The Department admits students into 100 level as well as 200 level for the B.Sc (Hons.) Mathematics based on their qualifications. In rare cases they may be admitted into upper levels.

**I. For 100 level:** Candidates must satisfy the general University and Faculty of Science requirements of five O’Level credits which must include: Mathematics, English, Physics and any other two relevant science subjects at Senior Secondary School Certificate level or examination in at most two sittings.

**II. For 200 level:** Candidates must in addition to (I) above have an Advanced level (A ‘Level) or its equivalence in Mathematics and any other science subject.

The duration of B.Sc. (Hons.) Mathematics programme is four years for candidates admitted into 100 level and three years for those admitted into 200 level. There are two semesters of formal University Studies in each academic session except at 300level where students are required to undergo Students’ Industrial Work Experience Scheme (SIWES) programme for 6 months. At the end of the programme, each student is required to write, present and defend a report on what he/she learned in the industry. At 400 level, students undertake a one year project in any field of interest.

For a student to graduate, he/she must pass all his/her core courses, earn at least 120 credit units (i.e. TECU ≥ 120) and have a Cumulative Grade Point Average of at least 1.50 (i.e. CGPA ≥ 1.50)

S/N |
Course Code |
Course Title |
Credit Units |
Prerequisite |
Semester |

1 | COSC101 | Introduction To Computing | 2 | O/Level Mathematics | First Semester |

2 | MATH101 | Sets and Number System | 2 | " | First Semester |

3 | MATH103 | Trigonometry and Coordinate Geometry | 2 | " | First Semester |

4 | MATH105 | Differential and Integral Calculus | 2 | " | First Semester |

5 | PHYS111 | Mechanics | - | " | First Semester |

6 | PHYS131 | Heat and Properties of Matter | - | " | First Semester |

7 | MATH102 | Algebra | 2 | " | Second Semester |

8 | MATH104 | Conic Sections and Applications of Calculus | 2 | " | Second Semester |

9 | MATH106 | Vectors and Dynamics | 2 | " | Second Semester |

10 | STAT102 | Introductory Statistics | 2 | " | Second Semester |

S/N |
Course Code |
Course Title |
Credit Units |
Prerequisite |
Semester |

1 | MATH201 | Mathematical Methods I | 3 | MATH105 | First Semester |

2 | MATH203 | Real Analysis I | 3 | MATH105 | First Semester |

3 | MATH205 | Abstract Algebra I | 3 | MATH101 | First Semester |

4 | MATH207 | Linear Algebra I | 3 | MATH102 | First Semester |

5 | MATH209 | Numerical Analysis I | 3 | MATH105 | First Semester |

6 | STAT201 | Discrete Probability Distributions | 3 | STAT102 | First Semester |

7 | MATH202 | Elementary Differential Equations I | 3 | MATH104 | Second Semester |

8 | MATH204 | Real Analysis II | 3 | MATH105 | Second Semester |

9 | MATH206 | Abstract Algebra II | 3 | MATH102 | Second Semester |

10 | MATH208 | Linear Algebra II | 3 | MATH102 | Second Semester |

11 | STAT202 | Continuous Probability Distributions and Distribution Techniques | 3 | STAT102 | Second Semester |

12 | COSC202 | Fortran and Structured Programming | 2 | COSC101 | Second Semester |

S/N |
Course Code |
Course Title |
Credit Units |
Prerequisite |
Semester |

1 | MATH301 | Mathematical Methods II | 3 | MATH201 | First Semester |

2 | MATH303 | Advanced Real Analysis I | 3 | MATH203 | First Semester |

3 | MATH305 | Theory of Rings & Fields | 3 | MATH206 | First Semester |

4 | MATH307 | Complex Analysis I | 3 | MATH203 | First Semester |

5 | MATH309 | Analytical Dynamics I | 3 | MATH201 | First Semester |

6 | MATH311 | Mathematical Modeling | 3 | MATH201 | First Semester |

7 | MATH313 | Axiomatic Set Theory | 3 | MATH205 | First Semester |

8 | MATH315 | Number Systems and Algebraic Structures | 3 | MATH205 | First Semester |

