Mathematics – Course Description

DESCRIPTION OF COURSES

MTH 111        Elementary Mathematics I                                             3 units

Elementary Set theory, subsets, union, intersection, complements. Venn diagrams, Real numbers, integers, rational and irrational numbers, mathematical induction, real sequences and series, theory of quadratic equations, binomial theorem. Circular measure, trigonometric functions of angles of any magnitude, addition and factor formulae. Complex numbers, algebra of complex numbers, the Argand Diagram, De Moivre’s theorem, nth roots of unity.

MTH 121        Elementary Mathematics II                                           3 units

Functions of a real variable, graphs, limits and continuity. The derivative as limit of rate of change. Techniques of differentiation. Curve sketching, integration as an inverse of differentiation. Methods of integration, definite integrals. Application of integration to areas and volumes.

MTH 122        Elementary Mathematics III                                         3 units

Geometric representation of vectors in 1-3 dimensions, Components, direction cosines. Addition of vectors and multiplication of vectors by a scalar, linear independence. Scalar and vector products of two vectors. Differentiation and integration of vectors with respect to a scalar variable. Two-dimensional coordinate geometry. Straight lines, circles, parabolas, ellipses, hyperbolas. Tangents and normals. Kinematics of a particle. Components of velocity and acceleration of a particle moving in a plane. Force and momentum. Newton’s laws of motion; motion under gravity, projectile motion, resisted vertical motion of a particle, elastic string, motion of a simple pendulum, impulse and change of momentum. Impact of two smooth elastic spheres. Direct and oblique impacts.

MTH 132        Elementary Mechanics I                                                 3 units

Vectors: Algebra of vectors; coplanar forces; their resolution into components, equilibrium conditions, moments and couples, parallel forces; friction; centroids and centres of gravity of particles and rigid bodies; equivalence of sets of coplanar forces. Kinematics and rectilinear motion of a particle, vertical motion under gravity, projection, relative motion. Dynamics of a particle. Newton’s laws of motion; motion of connected particles.

MTH 201        Advanced Mathematics I                                                3 units

Mathematics and symbolic logic: inductive and deductive systems. Concepts of sets;

mappings and transformations. Introduction to complex numbers. Introduction to vectors, Matrices and determinants.

MTH 202        Advanced Mathematics II                                             3 units

Discrete and continuous variables. The equation of a straight line in various forms. The circle. Trigonometric functions; logarithmic functions; exponential functions. Maxima, minima and points of inflexion. Integral Calculus: Integration by substitution and by parts. Expansion of algebraic functions. Simple sequences and series.

MTH 203 Advanced Mathematics III                                                  3 units

Matrices and determinants, introduction to linear programming and integer programming, sequences and series. Taylor’s and Maclaurin’s series. Vector Calculus, line integrals and surfce integrals. Gauss’ (divergence), Green’s and Stokes’ Theorems. Complex numbers and functions of a complex variable; conformal mapping; infinite series in the complex plane.

MTH 204        Advanced Mathematics IV                                            3 units

Translation and rotation of axes, space curves; applications of vector Calculus to space curves; the Gaussian and Mean curvatures, the geodesic and geodesic curvature. Differential equations: sECOd order ordinary differential equations and methods of solutions. Partial differential equations: sECOd order partial differential equations and methods of solution.

MTH 205        Advanced Mathematics V                                              3 units

Translation and rotation of axes, plane geometry of lines, circles and other simple curves; lines in space; equations of the plane, space curve. The Gaussian and mean curvatures; the geodesic and geodesic curvature.

MTH 206        Advanced Mathematics VI                                            2 units

Complex analysis – Elements of the algebra of complex variables, trigonometric, exponential and logarithmic functions. The number system; sequences and series. Vector differentiation and integration.

MTH 207        Advanced Mathematics VII                                           2 units

Elements of linear algebra. Calculus: Elementary differentiation and relevant theorems. Differential equations: Exact equations, methods of solution of sECOd-order ordinary differential equations; partial differential equations, with applications.

