eigenvalues and Eigenvectors. Diagonalization. Linear transformation, Kernel and range of a linear transformation. Matrices of Linear Transformations.

MAT 307: Scientific Computing
3 Credit(s)
Pre-requisite: MAT 101

This course provides an introduction to basic computer programming concepts and techniques useful for Scientists, Mathematicians and Engineers. The course exposes students to practical applications of computing and commonly used tools within these domains. It introduces techniques for problem solving, program design and algorithm development. MATLAB (approximately 24 lectures): Basic programming: introduction to the MATLAB environment and the MATLAB help system, data types and scalar variables, arithmetic and mathematical functions, input and output, selection and iteration statements. Functions: user defined functions, function files, passing information to and from functions, function design and program decomposition, recursion. Arrays: vectors, arrays and matrices, array addressing, vector, matrix and element-by-element operations. Graphics: 2-D and 3-D plotting. Other topics to be covered are coding in a High Level Language using MATLAB/OCTAVE. At least one Computer Algebra System (CAS): MAPLE, MAXIMA MATHEMATICA, DERIVE will also be covered.

3 Credit(s)
Pre-requisite: MAT 202 and 203

Limit and continuity of functions of several variables; partial derivatives, differentials, composite, homogenous and implicit functions; Jacobians, orthogonal curvilinear coordinates; multiple integral, transformation of multiple integrals; Mean value and Taylor’s Theorems for several variables; maxima and minima with applications.

### Second Semester

3 Credit(s)
Pre-requisite: MAT 301 and MAT 303

This course covers vector valued functions. It introduces students to the concept of change and motion and the manner in which quantities approach other quantities.  Topics include limits, continuity,  derivatives of vector functions, gradient, divergence, curl, formulae involving gradient, divergence, laplacian, orthogonal curvilinear coordinates,  line integrals, Green’s theorem in the plane, surface integrals. Other topics are the divergence theorem, improper integrals, Gamma functions, Beta functions, the Riemann Stieltjes Integral,  pointwise and uniform convergence of sequence and series, integration and differentiation term by term.

Limits, continuity and derivatives of vector functions; gradient, divergence and curl; formulae involving gradient, divergence, curl and laplacian and orthogonal curvilinear coordinates; line integrals; Green’s theorem in the plane; surface integrals; the divergence theorem; improper integrals; Gamma and Beta functions; The Riemann Stieltjes integral; pointwise and uniform convergence of sequence and series; integration and differentiation term by term.

MAT 306: Linear Algebra II
3 Credit(s)
Pre-requisite: MAT 305

This course introduces more algebraic methods needed to understand real world questions. It develops fundamental algebraic tools involving direct sum of subspaces, complement of subspace in a vector space and dimension of the sum of two subspaces. Other topics to be covered are one-to one, onto and bijective linear transformations, isomorphism of vector spaces, matrix of a linear transformation relative to a basis, orthogonal transformations, rotations and reflections, real quadratic forms, and positive definite forms.

MAT 308: Mathematical Modelling
3 Credit(s)
Pre-requisite: MAT 202

This course is designed to introduce students to basic concepts in mathematical modelling. It also equips the students with mathematical modelling skills with emphasis on using mathematical models to solve real- life problems. Topics to be covered in this course includes: methodology of model building, problem identification and definition, model formulation and solution, consideration of varieties of models involving equations like algebraic, ordinary differential equation, partial differential equation, difference equation, integral and functional equations, Single species models (exponential, logistic and, the Gompertz growth  models), interacting species models: (predator-prey models, competing species models, cooperating species models, multi-species models), the SI, SIR, SIS, SIRS and SEIR epidemic models, the basic reproduction number  R0: derivation, interpretation and application to stability analysis of disease-free and endemic equilibria, and case studies: Malaria, HIV-AIDS, TB.

MAT 310: Abstract Algebra I
3 Credit(s)
Pre-requisite: MAT 201 and MAT 202

This course focuses on traditional algebra topics that have found greatest application in science and engineering as well as in mathematics. The topics to be covered are: axioms for groups with examples, subgroups, simple  properties of groups, cyclic groups, homomorphism and isomorphism, axioms for rings, and fields, with examples, simple properties of rings, cosets and index of a subgroup, Lagrange’s theorem, normal subgroups and quotient groups, the residual class ring, homomorphism and isomorphism of rings, subrings.