9 | MATH317 | Numerical Analysis II | 3 | MATH209 | First Semester |

10 | MATH300 | SIWES | 6 | - | Second Semester |

S/N |
Course Code |
Course Title |
Credit Units |
Prerequisite |
Semester |

1 | MATH401 | Differential Equations | 3 | MATH201 | First Semester |

2 | MATH403 | General Topology I | 3 | MATH303 | First Semester |

3 | MATH405 | Theory of Finite Groups | 3 | MATH305 | First Semester |

4 | MATH407 | Advanced Real Analysis II | 3 | MATH303 | First Semester |

5 | MATH413 | Hydrodynamics I | 3 | MATH301 | First Semester |

6 | MATH415 | Quantum Mechanics | 3 | MATH309 | First Semester |

7 | MATH417 | Biomathematics | 3 | MATH311 | First Semester |

8 | MATH425 | Graph Theory and Combinatorics | 3 | MATH303 | First Semester |

9 | MATH402 | Functional Analysis | 3 | MATH303 | Second Semester |

10 | MATH404 | General Topology II | 3 | MATH303 | Second Semester |

11 | MATH406 | Group Representations and Characters | 3 | MATH305 | Second Semester |

12 | MATH408 | Measure and Intergration | 3 | MATH303 | Second Semester |

13 | MATH412 | Partial Differential Equations | 3 | MATH202 | Second Semester |

14 | MATH414 | Hydrodynamics II | 3 | MATH301 | Second Semester |

15 | MATH416 | Complex Analysis II | 3 | MATH307 | Second Semester |

16 | MATH422 | Differential Geometry | 3 | MATH301 | Second Semester |

17 | MATH424 | Electromagnetic Theory and Waves | 3 | MATH301 | Second Semester |

18 | MATH428 | Analytical Dynamics II | 3 | MATH309 | Second Semester |

19 | STAT412 | Operations Research | 3 | MATH311 | Second Semester |

S/N |
Course Code |
Course Title |
Credit Units |
Prerequisite |
Semester |

1 | MATH161 | Mathematics for Agriculture I | 2 | O/Level Mathematics | First Semester |

2 | MATH162 | Mathematics for Agriculture II | 2 | O/Level Mathematics | Second Semester |

3 | MATH241 | Calculus I | 3 | MATH105 | First Semester |

4 | MATH242 | Calculus II | 2 | MATH105 | Second Semester |

5 | MATH243 | Methods of Linear Algebra I | 2 | MATH102 | First Semester |

6 | MATH244 | Method of Linear Algebra II | 3 | MATH102 | Second Semester |

7 | MATH341 | Ordinary Differential Equations | 3 | MATH241 | First Semester |

8 | MATH342 | Functions of Several Variables | 2 | MATH241 | Second Semester |

9 | MATH441 | Complex Analysis | 3 | MATH341 | First Semester |

10 | MATH443 | Numerical Analysis | 3 | MATH241 | First Semester |

11 | MATH541 | Conformal Transformation and Applications | 3 | MATH441 | First Semester |

12 | MATH542 | Partial Differential Equations | 3 | MATH341 | Second Semester |

**MATH101 – Sets and Number System (2 Credit Units) :**

Sets: Definition of a set, finite and infinite sets, equality of sets, subsets, union, intersection, universal set, complements, empty set, Venn diagram. Symmetric difference, power sets and De-Morgan theorems. Inclusion-Exclusion principle. Elements of relations and functions.

Some Properties of number systems: Natural numbers, integers, rationals, irrationals and reals. Order relations in the set of real numbers. Open and closed intervals on the number line.

Complex Numbers: Definition of a complex number, addition, multiplication and division. Geometric interpretation modulus and conjugation. Polar representation, De- Moivre’s theorem, nth roots of a complex number, nth roots of unity.

