B.Sc. Mathematics

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What practical work in biology involves. Laboratory organization. Handling common laboratory equipment. Microscopic handling and maintenance. Making microscopic measurements. Procuring animal materials for practicals. Killing, preserving and maintaining animal materials.

Procuring plant materials. External features of plants (differences and similarities). Preparation of temporary slides. Preparation of stains and reagents. Techniques for microbial culture and grain staining. Setting up demonstration for physiological processes in plants. Setting up apparatus for demonstrating physiological processes in animals. Preparation required for dissection.

Observation and description of the morphological and diagnostic features as well as the differences among the different phyla of the plant, animal, archebacteria, eubacteria, fungi and protista kingdoms. Identification of the taxonomic hierarchy of the members of the above groups. Study of the structure and functions of their parts and habitats specifications

Hypothesis, theory and law with appropriate illustrations, Nature of matter – 3 states of matter, Atomic structure, electronic energy levels and orbital. Periodic classification of elements and its relationship to their electronic configurations, Chemical bonding, Survey of properties and trends in groups I, II, IV, VI and transition metal,

Simple reactions of hydrocarbons, alcohols, and acids. Petroleum chemistry, Oils and fats, hydrogenation of oils, polymer and biologically important molecule.

Mole concepts and calculations based on it, methods of expressing concentrations, Chemical Kinetics and equilibrium, and related calculations, Important application of equilibrium – pH, solubility products and solubility of ionic solids, Thermo chemistry and simple calculations based on Hess’s law, Electrochemistry and working of various cells, Brief mentions of corrosion; chemical thermodynamics; DG = DH – TDS

Practical based of CHM 101 and CHM 103: Cations and anions – identification, Acid- base titrations, Redox reactions and determinations

Practical based on general chemistry CHM 101 and introductory organic chemistry I CHM 102- Determination of melting and boiling points and reaction of functional groups.

Listening enabling skills, listening and comprehending comprehension,  note taking and information retrieval. Including data, figures, diagrams and charts. Listening for main idea, interpretation and critical evaluation.

Effective reading. skimming and scanning. Reading and comprehension at various speed levels. Vocabulary development in various academic contexts.

Reading diverse texts in narratives and expository. Reading and comprehension passages with tables, scientific texts. Reading for interpretation and critical evaluation.

Writing paragraphs: Topic sentence and coherence. Development of paragraphs: illustration, Description, cause and effect including definitions.

Formal letters; essential parts and stylistic forms, complaints and requests; jobs, ordering goods, letters to government and other organizations. Writing reports; reporting event, experiments.

Writing summaries: techniques of summarizing letters and sounds in English, vowels and consonants.

Interviews, seminar presentation, public speech making, articles, concord and sentences including tenses. Gerund, participles, active, passive and the infinitive. Modal auxiliaries.

Nature of science, scientific methods and theories; Law of nature,; History of science. Lost sciences of Africa, science, technology and inventions. Nature and scope of philosophy in science. Man, nature and his origin. Man, environment and resources. Great  Nigerian Scientists.

Getting started: How to use the book, why read about skills, getting yourself organised ; what is studying all about, reading and note taking; Introduction, reactions to reading, your reading strategy, memory, taking notes, conclusion.  Other ways of studying: Introduction, learning in groups, talks and lectures, learning from TV and radio broadcasts, other study media. Working with numbers; Getting to know numbers, describing the world, describing with the tables, describing  with diagrams and graphs; What is good writing? The Importance of writing, what does an essay look like, what is a good essay? Conclusion.

How to write essays: Introduction, the craft of writing, the advantages of treating essay writing as a craft, making your essay flow, making a convincing case, the experience of writing.  Preparing for examination.

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 numbers; the Argand Diagram. Re Moivre’s theorem, nth roots of unity. Circular measure, trigonometric functions of angles of any magnitude, addition and factor formalae.

Function of a real variable, graphs, limits and idea of continuity. The derivative as limit of rate of change, Techniques of differentiation, Extreme curve sketching. Integration as an inverse of differentiation, Methods of integration, Definite integrals; Application to areas and volumes

Geometric representation of vectors in 1-3 dimensions, components, direction cosines. Addition and Scalar multiplication of vectors and linear independence.