### First Semester

MAT 401: Real Analysis I
3 Credit(s)
Pre-requisite: MAT 302 and MAT 303

This course is designed as a basic introductory course in the analysis of metric spaces. It is aimed at providing the abstract analysis components for the degree course of a student majoring in mathematics. This course affords students an opportunity to gain some familiarity with the axiomatic method in analysis. The topics to be covered are: metric spaces, open spheres, open sets, limit points, closed sets, interior, closure, boundary of a set, sequences in metric spaces, subsequences, upper and lower limits of real sequences,  continuous functions on metric spaces, uniform continuity, isometry, homomorphism, complete metric spaces, compact sets in a metric space, Heine-Borel theorem, connected set, and the inter-mediate value theorem.

MAT 405: Ordinary Differential Equations
3 Credit(s)
Pre-requisite: MAT 301

The construction of mathematical models to address real-world problems has been one of the most important aspects of each of the branches of science. It is often the case that these mathematical models are formulated in terms of equations involving functions as well as their derivatives. Such equations are called differential equations. If only one independent variable is involved, often time, the equations are called ordinary differential equations. The course will demonstrate the usefulness of ordinary differential equations for modelling physical and other phenomena. Complementary mathematical approaches for their solution will be presented. Topics  covered include linear differential equation of order n with coefficients continuous on some interval J,  existence-uniqueness theorem for linear equations of order n, determination of a particular solution of non-homogeneous equations by the method of variation of parameters,  Wronskian matrix of n independent solutions of a homogeneous linear equation,  ordinary and singular points for linear equations of the second order,  solution near a singular point, method of Frobenius, singularities at infinity, simple examples of  Boundary value problems for ordinary linear equation of the second order, Green’s functions, eigenvalues, eigenfunctions, Sturm-Liouville systems, properties of the gamma and beta functions, definition of the gamma function for negative values of the argument; Legendre, Bessel, Chebyshev, Hypergeometic functions and  orthogonality properties.

MAT 406: Partial Differential Equations
3 Credit(s)
Pre-requisite: MAT 405

This course introduces students to the theory of boundary value and initial value problems for partial differential equations with emphasis on linear equations. Topics covered include first and second order partial differential equations, classification of second order linear partial differential equations, derivation of standard equation, methods of solution of initial and boundary value problems, separation of variables, Fourier series and their applications to boundary value problems in partial differential equation of engineering and physics, internal transform methods; Fourier and Laplace transforms and their application to boundary value problems.

MAT 409: Operations Research
3 Credit(s)
Pre-requisite: MAT 206

This course serves as an introduction to the field of operations research. It will quip students with scientific approaches to decision-making and mathematical modelling techniques required  to design, improve and operate complex systems in the best possible way. Topics covered include linear programming, the simplex method, duality and sensitivity analysis, integer programming , nonlinear programming, dynamic programming and  network models.

MAT 411: Classical Mechanics
3 Credit(s)
Pre-requisite: MAT 202

This is an introductory mechanics course designed to consolidate the understanding of fundamental concepts in mechanics such as force, energy, momentum etc. more rigorously as needed for further studies in physics, engineering and technology. Topcs covered include kinematics and dynamics of point masses, Newton’s laws,  momentum, energy, angular momentum and torque, conservation laws, motion under gravity, central force problem, Virial theorem, Kepler’s laws, Rutherford problem,  coupled oscillations, dynamics of rigid bodies, moment of inertia tensor, Euler’s equations, orthogonal transformation and Euler’s angle, Cayley Klein parameters, symmetric top,  Lagrangian dynamics, generalized coordinates and forces, Lagrange’s equation, Hamilton’s principle, and variational methods.

MAT 413: Optimization
3 Credit(s)
Pre-requisite: MAT 301 and MAT 302

This course introduces the principal algorithms for linear, network, discrete, nonlinear, dynamic optimization and optimal control. Emphasis is on methodology and the underlying mathematical structures. Topics include description of the problem of optimisation and the geometry of Rn, n > 1, convex sets and convex functions, unconstrained optimization: necessary and sufficient conditions for local minima/maxima,  constrained optimization: equality and inequality constraints, Lagrange multipliers and the Kuhn-Tucker conditions, computational methods  for unconstrained and constrained optimization, steepest descent and Newton's methods, quadratic programming, penalty and barrier methods, sequential quadratic programming (SQP) implementation in MATLAB/OCTAVE.