**Text books**

2. Set Theory and Related Topics, S. Lipschutz, (Schaum’s Outline Series), McGraw-Hill (1964).

**MATH103 – Trigonometry and Coordinate Geometry (2 Credit Units) :**

Plane Polar Coordinates: Relation between polar and Cartesian coordinates, plotting and sketching of simple curves whose polar equations are known.

Coordinate Geometry of lines and Circles: Pair of straight lines and system of circles. (Emphasis on concepts rather than formulae).

**Text books**

2. Pure Mathematics Books I & II, J.K. Backhouse et al, Longman (1980)

2. Calculus and Analytical Geometry, G.B. Thomas and R.L. Finney, Addison- Wesley, (1979).

3. Theory and Problems of Trigonometry, Frank Ayres, (Schaum’s Outline Series). (1954).

**MATH105 – Differential and Integral Calculus (2 Credit Units) :**

Differentiation: First principle, techniques of differentiation in general. Higher derivatives.

Integration: Integration as the inverse of differentiation, techniques of integration in general, definite integral (Evaluation only).

**Text books**

2. Pure Mathematics Books I & II, J.K. Backhouse, et al Longman (1980).

3. Calculus and Analytic Geometry, G.B. Thomas and R. L. Finney, Addison –Wesley (1979).

**COSC101 Introduction To Computing:**

Click here to download course material

**Suggested Lab work **
Lecturers should develop laboratory exercises and assignments targeted at providing hands-on practical experience on all topics in the syllabus.
The exercises should cover the typical tasks that students do with computers throughout their studies.

**Textbooks**

2. J.J. Parsons and D. Oja, Practical Computer Literacy, Thompson Learning, 2005

3. Curt Simmons, How to Do Everything with Windows XP, 2nd Edition McGraw-Hill/Osborne, 2003, ISBN 0-07-223080-0

4. Peter Norton’s, Introduction to Computers, 5th Edition McGraw-Hill/Glencoe, 2003, ISBN 0-07-826421-9

**PHYS111 Mechanics:**

**PHYS131 Heat and Properties of Matter:**

Structure of solids, liquids and gases. Kinetic theory of gases, Elasticity, surface tension, solid friction. Fluid in motion, Bernoulli’s law, Aerofoil; thermodynamics; thermal expansion. Heat transfer. EM radiation, prevost theory of heat exchange. Thermal radiation detectors; Optical pyrometer.

**MATH102 – Algebra (2 Credit Units) :**

Quadratic and other polynomial functions: Elementary properties of quadratic expressions, roots of quadratic equations, application to symmetric functions, polynomial functions of third and fourth degrees, remainder theorem, location of roots.

Permutation and combination: Notion of Factorials, nPr, nCr, and simple applications, mathematical induction principle and applications.

Binomial Theorem: Expansion of all rational index, interval of convergence, approximations and errors.

**Text books**

1. Mathematics for Fresh Undergraduates Vol. I, D. Singh, A. Mohammed, A.M. Ibrahim and I.A. Fulatan ABU press (2013)

2. Pure Mathematics Book I and II, J.K. Backhouse, et al, Longman (1980)

**MATH104 – Conic Sections and Applications of Calculus (2 Credit Units) :**

Rate of Change: Velocity, acceleration and other rates.

Curve Sketching: Asymptotes, maxima and minima. Small increments, approximations and errors.

Newton’s approximation,
simple application of integration to areas and volumes.

Differential equations: First order differential equations only.

**Text books**

2. Mathematics for Fresh Undergraduates Vol. III, J. Singh, H.M. Jibril, A.J. Alkali, Y.M. Baraya, A. Umar, ABU press (2013)

3. Pure Mathematics Books I & II , J.K. Backhouse, et al, Longman (1980)

4. Calculus and Analytic Geometry, G.B. Thomas and R.L. Finney, Addison-Wesley (1979).

**MATH106 – Vectors and Dynamics (2 Credit Units) :**

Dynamics: Kinematics of a particle. Components of velocity and acceleration of a particle moving in a plane. Force, momentum, laws of motion under gravity, projectiles, restricted vertical motion, elastic strings, simple pendulum, impulse. Impact of two smooth spheres, of a restricted sphere and a smooth sphere.