The Scalar and vector products of two vectors. Differentiation and integration of vectors with respect to a scalar variable. Two-dimensional co-ordinate geometry. Straight lines, circles, parabola, ellipse, hyperbola. Tangents, normals. 

Measures of Central Tendency and dispersion, (grouped and ungrouped); mean: – arithmetic and geometric, harmonic, median, mode quartiles, deciles, modes, relative and absolute dispersion, sample space and events as sets. Finite probability space properties of probability.  Statistical independenceand conditional probability. Tree diagram.  Bayes theorem. Discrete and continuous random variables. Expectation, independent Bernoulli trials. Binomial Poisson and Normaldistributions. Normal approximation to binomial and Poisson distribution, Hyper geometric.

Space and Time: Physical quantities: Units and dimensions of physical quantities; Kinematics:  Uniform velocity motion, uniformly accelerated motion; Dynamics:   Newton’s laws of motion; Impulse and Linear Momentum, Linear Collision, Newton’s universal law of gravitation; Work, energy and power; Conservation laws; Concept of mechanical equilibrium; Centre of mass and centre of gravity; Moment of a force; Rotational kinematics and dynamics: Torque; Moment of Inertia; angular momentum; Total mechanical energy. Simple harmonic motion

Heat and temperature, work and heat, Quantity of heat: heat capacities, latent heat; Thermal expansion of solids, liquids and gases; Gas laws, heat transfer; Laws of thermodynamics: Isothermal and Adiabatic changes, Carnot cycle; Application kinetic theory of gases; van der Waals gas.

Classification of matter into (solids, liquids and gases, forces between atoms and molecules, molecular theory of matter, Elasticity, plasticity, Hook’s Law, Young’s Shear and bulk Moduli) Crystalline and non-crystalline materials, Hydrostatics:   pressure,   buoyancy, Archimedes’ principle; Hydro-dynamics-streamlines, Bernouli and Continuity equations, turbulence, Reynold’s number, Viscosity, laminar flow, Poiseuille’s equation; Surface tension, adhesion, cohesion, capillary, drops and bubbles.

Electrostatics: Coulomb’s law, Gauss’s law, potential and capacitance, dielectrics, production and measurement of static electricity.

Current: Ohm’s law, resistance and resistivity, heating. Galvanometers, Voltmeters and Ammeters; D.C. circuits: sources of emf and currents, Kirchhoff’s laws; Electrochemistry; The Earth’s magnetic field; Magnetic fields and induction, Faraday’s and Lenz’s laws; Force on a current-carrying conductor. Biot-Savart law.

Flemming’s right and left-hand rules, motors and generators. A.C. Theory. Atomic structure; Production and properties of X-rays; Radioactivity; Photoelectric emission.

Graphs, Measurement, Error Analysis, Determination of Acceleration due to Gravity by Means of Simple Pendulum, Determination of force constant of a spiral spring, Determination of effective mass of a spiral spring and the constant, Determination of surface tension of water, Determination of specific latent heat of fusion of ice, Determination of the co-efficient of limiting static friction between two surfaces, Determination of the co-efficient of static friction on two surfaces using an inclined plane, Determination of Relative Density of kerosene using the specific Gravity Bottle, Determination of the Relative Density of a Granular substance not soluble in water using the specific gravity bottle.

Refraction through the glass block; Image formed by a concave mirror; Determination of the focal length of the convex mirror; Refraction through the triangular prism; Determination of the focal length of a converging lens and the refractive index of groundnut; Determination of resistance of resistors in series and in parallel in simple circuits; Determination of internal resistance of a dry cell using a potentiometer; To compare the E.M.F. of cells using potentiometer; Determine the unknown resistance of a resistor using Wheatstone Bridge; To determine the relationship between current through a Tungsten and a potential applied across it.

General introduction to logic; clarity of thought; expression and arguments as basis for conclusion.

Fundamentals of logic and critical thinking, types of discourse, nature of arguments; validity and soundness ; distinction between inductive and deductive inferences etc;  illustrations from familiar texts, including literature materials, novels, law reports and newspaper publications.

Complex number, the topology of complex plane. 

Limits and continuity of function of complex variables, properties and example of analytic functions, branch-points, Cauchy-Riemann equations. Harmonic function.

