MATHEMATICS COURSE OUTLINE

University of Nigeria, Nsukka.

Note: Depending on the lecturer teaching the course and nature of the text used for a particular course, there may be completely, little or no change to the following course outline:

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. Complex numbers; algebra of complex number, the Argand Diagram. De Moivre’s theorem, nth roots of unity. Circular measure, trigonometric functions of angles of any magnitude, addition and factor formulae.

MTH      121         Elementary Mathematics II | 3 Units

Function of a real variable, graphs, limits and continuity. The derivative as limit of a 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 variable. Two-dimensional co-ordinate geometry. Straight lines, circles, parabolas, ellipses, hyperbolas. Tangents and normal. Kinematics of a particle. Components of velocity and acceleration of a particle moving in a plane. Force and momentum; Newton’s laws 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; centriods and centres of gravity of particles and rigid bodies; equivalence of sets of coplanar forces. Kinematics and rectilinear motion of a particles, vertical motion under gravity, projection; relative motion. Dynamic 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 straight lines in various forms. The circle. Trigonometric functions; logarithmic functions; exponential functions. Maximal, 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 surface integrals. Gauss(divergence). Green’s and stokes’ Theorems. Complex numbers and functions of 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; second order ordinary differential equations and methods of solution. Partial differential equations: second order partial differential equations and methods of solution.

MTH      205         Advanced Mathematics VI | 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 V | 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 second order ordinary differential equations; partial differential equations, with application.

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 space over the real field. Subspaces, linear independence, bases and dimensions. Gramschmidt orthogonalization procedure. Linear transformations: range. Null space and rank. Singular and non-singular transformations.

MTH      213         Three-Dimensional analytic Geometry | 2 Units

Plane curves, parametric representations, length of plane arc, lines in three space. 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 diagonalisation. Canonical forms.

MTH      221         Real Analysis I | 3 Units

Bounds of real numbers, convergence of sequences of numbers. Monotone sequences, the theorem of nested intervals, Cauchy sequences, tests for convergence of series. Absolute and conditional convergence of series and re-arrangement. Completeness of real and incompleteness of rationals. Continuity and differentiability of functions. Rolle’s and mean-value theorems for differential functions. Taylor series.

MTH      222         Elementary Differential Equations I | 3 Units

First-order ordinary differential equations. Existence and uniqueness of solution. Second-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         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; 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, extreme, 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 cossets. Lagrange’s theorem and applications, Cyclic groups. 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.

MTH      312         Abstract Algebra II | 3 Units

Normal subgroups and quotient group Homomorphism, Isomorphism theorems. Cayley’s Theorems. Direct products. Groups of small order Group acting on sets. Sylow theorems, ideal and quotient rings, P.I.D.’s, U.F.D.’s, Euclidean rings. Irreducibility. Field and transcendental extensions. Straight-edge and compass constructions.

MTH      313         Geometry I | 2 Units

Coordinates in R³ . Polar coordinates; distance between points surface and curves in space. The plane and straight line.

MTH      314         Geometry II | 2 Units

Introductory projective geometry. Affined and Euclidean geometries.

MTH      314         Differential Geometry | 3 Units

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

MTH      321         Metric Space Topology | 3 Units

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

MTH      322         Elementary Differential Equations II | 3 Units

Series solution of second-order ordinary differential equations. Strum-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).

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, Stokes’ and divergence theorems. Tensor products of vector spaces. Tensor algebra. Symmetry. Cartesian tensors and applications.

MTH      325         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      326         Real Analysis II | 3 Units

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

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, equation, equations. Applications of mathematical models to physical, biological, social and behavioural sciences.

MTH      332         Optimization Theory I | 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      333         Optimization Theory II | 2 Units

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

MTH      334         Analytical Dynamics | 3 Units

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

MTH      335         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. Principle 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 Foucault’s pendulum. Euler’s dynamical equations of motion of a rigid body with one point fixed. The symmetric top. Processional motion.

MTH      336         Introduction to Operation Research | 3 Units

Phases 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      337         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 particle, Maxwell’s theory in a Relativistic form. Optical phenomena.

MTH      341         Discrete Mathematics I | 3 Units

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

MTH      342         Discrete Mathematics II | 3 Units

Lattices and Boolean algebra. Finite fields: Mini-polynomials, irreducible polynomials, polynomial roots. Applications (error correcting codes).

MTH      411         Abstract Algebra II | 3 Units

Splitting field. Separability. Algebraic closure. Solvable groups. Fundamentals theorem of Galois Theory. Solution by radicals. Definition and examples of modules, sub modules and quotient modules. Isomorphism theorems. Theory of group representations.

MTH      421         Ordinary Differential Equations | 3 Units

Existence and uniqueness of solution; dependence on initial conditions and on parameters, general theory for linear differential equations with constants coefficients. The two-point Sturn-Liouville boundary value problem; self-adjointness; sturn theory; stability of solutions of nonlinear equations; phase-plane analysis.

MTH      422         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 laws and polar identity in Hilbert space H; the natural embeddings of operators including the open mapping and closed graph theorem; the spaces C(X), the sequence (Banach) spaces, Ip and c (= space of convergent sequences).

MTH      423         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, second-order partial differential equation in two independent variables, existence and uniqueness of solution; normal forms. Boundary and initial-valued-problems for hyperbolic, elliptic and parabolic partial differential equations.

MTH      424         General Topology | 3 Units

Topological spaces, definition, open and closed sets, neighborhoods, 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, homomorphism, topological invariants, spaces of continuous functions. Point-wise and uniform convergence.

MTH      435         Lebesgue Measure and Integration | 3 Units

Lebesgue measure: measureable and non-measureable 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’’ and Stokes’ theorems. Curvilinear coordinates. Simple notion of tensors. The use of tensor notations.

MTH      431 General Theory of Relativity | 3 Units

Particles in a gravitational field: Curvilinear co-ordinates, 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      432         Electromagnetism | 3 Units

Maxwell’s field equations. Electromagnetic waves and electromagnetic theory of light; planes 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 vectors. Transformation of (E.H.) fields. The Lorentz force.

MTH      433         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 Stoke’s 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 aerofoil’s.

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      435         Quantum Mechanics | 3 Units

Particle-wave duality. Quantum postulates. Schrodinger’s equation of motion. Potential steps and wells in one-dimensional Heisenberg formulation. Classical limit of Quantum mechanics. Poission brackets. Linear harmonic oscillator. Angular momentum. Three dimensional square well potential. The hydrogen atom. Collision in three-dimensions. Approximation methods for stationary extremum problems.

MTH      436         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      437         Systems Theory | 4 Units

Lyapunov theorems. Solution of Lyapunov stability equations ATP+PA = Q. Controllability. Theorems on existence of solution of linear systems of differential equations with constant coefficient.

MTH      441         Mathematical Methods II | 3 Units

Calculus of vraitions: 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. Variation integral transforms. Laplace, Fourier and Hankel Transforms. Complex variable methods; convolution theorems; applications to solutions of differential equations with initial/boundary conditions.