# MTS COURSE SYNOPSIS

COURSE SYNOPSES

MTS 101        ALGEBRA (3 UNITS)

Elementary Set Theory:  Set notations, terminologies and operations.  Algebra of sets.  Venn
diagrams and applications.  Real Numbers:  Number systems from Natural to Reals.  Operations
on Real Numbers, indices, Logarithms and Surds.  Mathematical induction.   Polynomials and Rational Expressions:  Remainder and Factor Theorems, Partial Fractions.  Inequalities. Equations  in one variable:  Theory of quadratic equations, Cubic equations and equations reducible to quadratic type.  Simple simultaneous equations.  Sequences and Series:  AP and GP. Means, nth term and limits.  Binomial Theorem for any index.  Binomial series. Matrices (m, n, < 3): Notations.  Algebra of matrices Determinants.  Inverse of matrix and solution of linear
system of equations.  Complex Numbers:  Algebra of complex numbers.  Argand diagram.  De
Moivre’s Theorem, and nth roots of unity.

MTS 102      CACULUS AND TRIGONOMETRY  (3 UNITS)

Functions of a real variable and their graphs.  Limits, and idea of continuity of functions.  Removable discontinuity. Trigonometry:  Circular measures and Pythagorean identities.  Trigonometric  functions of any angles, compound angles.  Inverse Trigonometric functions, solution of trigonometric equations. The derivative as limit of rate of change.  Techniques of differentiation of elementary functions.  Application of derivative to errors and approximation, minima and maxima, curve sketching etc.  Integration as inverse of differentiation.  Methods of
integration.  Definite integrals.  Applications to areas and volumes etc.

MTS 103           VECTORS AND GEOMETRY   (2 UNITS)

Equations of straight lines, intersecting and perpendicular lines, equations of lines and planes in R3 conic sections; circle, parabola hyperbola and ellipse.  Geometric representation of vectors in
R2 and R3, components, direction cosines.  Addition, scalar, multiplication of vectors, linear independence.  Scalar and vector product of two vectors.  General equation of a conic in polar
coordinates.

MTS 104            MECHANICS   (3 UNITS)

(The vector approach should be used in what follows).  The notions of displacement, speed,
velocity and acceleration of a particle.  Newton’s laws of motion and application to simple problems.  Work, power and energy.  Application of the principle of conservation of energy to motion particles and those involving elastic strings and  springs.  Collision of smooth spheres.
Simple problems projectiles.  Simple pendulum and simple harmonic motion.  Resultant of any number of forces acting on particle.  Reduction of coplanar forces.  Equilibrium of coplanar forces, parallel forces, couples.  Laws of friction.  Applications of the principles of moments.
Moments of inertial of simple bodies.

MTS 105:       ALGEBRA AND TRIGONOMETRY FOR BIOLOGICAL SCIENCES  (3 UNITS)

Use of Mathematics in Agriculture.    Elementary Set Theory:  Set notations.   Set operations.
Algebra of sets.  Venn diagram.  Applications.   Operations with Real Numbers: Indices and Logarithms.  Surds.  Use of Logarithms in Agricultural Sciences.  Remainder and Factor theorems.  Partial Fractions.   Equations and Inequalities: Linear and Quadratic inequalities. Theory of quadratic equations.  Cubic Equations.  Equations reducible to Quadratic type. Sequences and Series: Arithmetic and Geometric Progression.  Arithmetic mean and Geometric
mean.  Arithmetic Series.  Geometric series.  nth term of a series.  Binomial theorem.  The General term. Binomial series.  Matrix Algebra: Matrices.  Algebra of Matrices.  Determinant of a Matrix.  Properties.  Inverse of a matrix.  Solution of Linear system of Equations.  Elementary Trigonometry: Degree and Radian Measures.  Pythagorean Identities.  Trigonometric functions of any Angle.  Graphs.  Inverse trigonometric functions.  Compound Angles.  Solution of
trigonometric Equations.

MTS 106         CALCULUS FOR BIOLOGICAL SCIENCES   (3 UNITS)

Functions in Agriculture.  Domain and Range of a function.  Graphs of Elementary functions.
One-to-one and Onto Functions.  Composite function.  Applications to Agricultural Sciences.
Limits and Continuity: Limits.  Algebra of Limits.  Continuity and discontinuity of functions.  Removable discontinuity.   Differentiation:  Geometrical meaning of derivatives.  Algebra of
differentiable functions.  Implicit differentiation.  Logarithmic differentiation.  Higher derivatives etc.  Applications of derivatives:  Errors and Approximations.  Minima and Maxima.  Curve sketching etc.  Applications to Agricultural Sciences.   Integration  as Inverse of Differentiation.  Indefinite and Definite Integrals.  Methods of Integration. Applications to area under a curve.  Surface Areas and volumes etc.  Co-ordinate Geometry: Slope and midpoint of a line.  Equations of a straight line.  Parallel and perpendicular lines.  Equations of a circle, Parabola, Ellipse and hyperbola.  Tangents and Normal