Set: Binary operations, mapping, equivalence relations integers:  Fundamental theorem of arithmetic, congruence equations, Euler’s function (n).

Group Theory: Definition and examples of groups.  Subgroups, coset decomposition, Lagrange’s theorem.  Cyclic groups.  Homeomorphisms, isomorphism.  Odd and even permutations. Cayley’s theorem. 

Rings:  Definition and examples of rings.  Commutative rings.  Integral domain.  Order, well-ordering principles.  Mathematical induction.

Vector spaces.  Liner independence.  Basis, change of basis and dimension. 

Linear equations and matrices.  Linear maps.  The diagonal, permutation, triangular matrices.  Elementary matrix.  The inverse of a matrix.  Rank and nullity.  Determinants. 

Adjoint, cofactors, inverse matric.  Determinant rank.  Crammer’s rule.  Canonical forms, similar matrices, Eigen values and vectors, quadratic forms.

Interpolation:  Lagrange’s and Hermite interpolation formulae, divided differences and difference schemes.  Interpolation formulas by use of divided differences.  Approximation: Least-square polynomial approximation, Chebychev polynomials continued fraction and rational fraction orthogonal polynomials.

Numerical Integration: Newton’s-cotes formulae, Gaussian Quadrature. Solution of Equations: Graffe’s method (iterative method) Matrices and Related Topics: Definitions, Eigenvalue and Eigenvectors, Algebraic Eigenvalue problems-power method, Jacobi method.

Systems of linear Equations: Gauss elimination, Gauss-Jordan method. Jacobi iterative method, Gauss-field iterative method.

Introduction, equation of first order and first degree, separable equations, homogeneous equations, exact equations, linear equations, Bernoulli’s and Riccati equations. 

Applications to mechanics and electricity. Orthogonal and oblique trajectories.  Second order equations with constant coefficients.

Sets: Cartesian products, functions and mappings direct and inverse images.  Countable sets.

Limits: Elementary properties of limits.  Upper and lower bounds, supremum, infimum, convergence of sequences.  Limit of monotone functions and sequences.  Cauchy convergence principles. 

Continuity: Real-Valued functions of a real variable Monotone functions, periodic functions, bounded functions.  Continuity of functions using neighborhood.  Elementary properties of continuous functions.  Uniform continuity.  Series: convergence of series, tests for convergence, absolute convergence, power series, uniform convergence.

 Static:  System of live vectors.  Coyoles and wrenches.  Principles of virtual work.  Stability of equilibrium. 

Dynamics of systems of particles: Elastic strings.  Hooke’s law.  Motion in resisting media.  Changing mass.  Motion along a curve. Frenet’s formulae.

Coplanar Motion:  Energy equation.  Motion in a vertical circle.  Simple pendulum.  The cycloid and cycloidal motion.  Orbital motion-disturbed orbits and stability.

Sequences and Series:  Limits, continuity, Differentiability, implicit functions, sequences.  Series, test for convergence sequences and series of functions. 

Calculus: partial differentiation, total derivatives, implicitly functions, change of variables. Taylor’s theorem and maxima and minima functions, of two variables.  Lagrangian multiplier. 

Numerical Methods: Introduction to iterative methods, Newton’s method applied to finding roots.  Trapezium and Simpson’s rules of integration.

Vector Theory:  Vector and scalar field functions. Grad, divi, curl, directional derivatives.  Orthogonal curvilinear coordinates.

Complex Numbers:  The algebra and geometry of complex numbers; de’moivre’s theorem.  Elementary transcendental functions.  The n^th root of unity and of a general complex number.

Atomic structure: Experimental basis of quantum theory: Black body radiation; electrons and quanta; Charge quantization, Mass spectra, the plum pudding model, Rutherford model and Bohr models of the atom, Hydrogen spectra, Magnetic moment and Angular momentum of an atom, Electron spin, Pauli Exclusion Principle and electronic configuration, X-ray spectra, De Broglie hypothesis, the uncertainty principle; Wave-particle duality, Schrodinger’s equation and simple applications; Nuclear Structure: nomenclature, binding energy and stability, Radioactivity, The radioactive series, Accelerators, Detectors. Bohr’s theory of atomic structure.