MTH 208        Advanced Mathematics VIII                                        2 units

Numerical analysis: Linear equations, non-linear equations; finite difference operators. Introduction to linear programming.

MTH 211        Sets, Logic and Algebra                                                   3 units

Introduction to the language and concepts of modern mathematics; topics include: Basic set theory, mappings, relations, equivalence and other relations, Cartesian product. Binary logic, methods of proof. Binary operations, algebraic structures, semi-groups, rings, integral domains, fields. Homomorphism. Number systems; properties of integers, rationals, real and complex numbers.

MTH 215        Linear Algebra I                                                                2 units

System of linear equations. Matrices and algebra of matrices. Vector spaces over the real field. Subspaces, linear independence, bases and dimensions.Gram-Schmidt orthogonalization procedure. Linear transformations: range, null space and rank. Singular and non-singular transformations.

MTH   218      Three-Dimensional Analytic Geometry                  2 units

Plane curves, parametric representations, length of a plane arc, lines in three-space, surfaces,

cylinders, cylindrical and spherical coordinates, quadratic forms, quadrics and central quadrics.

MTH   216      Linear Algebra II                                                             2 units

Representations of linear transformations by matrices, change of bases, equivalence and similarity. Determinants. Eigenvalues and eigenvectors. Minimum and characteristic polynomials of a linear transformation. Cayley- Hamilton theorem, bilinear and quadratic forms, orthogonal diagonalization. Canonical forms.

MTH   221      Real Analysis I                                                                     3 units

Bounds of real numbers, convergence of sequences of numbers. Monotone convergence of series. Absolute and conditional convergence of series, and rearrangements. Completeness of reals and incompleteness of rationals. Continuity and differentiability of functions. Rolle’s and mean-value theorems for differentiable functions. Taylor series.

MTH 222        Elementary Differential Equations I                          3 units

First-order ordinary differential equations. Existence and uniqueness of solution. SECOd-order ordinary differential equations with constant coefficients. General theory of nth-order linear ordinary differential equations. The Laplace transform. Solution of initial- and boundary-value problems by Laplace transform method. Simple treatment of partial differential equations in two independent variables. Applications of ordinary and partial differential equations to physical, life and social sciences.

MTH 224        Introduction to Numerical Analysis                          3 units

Solution of algebraic and transcendental equations. Curve fitting, error analysis. Interpolation, approximation, zeros of non-linear equations of one variable. Systems of linear equations. Numerical differentiation and integration. Numerical solution of initial-value problems for ordinary differential equations.

MTH 231        Elementary Mechanics II                                                2 units

Impulse and Momentum, conservation of momentum; work, power and energy; work and energy principle, conservation of mechanical energy. Direct and oblique impact of elastic bodies. General motion of a particle in two dimensions, central orbits, motion in horizontal and vertical circles, simple harmonic motion, motion of a particle attached to a light inelastic spring or string. Motion of a rigid body about a fixed axis; moments of inertia

calculations; perpendicular and parallel axes theorems, principal axes of inertia and directions. Conservation of energy. Compound pendulum. Conservation of angular

momentum .

MTH 242        Mathematical Methods I                                                3 units

Real-valued functions of a real variable. Review of differentiation and integration and their applications. Mean-value theorem. Taylor series. Real-valued functions of two or three variables. Partial derivatives. Chain-rule, extrema, Lagrange’s multipliers, increments, differentials and linear approximations. Evaluation of line-integrals. Multiple integrals.