MTS  201      MATHEMATICAL FOUNDATIONS (3 UNITS)   PRQT, MTS 101 OR MTS 102

Review of integration and its applications to area, volume, controids and center of mass. Sequence, series, power series and their convergence.  Equations of line and circles.  Conic sections.  First and second order ordinary differential equations.  Partial derivatives, total derivatives and applications.  Double and multiple integrals.  Introduction to vector spaces, linear algebra and matrices.  Eigenvalues and eigenvectors and application of matrices to a system of linear equations.

MTS  211          ABSTRACT ALGEBRA (2 UNITS) PRQT MTS 101

Set: Binary operations, mappings, equivalence relations.  Cartesian products.  Number theory:
Divisibility and primes, Fundamental theorem of arithmetic, congruencies, linear congruence
equations, Euler’s function.  Group theory: Definition and examples of groups.  Subgroup, coset
decomposition, Lagrange’s theorem, cyclic groups.  Homomorphism, isomorphism,
Permutations groups.  Cayley’s theorem.
Rings: Definition and examples of rings.  Commutative rings.  Integral domain, division rings.
Fields, construction of the field of fractions of an integral domain and the embedding theorem.

MTS  212  LINEAR ALGEBRA (3 UNITS)  PRQT MTS 101 OR MTS 103

Vector spaces over the real field, subspaces, sum and direct sum of spaces. Linear independence, Basis, and dimension.  Linear transformations and matrix representations; range, null space and rank.  Singular and non-singular transformations and matrices.  Matrix algebra. Triangular matrices.  Elementary matrix.  Rank and nullity.  Determinants.  Adjoints, Cofactors, inverse matrix and solution of system of linear equation.  Determinant and rank.  Cramer’s rule.
Equivalent and similar matrices.  Minimum and characteristic polynomials.  Eigenvalues and eigenvectors vectors, bilinear and quadratic  forms.

MTS  223         REAL ANALYSIS  (3 UNITS)  PRQT MTS 101 OR MTS 103

Sets.  Real and complex numbers.  Convergence and divergence of sequences and series of complex numbers.  Mapping, functions of  real variables, continuity and differentiability. Taylor’s theorem extensions and applications.  Introduction to Reimann integration.

MTS 232      ORDINARY DIFFERENTIAL EQUATIONS (2 UNITS)  PRQT MTS 102

Derivations of equations from physics, geometry, biology, etc.  Techniques for solving first and second order linear and non-linear equations  and for  solving the n-th order linear equations.
Finite differences and difference equations.  Interpolation, error, solution of  equations, elementary numerical integration.

MTS   242      MATHEMATICAL METHODS I  (3 UNITS)  PRQT MTS 101 OR MTS 102

Real-valued functions of real variable.  Review of differentiation and integration and their applications.  Mean value theorem.  Taylor’s series.  Functions of several variables, Jacobian function, dependence and independence, multiple integrals, line integrals, improper integrals .Relations between vector field functions, integral theorems,.  Gauss’s Stoke’s and Green’s heorems.  Elementary tensor calculus.  Fourier and Laplace transforms, convolution properties,
linear integral equations.

MTS 311         GROUPS AND RINGS  (3 UNITS)  PRQT MTS 211

Groups, subgroups, cyclic groups.  Lagrange’s theorem and applications, normal subgroups, quotient groups.  Cayleys theorem.  Groups acting on sets.  Sylow theorems.  Commutators, direct product, composition series.  Rings, quotient rings, Isomorphism theorems, prime and maximal ideals, principal ideal domains, Euclidean domain, unique factorization domains.

MTS 314  THEORY OF MODULES (2 UNITS)  PRQT MTS 211 OR MTS 212

Modules, sub modules quotient modules, isomorphism theorems.  Polynomial and power series in several variables, symmetric polynomials.  Finitely generated modules over  principal ideal domains with applications to abelian groups.  Bilinear and quadratic forms.  Multilinear algebra, tensors, exterior and symmetric products.