Macroscopic properties of dielectrics: polarisation, Gauss’s law in a dielectric, the displacement vector, boundary conditions on D and E, dielectric strength and breakdown; Capacitor: capacitance, the parallel plate capacitor, effect of a dielectric, energy stored in a dielectric medium, capacitors in series and parallel, practical capacitors; Microscopic properties of dielectrics: microscopic picture of a dielectric in a uniform electric field, determination of local field, Clausius-Mossotti equation, behaviour of dielectric in alternating fields; Magnetism of materials: response of various substances to a magnetic field, magnetic moment and angular momentum of an atom, diamagnetism and paramagnetism, Lamor precession, magnetization of paramagnets, ferromagnetism, magnetic field due to a magnetized material, magnetic intensity, relationship between E and H for magnetic material, magnetic circuits.

Nature of light: the corpuscular model, the wave model, light as an electromagnetic wave;

Reflection and refraction of light: electromagnetic waves at the interface separating two media, idealisation of waves as light rays, Fermat’s principle;

Perception of light: human vision, colour vision;

Polarisation of light: simple states of polarised light, principles of producing linearly polarised light, wave plates.

Discrete sample spaces: Algebra and probability of events, combinatorial analysis.  Sampling with and without replacement. 

Conditional probability, Bayes theorem and stochastic independence. Discrete distributions: Binomial, Poisson, negative binomial-hyper geometric and multinomial. 

Normal approximation to binomial and Poisson, Poisson approximation to binomial.

Random variables and expectations: mean, variance, covariance.  Probability generating function and moment generating function.  Cheycher’s inequality. 

Continuous joint distributions: marjind as conditional density.

Expectations: movement, movement generating functions.  Uniform normal, beta Cauchy and hop-normal distributions.

Series, solution of second order linear equations.  Bessel, legendry and hyper geometric equations and functions.  Gamma  and Beta functions. 

Storm Lionvelle problems.  Orthogonal polynomial and functions, Fourier, Fourier, Bessel and Fourier – legendry series.  Expansion in series of orthogonal functions. 

Fourier transformation.  Laplace transforms solution of wave and heat equations by Fourier method.

Vector algebra, Vector dot and cross products.  Equation of curves and surfaces.  Vector differentiation and application.  Gradient, divergence and curl. 

Vector integration, line, surface and volume integrals, Green stoke’s and divergence theorems. 

Tensor products and vector spaces tensor algebra, symmetry, Cartesian tensors.

Functions of a complex variable.  Limits and continuity of functions of a complex variables.

Deriving the Cauchy-Riemann equations. Analytic functions.  Bilinear transformations, conformal mapping.

Contour Integrals, Cauchy’s theorems and its main consequences. Convergence of sequences and series of functions of complex variables. Power Series, Taylor Series.

Laurent expansions, isolated singularities and residues, residue theorem, calculus of residue and application to 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.

Polynomial and Splines approximations:  Orthogonal polynomials and chebychev approximations, least squares, cubes spline, Hermits approximations, Numerical Integration.  Boundary value problem.  Introduction to numerical solution of partial differential equations.

Polynomial and Splines approximations:  Orthogonal polynomials and chebychev approximations, least squares, cubes spline, Hermits approximations, Numerical Integration. 

Boundary value problem.  Introduction to numerical solution of partial differential equations.

Methodology of the Model building.

Identification, formulation and solution of problems, cause – effect diagrams, equation types, algebraic, ordinary differential, partial differential, difference, integral and functional equations.

Application of Mathematical model to physical, biological, social and behavioural sciences.

Linear programming models.  The simplex method, formulation and theory.  Duality, integer programming. Transportation problem, two-person zero-sum  games.

Non – linear programming, quadratic programming Kuhn tucker methods, optimality criteria simple variable optimization.  Multivariable techniques, Gradient methods.

Value, Limit and Continuity of functions of several variables. Partial derivatives of functions of several variables.  Total derivative of a function. Partial Differentials and Total Differentials of f(x,………x _n).