**Text books**

2. Textbook of Dynamics, F. Charlton, Ellis Horwood, 1977.

3. Vector Analysis, Murray R. Spiegel, Schaum’s Outline Series (1974)

**STAT102 Introductory Statistics (2 Credit Units):**

**Text Books**

2. Mathematical Statistics, Ray, M., Sharma, H.S. and Choudhary, S., Ram Prakash and Sons Agra - 3, India.

3. Fundamentals of Mathematical Statistics, Gupta S.C. and Kapoor, V.K., Sultan Chand and Sons, New Delhi, India.

**MATH201 – Mathematical Methods I (3 Credit Units):**

Partial Differentiation: Real valued functions of two and three variables. Partial derivatives, chain rule, Jacobian. Extrema, Lagrange’s multipliers, increments, differentials and linear approximations.

**Text books :**

2. Advanced Engineering Mathematics, E. Kreyszig, Wiley, (1987).

**MATH203 – Real Analysis I (3 Credit Units) :**

Boundedness: Boundedness and related simple results.

Relations and Functions: Cartesian products of sets.

Relations: relations, classes.

Functions: injective, surjective, bijective, inverse, composition of functions, monotone functions, graph of

functions, algebraic operations on functions.

Sequences and series of Real Numbers: Sequences of real numbers, subsequences, bounded and unbounded sequences.

Limit of a sequence; limit superior and limit inferior, improper limits. Algebraic operations on sequences and their limits; Monotone sequences and properties. Cauchy sequences and related results.

Series of real numbers: partial sums, convergence, absolute and conditional convergences.

Convergence tests: comparison, ratio, Ra’abe, De-Morgan and Betrand, logarithmic, Cauchy root test. Cauchy condition for the convergence of series, rearrangement of series.

**Text books:**

2. Real Analysis (An introduction) White A.J, Addison – Wesley, (1968)

1. Foundation of Mathematical Analysis, C.E Chidume

**MATH205 – Abstract Algebra I (3 Credit Units) :**

Boolean algebra; basic definitions and some simple theorems.

Elementary Notion of Groups: Binary operations, closure, associativity, semigroup, identity, inverse, group axioms, commutativity. Elementary properties of a group. Abelian groups, symmetric group of degree n, permutations and permutation groups, symmetric group of regular polygon.

**Text books:**

2. Set theory and related Group Theory, B. Baumslag and B. Chandler.

3. Introduction to Abstract Algebra, F.M. Hall University Press (1966).

4. First Course in Abstract Algebra, J.B. Fraleigh, Addison – Wesley (1977)

**MATH207 – Linear Algebra I (3 Credit Units) :**

Inverse of a matrix.

Systems of linear equations and matrices: Systems of m linear equations in n unknowns and their solutions. Gaussian elimination by pivot method and matrix representation. Solution of the system using Gaussian elimination and Gauss-Jordan reduction.

Determinants: Definition, evaluation of determinants. Cofactor expansion, inverse of a non-singular matrix. Solution of systems of linear equations using Cramer’s rule.

**Text books:**

2. Linear Algebra and Matrix Theory, E.D. Nerring, John Wiley, (1967).

**MATH209 – Numerical Analysis I (3 Credit Units) :**

Evaluation of functions: using series approximation, solution of polynomial, algebraic and transcendental equations, curve fitting.