MTS 321     COMPLEX ANALYSIS I (3 UNITS)  PRQT MTS 223 OR MTS 212

Function of a complex variable.   Limits and continuity of function of a complex variable, analytic functions, complex integrations, Cauchy’s integral.  Derivative theorems.  Taylor’s and Laurent’s theorems.  Classification of singularities.  Convergence of sequence and series of complex  functions (including power series and characterization of analytic functions by power series).  Isolated singularities and residues.  Residuce theorem.  Rouche principle.  Argument principle of theorem.  The fundamental theorem of algebra.  Principle of analytic continuation.  Multiple valued functions and Riemann surfaces.

MTS 322      VECTORS AND TENSORS ANALYSIS  (2 UNITS) PRQT MTS 212

Revision of elementary vector algebra.  Index notation.  Equation of curves surfaces.  Differentiation of vectors and applications.  Gradient, divergence and curl.  Lines, surface and volume integrals.  Divergence.  Theorem, Green’s and stoke’s theorems.  Tensor products of vector spaces.  Tensor algebra.  Symmetry.  Cartesian tensors, transformation law, Gauss’ theorem, the quotient rule.

MTS 323  REAL ANALYSIS II  (3 UNITS)  PRQT MTS 223

Differentiation, directional derivatives, partial derivatives and higher order derivatives, Taylor’s theorem, inverse function theorem.  Implicit function theorem.  Extrema and method of lagrange multipliers.  Riemann integral, Reimann-Stieltjes integral functions of bounded variation.  Partial integration formula.  Mean value theorems.  Integration of function of several variables.

MTS 342       MATHEMATICAL METHODS II (3 UNITS)  PRQT MTS 232, MTS 242

Boundary value problem  –  eigenvalues and eigenvectors, linear dependence.  Wronskian, reduction of order, variation of parameters, series solutions near an ordinary and a regular, singular points, special functions, Bessel, Legendre and  hypergeometric equations and functions, Gama and Beta functions.  Laplace transforms and applications to initial value problems.

MTS 354  MATHEMATICAL MODELLING I (3 UNTIS) PRQT MTS 212 OR MTS 232

Introduction and Methodologies. Dimensional  analysis, scaling, approximation and reasonableness of answers.  Elements of dynamical system.  Solutions to linear state equations,
controllability and observability.  Application to deterministic and stochastic processes, examples from life, physical, biological, social and behavioural sciences.  Simple algorithms to handle simple discrete models.

MTS 362  METRIC SPACES  (3 UNITS)  PRQT MTS 223

Metric Spaces: Definition and examples, open sets, neighbourhoods, closed sets, interior, exterior, frontier, limit points, closure, dense subset, separable spaces, continuity, compactness
and connectedness.  Point set topology.  The space Rn with Euclidean metric, metric topologies.  Equivalent metric.  Heine-Borel theorem, Bolzeno – Wierstrass theorem.  The Cantor set.

MTS 363    INTRODUCTION TO OPERATIONS RESEARCH (2 UNITS) PRQT MTS 223 OR MTS 212

Phases of operations research, study, Modelling, linear, dynamic and integer programming.  Probabilistic models.  Decision theory and games traffic flow, network,  flow project controls,
inventory models.

MTS 401        INDUSTRIAL ATTACHMENT (3 UNITS)

Every student in the Department of Mathematics must undergo an Industrial Attachment in a
place relevant to the student’s area of interest during the long vacation of  the penultimate year.  Report of the acquired experience will be typed, bound, submitted and presented in form of a seminar at a time to be announced during the first semester.

MTS 411        ADVANCED ALGEBRA I (3 UNITS)  PRQT MTS 311 OR MTS 314

Lattice  theory, Neotherian and Artinian modules and rings.  Hilbert basis theorem.  Chinese remain theorem.  Semi-simple modules and rings.  Prime spectrum of a commutative ring.

MTS 412      ADVANCED ALGEBRA II (3 UNITS)  PRQT MTS 311 OR MTS 314

Polynomials over fields. Irreducibility criterion especially over Q, the rational number fields.  Fields extensions.  Finitely generated. Finite and simple extensions. Algebraic extensions, Splitting fields. Derivatives and Separability, Normal extensions. Automorphosims of fields.  F-automorphisms, and normal extensions. Fundamental theorem of Calois theory. Applications

MTS 421      COMPLEX ANALYSIS II ( 3 UNITS)  PRQT MTS 311 OR MTS 314

Meromorphic functions.  Zeros and poles.  Argument principle.  Rouche’s theorem.  Summation
of series.  Mittag-Laffler’s theorem.  Maximum principle.  Principle of analytic continuation.  Schartz-Christoffel transformation.  Boundary value problem.