Composite differentiation. Fuller’s   Theorem. Implicit Differentiation.   Taylor’s Series for functions of two variables.  Maxima and Minima of functions of several variables.  Lagrange’s  Multipliers.  Differentials  under the integral sign, The Jacobians

Normal subgroups and quotient groups.  The isomorphism theorem.  Symmetric groups, automorphism, conjugate classes, Normalisers. 

The sylow theorems.  Normal and composition series.  The Jordan-Holder theorem.  Direct product. 

Solvable group.  Isomorphism theorems for rings.  Ideals and quotient rings.  Commutative ring, maximal ideals.  Euclidean rings, principal ideal domain and unique factorization domain.

Degrees of freedom, Holonomic and non-holonomic constraint.  Generalized coordinates.  LaGrange’s equation for holonomic systems, force dependent on coordinates only, force obtainable from a potential, Impulsive force, variational principles, calculus of variation, Hamilton principles. 

Canonical transformation, normal modern of variation, Hamilton Jacobi equation.

The notion of displacement, speed, velocity and acceleration of a particles.  Newton’s law of notions and applications to simple problems. 

Work, power and energy.  Application of the principle of conservation of energy to notion of particles and those involving elastic string and springs.  Simple Harmonic motion.

Resultant of any number of forces acting on a particle. Reduction of coplanar forces acting on a rigid body to a force and a couple.  Equilibrium of coplanar forces, parallel forces, couples Laws of friction.  Application of the principle of moments.  Moments of Inertia of simple bodies.

Real and Ideal fluid. Differentiation following the motion of fluids particles, Equation of continuity. Equation of motion for incompressible in viscid fluids.

Velocity potential and stoke’s stream function. Bernoulli’s equation with applications.

Kinetic Energy. Sources, sinks, doublets in 2 and 3 dimensions stream lines. Images. Use of conformal transformation.

Integration:  The integral as the area of the ordinate set of a function.  Definition of the Riemann integral of bounded functions.  Conditions for integrality. Properties of the integral.  Relations between integrals and derivatives.  Approximation to integrals by sum.

The Riemann Integral: Riemann-Sieltejes integral.  Properties, functions of bounded variation and extension to the notion of integration.

Sequences and Series of Functions:  Convergence of sequences and series of functions.  Uniform convergence.  Continuity of sum of a uniform convergent series of continuous functions.  Terms by term integration and differentiation of a series of continuous functions.  Applications to power spaces metric spaces.

Functions of several variables: Jacobian, functional dependence and independence.  Multiple integrals, line integrals.  Improper integrals. 

Vector Field theory: Relations between vector field functions.  Integral theorems.  Gauss’s. Stoke’s and Green’s theorems.  Elementary tensor calculus.

Functions of a complex variable:  The Cauchy-Riemann equations.  Integration of complex plane. Cauchy’s theorem Cauchy’s inequality.  The residue theorem and the evaluation of integrals.

Integral Transforms:  Fourier and Laplace transforms.  Convolution properties and their applications.

Ordinary Differential Equations: The concept of existence and uniqueness of solutions.  Operational methods of solution of linear equations.  Sturm-Lionvelle theory, Green’s functions, series solution.  Special functions and some of their elementary properties; Gamma and Beta functions.  Partial Differential Equations:  Solutions of boundary and eigenvalue problems of partial differential equations by various methods which include: Separation of variables, transform techniques.  Sturn-Liouville theory; Green’s functions; method of characteristics.

Probability spaces measures and distribution. Distribution of random variable spaces.  Product probabilities. Independence and expectation of random variables. 

Convergence of random variables. Week convergence almost everywhere, laws of large numbers. Characteristic function and inversion formula.

Vector random variables.  Expectations of random vectors and matrices. 

Multivariate normal distribution and distribution of quadratic forms.

Application to linear models:  Tests of general linear hypothesis and estimation.

Least square theory: Guass-Markoff and general linear hypothesis with applications to regression and experimental design models. 

Estimation:  partial and multiple correction coefficients, mean vector and co-variance matrix.  Hatelting’s T² and Wishart distribution: multivariate ANOVA.

Separation axioms: T-spaces, Hausdorff spaces, Regular spaces.  Normal spaces, Urgsohn’s lemma. 

Category and seperability: Dense sets, nowhere dense sets.  Sets of the first and second categories.  Perfectly separable spaces. Separable spaces.  The axiom of count ability. 