Interpolation: Newton’s difference formulae, central difference formulae, Lagrange’s formula. Numerical differentiation. Numerical Integration

**Text books:**

2. Theory and Problems of Numerical Analysis, Francis Scheid, Schaum’s Series (1968).

3. Numerical Analysis: An Introduction, S.A. Bhatti, Mathematics Departmental Library, (Lecture Notes, 1980’s).

4. Calculus of Finite differences and Numerical Analysis, P.P. Gupta & G.S. Malik.

**STAT201 – Discrete Probability Distributions (3 Credit Units) :**

**Text Books:**

1. Introduction to the theory of Statistics, Mood, A.M., Graybill, F.A. and Boes, D.C. Mc-Graw-Hill, New York, USA.

2. Fundamentals of Mathematical Statistics, Gupta S.C. and Kapoor, V.K., Sultan Chand and Sons, New Delhi, India.

**MATH202 –Elementary Differential Equations1 (3 Credit Units) :**

Differential Equations: Concept of differential equations. First order ordinary differential equations of the forms; variable separable, homogeneous, exact and linear. Second order ordinary linear differential equations with constant coefficients, auxiliary equation, and cases of auxiliary equations having distinct, equal, and complex roots, complementary functions and particular integrals in connection with non-homogeneous equations. Uses of the operator D = d/dx and the method of undetermined coefficients for calculating particular integrals. Differential equations of Euler’s type of second order. Solutions of systems of two linear differential equations. Second order Ordinary Linear Differential Equations with variable coefficients; reduction of order, variation of parameters.

Series solution. Frobenius method. System of linear first order equations. Applications of differential equations to geometry and Physics. Partial differential equations: Solutions of one dimensional heat and wave equations.

**Text books:**

2. Theory and Problems of Differential Equations, Frank Ayres Jr. (Schaum’s Outline Series). McGraw-Hill, (1972).

3. Advanced Engineering Mathematics, E. Kreyszig, Wiley, (1987).

**MATH204 – Real Analysis - II (3 Credit Units) :**

Derivatives: derivative of functions derivative of composition of functions. Higher order derivatives. Algebraic operations on derivatives of functions. Differentiability and some related results. Rolle’s and Mean value theorems, Taylor’s formula, L’Hospital’s rule, local and global extrema, saddle points, monotonicity, and geometrical interpretations.

Riemann Integration: Partition of an interval, refinement¬¬¬¬, Riemann sums, Riemann integrals, uniqueness of Riemann integral, Darboux integral of a real valued function, relation between Riemann and Darboux integrals.

**Text books :**

2. Introduction to real Analysis, A. Olubumo, Heinemann (1979).

**MATH206 – Abstract Algebra II (3 Credit Units) :**

Homomorphism of groups: Definition and examples of homormorphism, kernel, image, epimorphism, monomorphism, isomorphism. Isomorphic groups. Fermat and Cayley theorem. First isomorphism theorem.

Algebraic structures: Definition and examples of rings, fields, vector spaces and modulus.

**Text books:**

2. Set theory and related Group Theory, B. Baumslag and B. Chandler.

3. Introduction to Abstract Algebra, F.M. Hall University Press (1966).

4. First Course in Abstract Algebra, J.B. Fraleigh, Addison – Wesley (1977)

**MATH208 – Linear Algebra Ii (3 Credit Units) :**

Linear Mappings and Matrices: General linear transformation of n-dimensional into m-dimensional space, matrix representation of a linear map, similar matrices and change of basis. Eigenvalue and eigenvectors. Characteristic polynomial and characteristic equation. Caley-Hamilton theorem. Orthogonal diagonalization.

Canonical Forms: Primary decomposition theorem, Triangular
Jordan and

Rational forms for linear operator (square matrices). Quadratic and
bilinear forms.

**Text books:**

2. Linear Algebra and Matrix Theory, E.D. Nerring, John Wiley, (1967).

**STAT202 - Continuous Probability Distributions and Distribution Techniques (3 Credit Units) :**

**Text Books:**

2. Fundamentals of Mathematical Statistics, Gupta S.C. and Kapoor, V.K., Sultan Chand and Sons, New Delhi, India.

**COSC202: Fortran and Structured Programming:**

DECLARATIVE STATEMENTS: the DATA statement, the COMMON statement, the Statement, MATLAB Package

**Text Books:**

**MATH301 – MATHEMATICAL METHODS II (3 CREDIT UNITS):**

Fourier series: Definition, computation of Fourier coefficients, expansions of even and odd functions, change of period, half period expansion, Fourier transform.