MTS 423  FUNCTIONAL ANALYSIS (3 UNITS )  PRQT MTS 362

A survey of the classical theory matrix spaces (including Baire’s category theorems compactness, separability, isometrics and completion), elements of Banch and Hilbert spaces, parallelogram law and linear spaces into second dual, and Hongo H, properties of operators (including the  open mapping and closed graph theorem), the spaces C(X), the sequences (Banach) spaces Lnp, Lp and C (spaces of convergent sequences).

MTS 424   LESBEGUE MEASURE AND THEORY OF INTEGRATION (3 UNITS) PRQT 323

Lesbegue measure for subsets of Rn.  Lesbegue integration of real and complex-valued functions defined on subsets of Rn.  General measure space (X,  σ,  µ  )  and Lesbegue integration with respect to µ of real and complex-value function on X.  The classical Banach spaces.

MTS 441      ORDINARY DIFFERENTIAL EQUATIONS (3 UNITS)  PRQT MTS 342

Existence and uniqueness theorem, dependence of solution on initial data and parameter.  Properties of solutions.  Sturm comparison and Senin-Polye theorems.  Linear system, Floguet’s
theory.  Non-linear system, stability theory.  Integral equations, classification.  Fredholm’s alternative, method of successive approximations, Neu-man’s series, resolvent kernel.  Volteral
equations.  Application to ordinary differential equations.

MTS 442  PARTIAL DIFFERENTIAL EQUATIONS (3 UNITS)  PRQT MTS 342

Theory and solution of first order equations.  Second order linear equation, classifications, characteristics, canonial forms, Cauchy problem.  Elliptic, Laplace’s and Possion equations. Fundamental solutions; Green’s functions, Possion’s formula, properties of harmonic functions, hyperbolic functions, the wave equation, retarded potential, transmission linear equations.  Remain method parabolic equations, Diffusion equation, singularity function.  Boundary and initial value problems.

MTS 443        MATHEMATICAL METHODS II (3 UNITS )  PRQT  OR MTS 342

Stum-Liouville problem.  Orthogonal polynomial and functions.  Fourier series and integrals, partial differential equations, first and second order equations.  Classification of second order liner equations, solution of heat, wave and Laplace equations by the methods of separable variables, eigen function expansions and Fourier transforms.

MTS 453      MATHEMATICAL MODELLING II (3 UNITS)  PRQT MTS 354

Dynamic programming and sequential multistage decision processes,  Bellam’s principle of optimality.  Further dynamic programming models, the Markovian decision problem, finite-stage model, exhaustive enumeration and policy inerration with and without discounting, linear programming, solution of the Markovian decision problem.

MTS 461  GENERAL TOPOLOGY  (3 UNITS)   PRQT MTS 362

Topological spaces:  Neighborhoods and neighborhood systems. Subspace induced topology,
Bases, sub-bases continuity.  Matirc and normed spaces.  First and second countable, separable
spaces, Hausdorff, regular, normal spaces, T1, T2, T3, T4  spaces, compactness and  product
spaces.  Path.  connectedness.

MTS 463        GENERAL RELATIVITY (3 UNITS)  PRQT MTS 322

Particles in a gravitational field.  Curvilinear coordinates, intervals, covariant differentiation.  Christofell symbol and metric tensor.  The constant gravitational field.  Rotation, The curvature tensor.  The Action function for the gravitational field.  The energy momentum tensor. Newton’s law.  Motion in  a centrally symmetric gravitational field.  The energy momentum tensor.  Newtom’s law.  Motion in a pseudo-tensor.  Gravitational waves.  Gravitational fields at large distance from bodies.  Isotopic space.  Space-time metric in the closed and in one open  isotopic model.  The red shift.

MTS 464     CONTINUUM MECHANICS (3 UNITS)  PRQT MTS 242 OR MTS 322

Tensor calculus of double field.  Deformation and its derivative tensors.  Kinematics of lines, surface and volumes.  Analysis of stress Isotropy group of material.  Equation of balance with and without discontinuity surfaces.  Thermomenchanical constitutive, relations for various materials with and without kinematics constraints.  Solution of simple equations of selected materials.

MTS 465      OPTIMIZATION THEORY (3 UNITS)  PRQT MTS 363 OR MTS 351

Review of linear programming models.  The general non-linear programming: direct search and gradient methods, golden search and Fibenacci methods, conjugate gradient methods of Fletcher, Power and Reeves, cutting place methods, unconstrained optimization,  Langrange multipliers method of constrained optimization, convex programming, penalty function methods, sequential unconstrained techniques, Kuhn-Tucker theory, quadratic programming algorithms of Scale and Wolfe, the complimentary problem.

MTS 466        CACULUS OF VARIATION (3 UNITS) MTS 342

Functional Euler’s equations, problems with moving boundaries, extremes with corners, conditional extreme; principles of least action, isoperimetric problems.  Hamilton’s principle; direct methods of Euler, Rits, Kantirivich, etc.