MTH 311        Abstract Algebra I                                                              3 units

Group: definition; examples, including permutation groups. Subgroups and cosets. Lagrange’s theorem and applications. Cyclic groups. Normal subgroups and quotient groups. Homomorphism, Isomorphism theorems. Cayley’s theorems. Direct products. Groups of small order. Group acting on sets. Sylow theorems,

MTH 312        Abstract Algebra II                                                            3 units

Rings: definition; examples, including Z, Zn; rings of polynomials and matrices, integral domains, fields, polynomial rings, factorization. Euclidean algorithm for polynomials, H.C. F. and L.C.M. of polynomials.ideals and quotient rings, P.I.D.’s, U.F.D’s, Euclidean rings. Irreducibility. Field theorems, degree of an extension, minimum polynomial. Algebraic and transcendental extensions. Straight-edge and compass constructions.

MTH 313        Geometry I                                                                           2 units

Coordinates in Â3. Polar coordinates; distance between points, surfaces and curves in space. The plane and straight line.

MTH 314        Geometry II                                                                         2 units

Introductory projective geometry. Affine and Euclidean geometries.

MTH   316      Differential Geometry                                                     3 units

Concept of a curve, regular, differentiable and smooth curves, osculating, rectifying and normal planes, tangent lines, curvature, torsion, Frenet-Serret formulae, fundamental, existence and uniqueness theorem, involutes, evolutes, spherical indicatrix, developable surfaces, ruled surfaces, curves on a surface, first and sECOd fundamental forms, lines of curvature, umbilics, asymptotic curves, geodesics. Topological properties of simple surfaces.

MTH 321        Metric Space Topology                                                     3 units

Sets, metrics and examples. Open spheres or balls. Open sets and neighbourhoods. Closed sets. Interior, exterior, frontier, limit points and closure of a set. Dense subsets and separable space. Convergence in metric space, homeomorphism. Continuity and compactness, connectedness.

MTH   327      Elements of Differential Equations II                       3 units

Series solution of sECOd-order differential equations. Sturm-Liouville problems. Orthogonal polynomials and functions. Fourier series, Fourier-Bessel and Fourier-Legendre series. Fourier transformation, solution of Laplace, wave and heat equations by the Fourier method. (separation of variables). Special functions:Gamma,Beta, Bessel, Legendre and Hypergeometric

MTH 323        Complex Analysis I                                                          3 units

Functions of a complex variable: limits and continuity of functions of a complex variable. Derivation of the Cauchy-Riemann equations; Bilinear transformations, conformal mapping, contour integrals. Cauchy’s theorem and its main consequences. Convergence of sequences and series of functions of a complex variable. Power series. Taylor series.

MTH 324        Vector and Tensor Analysis                                            3 units

Vector algebra. The dot and cross products. Equations of curves and surfaces. Vector differentiation and applications. Gradient, divergence and curl. Vector integrals: line, surface and volume integrals. Green’s, Stoke’s and divergence theorems. Tensor products of vector spaces. Tensor algebra. Symmetry. Cartesian tensors and applications.

MTH 326        Real Analysis II                                                                   3 units

Riemann integral of real function of a real variable, continuous monopositive functions. Functions of bounded variation. The Riemann-Stieltjes integral. Point-wise and uniform convergence of sequences and series of functions ®Â. Effects on limits (sums) when the functions are continuously differentiable or Riemann integrable power series.

MTH 328        Complex Analysis II                                                           3 units

Laurent expansions, isolated singularities and residues. The Residue theorem, calculus of residues and application to the evaluation of integrals and to summation of series. . Maximum modulus principle. Argument principle. Rouche’s theorem. The fundamental theorem of algebra. Principle of analytic continuation. Multiple-valued functions and Riemann surfaces.

MTH 331        Introduction to Mathematical Modelling                   3 units

Methodology of model building; identification, formulation and solution of problems; cause-effect diagrams. Equation types. Algebraic, ordinary differential, partial differential, difference, integral and functional equations. Applications of mathematical models to physical, biological, social and behavioural sciences.

MTH 334        Special Theory of Relativity                                           4 units

Classical mechanics and principles of Relativity, Einstein Postulates; Interval between events, Lorentz transformation and its consequences; Four-Dimensional Space-time, Relativistic Mechanics of a particle, Maxwell’s theory in a Relativistic form. Optical phenomena.