Laplace Transform: Definition, elementary formulae, convolution theorem, Applications to solutions of ordinary differential equations.

**Text books:**

2. Theory and Problems of Advanced Calculus, M. R. Spiegel (Schaum’s Outline Series), Mc Graw-Hill (1974).

3. Theory and Problems of Vector Analysis, M.R. Spiegel (Schaum’s Outline Series), Mc Graw-Hill (1974).

**MATH303 – Advanced Real Analysis I (3 Credit Units) :**

Metric Spaces: Definition and examples of metric spaces, bounded sets, diameter of a set and distance between sets, open spheres or balls, open sets, neighbourhoods, interior, exterior, frontier, closed sets, accumulation point, closure of a set, convergence in metric spaces; Equivalence matrices, Cauchy sequence in metric spaces, complete metric spaces.

Real Functions of Several Variables: Introduction to n-dimensional Euclidean space Rn or En, neighbourhoods in Rn, Norm of a point in Rn, Minkowski’s and Cauchy-Schwartz’s inequalities, open balls and boundedness in Rn . Functions from Rn to Rm (m < n), component

functions, linearity and related results. Limit of a function in Rn , notion of inner limits, simultaneous and repeated limits, existence theorems, uniform limit and Moore-Osgood theorem, continuity of functions in Rn, partial derivatives and differentiability, uniform partial derivatives. Higher order partial derivatives, Young’s and Schwartz’s theorems, Implicit function and Inverse function theorems. Mean Value theorems and Taylor’s formula.

**Text books:**

2. Mathematical Analysis, Apostle T.M.,Addison-Wesley (1974)

**MATH305 – Theory of Rings & Fields (3 Credit Units) :**

Modules: Review of Definition, Abelian group as module over ring of integers. Submodules direct sum of submodules. Homomorphism of modules.

Rings: Review of basic definitions, types of rings, integral domains, skew fields and fields, subrings. Concrete examples of rings, homomorphism, isomorphism of rings, kernels, ideals, quotient rings. Integral domain, characteristics of an integral domain, finite integral domain as a field, example of an integral domain that is not a field. Prime fields and field of fractions. Euclidean rings, polynomial rings, division algorithm for polynomial rings, factorization theorems, unique factorization domains, rational test and Einstein criterion for irreducibility.

Quotient Rings and Fields: Ideals, Principal ideal domains. Quotient rings. Maximal ideals, Prime ideals.

**Text books:**

**MATH307 – Complex Analysis I (3 Credit Units) :**

Laplace and Cauchy-Riemann equations. Elementary functions (exponential, trigonometric, logarithmic, rational, power and hyperbolic functions). Harmonic functions. Complex Integration: Definition and properties of complex integration. Contour integration: Integration of complex functions along a continuously Differentiable arc, along a piece wise differentiable arc and along a rectifiable arc. Cauchy integral theorem, Cauchy formulae and Cauchy-Goursat theorem.

**Text books:**

2. Elements of complex variables, Gordon L. I. & Lasher S., Rinehart (1963).

**MATH309 – Analytical Dynamics I (Credit Units) :**

Newtonian Mechanics: Newtonian law, Equation of motion for a particle. Conservation theorems for a system of particles. Dynamical System: Representation of motion; constants; rigid body kinematics mechanics problems, the nature of Newtonian Mechanics.

Variational Principle: The principle of virtual displacement and virtual work. D’Alemberts principle. Constraints: holomonic and non-holomonic constraints. Hamilton’s principle.

Generalized coordinates; Lagrange’s equations, embedding constraints, formulation and Solution of Problems by using Lagrange.

**Text books:**

2. Classical Mechanics, Gupta, et al, Pragati Pakistan (1990).

**MATH311 – Mathematical Modeling (3 Credit Units) :**

**Text books:**

2. Mathematical Modeling for Industrial Processes, Lassi Hyvaarinen, Springer-verlag (1970).

3. Mathematical Methods of Operations Research, T.L. Saaty, Dover Publications, Inc. (1988).

**MATH313 – Axiomatic Set Theory (3 Credit Units) :**

Axioms of Set Theory: Extensionality, Pairing, Comprehensions, Power Set, Infinity, Replacement, Regularity, and Choice.