MTS 468     MATHEMATICAL THEORY OF FLUID DYNAMIC (3 UNITS)  PRQT MTS
242 OR PHS 311 OR PHS 312

Real and ideal fluids.  Differentiation following the motion of fluid particles.  Equations of motion and continuity for incompressible inviscid fluids.  Velocity potentials and stroke’s stream functions.  Bernoulli equation with application to flow along curve paths.  Kinetic energy: sources, sinks, doublets in 2 and 3 dimensions, limiting stream lines, images and rigid planes.

MTS 471    SYSTEMS THEORY  (3UNITS)  PRQT MTS 342, MTS 342 OR MTS 354

Lyapunov theorem, solution of Lyanpunov stability equation.  Controlability and observability.
Theorem on existence of solution of linear systems of differential operations with constant coefficients.

MTS 472          GRAPHS AND MATROIDS (3 UNITS) PRQT MTS 311

Matrix representation of graphs, planarity and connectivity, complete graphs, bipartite graphs, diagraphs, gravy-isomorphisms, distances in a graph, tournaments, detection of cliques in a communication network, Euler graphs, Hamiltonian circuits, the Konigberg bride problem and in generalizations, trees, cycles, cocycles, spanning treses, Menger’s theorem, Kuratowaki’s theorem, colorability, graph enumeration, distinct representations, Hall’s theorem, introduction to the theory of matroids.
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MTS 499           PROJECT (6 UNITS)

Independent research works in mathematics

(5)  ON-GOING DEPARTMENTAL RESEARCH
(1)  Theory of Loops
(2)  Mathematical Analysis
(3)  Theory of Ordinary Differential Equation
(4)  Bi – Algebraic Structures
(5)  Vibration Theory
(6)  Fluid Dynamics

Five (5) Credits at SSCE in English Language, Physics, Chemistry, Mathematics,
Biology or Agriculture and Biology

To be eligible for admission into the Masters Degree programme, candidate must be
graduate of this University or any other University recognized by Senate and shall
normally have obtained a minimum of Second Class (Upper Division) degree in the
relevant field.  In exceptional cases, candidates with Second Class (Lower Division) may
be considered.

To be eligible for admission into the Doctor of Philosophy Degree programme, a student
must have obtained a Master’s Degree from this University or its equivalent from any
University recognized by Senate.  For holders of one year Degree of Master’s the
minimum duration on the Ph.D programme shall be three years.

(7)  LABORATORY STATUS WHERE APLICABLE:  (i)  LABORATORY
EQUIPMENT  (ii)  STAFF (TECHNICAL)

NOT APPLICABLE

(8)  NAME OF DEPARTMENT (ACRONYM)
DEPARTMENT OF MATHEMATHETICS (MTS)

(9)  FOUNDATION YEAR

1983

(10)  ACCREDITATION STATUS

FULL

(11)  DEPARTMENTAL AFFLIATION WITH PROFESSIONAL BODIES

(1)  Nigerian Mathematical Society (NMS)
(2)  Mathematical Association of Nigeria (MAN)
(3)  Nigerian Association of Mathematical Physics (NAMP)
(4)  London Mathematical Society (LMS)
(5)  American Mathematical Society (AMS)

(12)  UNIVERSITY COURSE LEVEL AND AREA OF STUDIES (For Undergraduate,
Certificate/Diploma and Postgraduate:  Master and Ph.D)

Course Listing

100 Level  -  First Semester

Course Code    Course Title
MTS 101    Algebra
MTS 103    Vector and Geometry
MTS 105    Algebra and Trigonometry for Biological Sciences

100 Level – Second Semester

Course Code    Course Title

MTS 102    Calculus and Trigonometry
MTS 104    Mechanics
MTS 106    Calculus for  Biological  Sciences

200 Level  -  First Semester

Course Code    Course Title
MTS 201*    Mathematical Foundation
MTS 211    Abstract Algebra
MTS 223    Real Analysis I

200 Level – Second Semester

Course Code    Course Title

MTS 212    Linear Algebra
MTS 232    Ordinary Differential Equations
MTS  242    Mathematical Methods I

300 Level – First Semester

Course Code    Course Title

MTS 311    Groups and Rings
MTS 321    Complex Analysis
MTS 323    Real Analysis II
MTS 363    Introduction to Operation Research

300 Level – Second Semester

Course Code    Course Title
MTS 322    Vectors and Tensors Analysis
MTS 342    Mathematical Methods II
MTS 354    Mathematical Modelling I

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