MTH 335        Introduction to Operations Research                        3 unitsPhases of operations research study. Classification of operations research models; linear, dynamic and integer programming. Decision theory. Inventory models. Critical path analysis and project controls.

MTH 336        Dynamics of a Rigid body                                               3 units

General motion of a rigid body as a translation plus a rotation. Moment of inertia and product of inertia in three dimensions. Parallel and perpendicular axes theorems. Principal axes, angular momentum, kinetic energy of a rigid body. Impulsive motion. Examples involving one- and two-dimensional motion of a simple systems. Moving frames of reference; rotating and translating frames of reference. Coriolis force. Motion near the earth’s surface. The Foucauli’s pendulum. Euler’s dynamical equations of motion of a rigid body with one point fixed. The symmetric top. Precessional motion.

MTH 337        Optimization Theory II                                                   2 units

Linear programming models. The simplex method: formulation and theory, duality, integer programming; transportation problem. Two-person-zero-sum games. Nonlinear programming. Quadratic programming.

MTH 338        Optimization Theory II                                                  2 units

Kuhn-Tucker methods. Optimality criteria. Single variable optimization. Multivariable techniques. Gradient methods.

MTH 339        Analytic Dynamics                                                           3 units

Degrees of freedom. Holonomic and non-holonomic constraints. Generalized coordinates. Lagrange’s equations of motion for holonomic systems; force dependent on coordinates only; force obtainable from a potential. Impulsive force.

MTH   341      Discrete Mathematics I                                                   2 units

Groups and subgroups, group axioms, permutation groups, cosets, graphs, directed and undirected graphs, subgraphs, cycles, connectivity. Applications (flow charts) and state-transition graphs.

MTH 342        Discrete Mathematics II                                                 2 units

Lattices and Boolean algebra. Finite fields: Mini-polynomials, irreducible polynomials,

polynomial roots. Applications (error-correcting codes)

MTH 344       Numerical Analysis I                                                        3 units

Polynomials and splines approximation. Orthogonal polynomial and Chebysev approximation. Direct and iterative methods for the solution of system of Linear equations. Eigenvalue problem – power methods, inverse power method, pivoting strategies.

MTH 412        Abstract Algebra III                                                          3 units

Splitting fields. Separability. Algebraic closure. Solvable groups. Fundamental theorem of Galois theory. Solution by radicals. Definition and examples of modules, submodules and quotient modules. Isormorphism theorems. Theory of group representations.

MTH 421        Ordinary Differential Equations                                 3 units

Existence and uniqueness of solutions; dependence on initial conditions and on parameters, general theory for linear differential equations with constant coefficients. The two-point Sturm-Liouville boundary value problem; self-adjointness; Sturm theory; stability of solutions of nonlinear equations; phase-plane analysis. Floquet Theory. Integral equations classifications – Voltera and Freedhom types. Reduction of ordinary differential equations to integral equations.

MTH   424      General Topology                                                              3 units

Topological spaces, definition, open and closed sets, neighbourhood. Coarser and finer topologies. Bases and sub-bases. Separation axioms, compactness, local compactness, connectedness. Construction of new topological spaces from given ones. Subspaces, quotient spaces, continuous functions, homomorphisms, topological invariants, spaces of continuous functions. Point-wise and uniform convergence.

MTH 425        Lebesgue Measure and Integration                            3 units

Lebesgue measure: measurable and non-measurable sets. Measurable functions. Lebesgue integral: integration of non-negative functions, the general integral convergence theorems.

MTH 426        Measure Theory                                                                 4 units

Abstract Lp– spaces.

MTH 427        Field Theory in Mathematical Physics                      3 units

Gradient, divergence and curl. Further treatment and application of the definitions of the differential. The integral definition of gradient, divergence and curl. Line-, surface- and volume- integrals. Green’s, Gauss’s, and Stokes’ theorems. Curvilinear coordinates. Simple notion of tensors. The use of tensor notations.