MAT 413: Optimization
3 Credit(s)
Pre-requisite: MAT 301 and MAT 302

This course introduces the principal algorithms for linear, network, discrete, nonlinear, dynamic optimization and optimal control. Emphasis is on methodology and the underlying mathematical structures. Topics include description of the problem of optimisation and the geometry of Rn, n > 1, convex sets and convex functions, unconstrained optimization: necessary and sufficient conditions for local minima/maxima,  constrained optimization: equality and inequality constraints, Lagrange multipliers and the Kuhn-Tucker conditions, computational methods  for unconstrained and constrained optimization, steepest descent and Newton's methods, quadratic programming, penalty and barrier methods, sequential quadratic programming (SQP) implementation in MATLAB/OCTAVE.

MAT 420: ABSTRACT ALGEBRA II
3 Credit(s)
Pre-requisite: MAT 304 and MAT 305

This course focuses on traditional algebra topics that have found greatest application in science and engineering as well as in mathematics. The topics covered include: Ideals and quotient rings, axioms for the integral domains, with examples, subdomains and subfields, ordered integral domains and fields, polynomial rings and field of quotients of an integral domain.

### Second Semester

MAT 402: Real Analysis II
3 Credit(s)
Pre-requisite: MAT 401

This course is designed to offer a basic introduction to measure theory and Lebesgue’s integral. The topics to be covered are: countable and uncountable sets, countability of the rationals,  uncountability of the reals, measurable sets and functions, the Lebesgue’s integral where E is a measurable subset of the real line and f is measurable on E, the  spaces as metric spaces, Cauchy sequences in  spaces, completeness of spaces, the Riesz-Fischer theorem and Mean convergence in the space .

MAT 404: Complex Analysis
3 Credit(s)
Pre-requisite: MAT 302 and MAT 303

This course provides an introduction to complex analysis which is the theory of complex functions of a complex variable. The topics to be covered in the course are: complex numbers, sequences and series of complex numbers, limits and continuity of functions of complex variables, elementary functions of a complex variable, Cauchy-Riemann criterion for differentiability,  analytic functions, complex integrals, Taylor’s and Laurent’s series, calculus of residues, contour integration and conformal mapping.

MAT 408: Introductory Functional Analysis
3 Credit(s)
Pre-requisite: MAT 401

This course is intended to introduce the student to the basic concepts and theorems of functional analysis and its applications. Topics covered include linear spaces, topological spaces, normed linear spaces & Banach Spaces, inner product spaces,  Hilbert spaces, linear functional and the Hahn-Banach theorem.

MAT 410: Quantum Mechanics
3 Credit(s)
Pre-requisite: MAT 411

This course develops concepts in quantum mechanics such that the behaviour of the physical universe can be understood from a fundamental point of view. It provides a basis for further study of quantum mechanics. Content will include: Historical origin of Quantum Theory: Blackbody radiation, Photoelectric effect, Compton effect, Optical Spectra of atoms. General formalism of Quantum theory: operators, wavefunctions and their physical significance, expectation value, commutation relations, uncertainty principle. The Schroedinger equation, infinite square well, the square well in three dimensions, central potential, step potential. The Harmonic Oscillator, Angular momentum in quantum mechanics. Approximation methods: Stationary Perturbation theory, Variational method, WKB approximation, Theory of Scattering.

MAT 412: Numerical Analysis II
3 Credit(s)
Pre-requisite: MAT 407

This course is designed to equip students with the basic techniques for the efficient numerical solution of problems in science and engineering. Topics will include: Curve fitting and function approximation.  Approximation formulae for kth derivatives.  Composite rules and Romberg integration, Gauss quadrature, numerical method for multiple integrals. Numerical methods for ordinary differential equations.  Numerical methods for Eigenvalues, the power method for finding dominant eigennvalues, the inverse power method for finding smallest eigenvalues, the shifted inverse power method, for finding an eigenvalues closest to a given approximate eigenvalue. Piece-wise polynomial interpolation, cubic splines.

MAT 414: Introduction to Topology
3 Credit(s)
Pre-requisite: MAT 301 & MAT 303

This course introduces topology, covering topics fundamental to modern analysis and geometry. It also deals with subjects like topological spaces and continuous functions, connectedness, compactness, separation axioms, and further topics such as open and closed sets, neighbourhood, basis, convergence, limit point, completeness, subspaces, product spaces, quotient spaces.