Compactness: Covers, compact sets, subsets of compact spaces.  Sequentially, count ably and locally sets.  Compactification. 

Product spaces: product topology.  Base for a finite product topology. Tychonoff product theorem.

Connectedness: separated sets, connected sets, connected spaces.  Connectedness of the real line.  Components.  locally-connected spaces.  Homotopic paths.  Homotopy relations.  Simple connected spaces.

Measure Theory: Measure of open, closed sets.  Outer and inner measure.  Measurable sets.  Properties of measure.  Non-measurable sets.  Measurable in the scene of Borel.  Measurable space.  Measurable functions.  Simple function Algebra. 

The Lebesgue integral:  Lebesgue measure.  Integral of non-negative function.  Integral as measure of ordinate set, as a limit of approximate sums. 

Integral of an unbounded function.  Integral over an infinite range.  Simple properties of the integral.  Sequences of integral (Positive functions; functions with positive and negative values). 

Lesbesgue monotone convergence theorem.  Fatou’s Lemma, Dominated convergence.  Bepo’s Lemma-Bounded Convergence.  Sets of measure zero.  Integration by parts.  Fubini’s theorem and applications to multiple integrals.

Normal Linear Space: Definition and examples.  Convex sets.  Norms. Holder’s minkowski’s inequalities.  Riese-Fisher theorem. 

Linear operations on finite dimensional spaces.  Linear functionals spaces.  Banach spaces, examples.  Quotient spaces.  Inner product spaces.  Topological linear spaces.  Hilbert space, examples.  Linear operators in Hilbert spaces.  Adjoint operators.  Hermitian operators.  Orthogonality; orthogonal complement and projections in Hilbert spaces.

Lagrange’s equations for non-holonomic systems.  Lagrangran multipliers.  Variational principles. 

Calculus of variation, Hamilton’s principle, Lagrange’s equation from Hamilton’s principles.  Canonical transformation Normal modes of vibrations.  Hamilton-Jacobian equations.

Lyapunov theorems.  Solution of the Lyapunov stability equation

A^TP + PA= -Q.  Controllability and observability.  Theorems on existence of solution of linear systems of differential operations with constant coefficient.

Algebraic numbers; quadratic and cyclotomic fields. 

Factorization into irreducible, ideals, Murkowski’s theorem, class-group and class number, Fermat’s last theorem Dirichilet’s unit theorem.

Maxwell’s field equations.  Electromagnetic waves and electromagnetic theory of lights.  Plane detromagnetic waves in non-conducting media, reflection and refraction of plane boundary.  Wave guide and resonant cavities.  Simple radiating systems.  The Lorentz-Einstein transformation.  Energy and momentum.  Electromagnetic 4-Vectors.  Transformation of (E.H) fields.  The Lorentz force.

Existence and uniqueness theorems, dependence of solution on initial data and parameters. 

Properties of solutions.  General theory for linear differential equation with constant coefficients, the two-point Sturm-Liouville boundary value problem, self-adjointness, linear and non-linear equations, Theorems and solution of Lyapunov equation.  Controllability and observability.

Theory and solutions of first order equations.  Second order linear equations.  Classification, characteristics canonical forms, Cauchy problem. 

Elliptic equations.  Laplace’s and Poisson’s formulae, properties of harmonic functions. 

Hyperbolic equations, retarded potential transmission line equation, Riemann methods, parabolic equation, diffusion equation, singularity function boundary value and initial value problems.

Integral Equation: Classification – Volterra and Fredholm types.  Transformation of Differential Equations.  Neumann series. Fredholm alternative for degenerate Hilbert – Schmidt Kernels.  Reduction of ordinary differential equation to Integral equations.  Symmetric Kernels, eigen function expansion with applications.

Minimal polynomial of an algebraic number. Eisenstein’s irreducibility criterion. Splitting fields and normal extension.

Primitive element theorem. Galois group of a polynomial. Field degrees and group orders. The Galois group of a polynomial. Field degrees and group orders. The Galois correspondence. The fundamental theorem

Governing equations of viscous flow, exact solutions, Low Reynolds’s number solutions, Boundary layers, compressible flows.

Individual or Group projects of approved topics related to the current research interests in the department.