MTH 428        Partial Differential Equations                                      3 units

Partial differential equations in two independent variables with constant coefficients: the Cauchy problem for the quasi-linear first- order partial differential equations in two independent variables; existence and uniqueness of solutions. The Cauchy problem for the linear, sECOd- order partial differential equation in two independent variables, existence and uniqueness of solution: normal forms. Boundary- and initial – value problems for hyperbolic, elliptic and parabolic partial differential equations.

MTH 429        Functional Analysis                                                           3 units

A survey of the classical theory of metric spaces, including Baire’s category theorem, compactness, separability, isometries and completion.; elements of Banach and Hilbert spaces; parallelogram law and polar identity in Hilbert space H; the natural embeddings of normed linear spaces into sECOd dual, and H onto H; properties of operators including the open mapping and closed graph theorem; the spaces C(X), the sequence (Banach) spaces, lpn, lp and (c=space of convergent sequences).

MTH 432        General Theory of Relativity                                          3 units

Particles in a gravitational field: Curvilinear coordinates, intervals. Covariant differentiation: Christoffel symbols and metric tensor. The constant gravitational field. Rotation. The curvature tensor. The action function for the gravitational field. The energy- momentum tensor. Newton’s laws. Motion in a centrally symmetric gravitational field. The energy- momentum pseudo- tensor. Gravitational waves. Gravitational fields at large distances from bodies. Isotropic space. Space- time metric in the closed and open isotropic models.

MTH 434        Elasticity                                                                                 3 units

Stress and strain analysis, constitutive relations, equilibrium and compatibility equations, principles of minimum potential and complementary energy, principles of virtual work, variational formulation, extension, bending and torsion of beams; elastic waves.

MTH 436        Fluid Dynamics                                                                     3 units

Real and ideal fluids; differentiation following the motion of fluid particles. Equations of motion and continuity for incompressible inviscid fluids. Velocity potentials and Stokes’ stream function. Bernoulli’s equation with applications to flows along curved paths. Kinetic energy. Sources, sinks and doublets in 2- and 3- dimensional flows; limiting stream lines. Images and rigid planes, streaming motion past bodies including aerofoils.

MTH 437        Systems Theory                                                                     4 units

Existence, boundedness and periodicity for solutions of linear systems of differential equations with constant coefficients. Lyapunov theorems. Solution of Lyapunov stability equations. ATP+PA=Q. Controllability and observability. Theorems on existence of solution of linear systems of differential equations with constant coefficients.

MTH 438        Electromagnetism                                                               3 units

Maxwell’s field equations. Electromagnetic waves and electromagnetic theory of light; plane electromagnetic waves in non- conducting media; reflected and refractional place- boundary. Wave- guides and resonant cavities. Simple radiating systems. The Lorentz-Einstein transformation. Energy and momentum. Electromagnetic 4- vectors. Transformation of (E.H.) fields. The Lorentz force.

MTH 439        Analytical Dynamics II                                                       3 units

Lagrange’s equations for non- holonomic systems. Lagrange’s multipliers. Variational principles. Calculus of variations. Hamilton’s principle. Lagrange’s equations of motion from Hamilton’s principle. Contact or canonical transformations. Normal modes of vibration. Hamilton- Jacobi equations for a dynamical system.

MTH 441        Mathematical Method II                                                      3 units

Calculus of variations: Lagrange’s functional and associated density. Necessary condition for a weak relative extremum. Hamilton’s principle. Lagrange’s equations and geodesic problems. The Du Bois Raymond equation and corner conditions. Variable end points and related theorems. Sufficient conditions for a minimum, isoperimetric problems. Variational integral transforms. Laplace, Fourier and Hankel transforms. Complex variable methods; convolution theorems; applications to solutions of differential equations with initial/boundary conditions.