**Text books:**

2. Foundations of Mathematics, R. L. Wilder. (Rev. ed.), John Wiley, 1965.

3. Introduction to Foundation and Fundamental concepts of Mathematics, H. Eves and V.C. Newson: Holt, Rinehart, et al (Rev. ed.), 1995.

**MATH315 – Number Systems and Algebraic Structures (3 Credit Units) :**

Algebraic congruences, primitive roots, indices with respect to a primitive root.

Quadratic and high power residues. The Legendre and Jacobi symbols. Gauss Law of quadratic reciprocity. Representative of integers by binary quadratic forms.

Diophantine equation like ax +by =c, x2 +y2 =z2 , x4 +y4 =z4, e.t.c.

**Text Books :**

**MATH317 – Numerical Analysis II (3 Credit Units) :**

**Text books:**

2. Elements of complex variables, Gordon L. I. & Lasher S., Rinehart (1963).

**MATH401 – Differential Equations (3 Credit Units) :**

Ordinary Differential Equations; Series solutions of second order linear differential equations. Solutions of Legendre and Bessel (first kind only) equations; Legendre and Bessel’s polynomials. Problem of existence of solutions, existence and uniqueness theorems. Dependence of solutions on initial data parameters. Properties of solutions.Sturm comparison Linear and non-linear systems. Integral Equations; classification, Volterra and Fredholm types. Solutions using Laplace and Fourier transform. Reduction of an O.D.E. to an Integral Equation. Stability; Lyapunov function. Symmetric Kernels. Eigenfunction with application.

**Text books:**

2. Theory and Problems of Differential Equations, Frank Ayres Jr. (Schaum’s Outline Series). McGraw-Hill, (1972).

3. Advanced Engineering Mathematics, E. Kreyszig, Wiley, (1987).

**MATH403 – General Topology I (3 Credit Units) :**

Continuous mappings: Continuity in metric spaces, Open and closed mappings, Homeomorprohism. Topological invariants.

Connectedness: Union, product, closure of connected sets Intervals as connected subsets of the real line. Image of connected sets under continuous mappings. Connected components.

**Text books:**

2. Topology (A first course), James R. Munkres, Prentice-Hall, Inc (1975)

3. Topology, M. Eisenberg, Rinebort, (1974).

**MATH405 – Theory of Finite Groups (3 Credit Units) :**

Isomorphism theorems: First, second and third isomorphism theorems, Free groups, Groups of automorphisms. Group action on a set. Burnside lemma. Structure theory of Abelian groups. Free Abelian groups.

Normal Structure of groups: Composition series, derived series, Jordan – Hölder theorem. Soluble and Nilpotent groups.

**Text books:**

2. Abstract Theory of Groups, O. U. Schmidt, Freeman, (1966).

**MATH407 – Advanced Real Analysis II (3 Credit Units) :**

Uniform convergence and differentiability. Weierstrass’s continuous non-differentiable function.

Uniform convergence of power series.
Functions of Bounded Variation and their Properties: Variation function of a function of bounded variation, Jordan’s theorem.

Riemann – Stieltjes Integral: Stieltjes integral and its various generalizations, conditions of integrability, integration by parts. First mean value theorem, second mean value theorem. Differentiation under the integral sign.

**Text books:**

2. Mathematical Analysis, Apostle T.M., Addison-Wesley (1974)

**MATH413 – Hydrodynamics I (3 Credit Units) :**

Conservation of Mass: Principle of conservation of mass of a fluid element. Equation of continuity in Cartesian, cylindrical and spherical polar coordinates. The Laplacian equation ∇2φ= 0 for steady, irrotational and incompressible flows. The concepts of stream function for steady two-dimensional, incompressible flows. Cauchy-Riemann relations and the complex potential w = + i. Equation of streamlines as = constant. Circulation.

Sources, Sinks, Doublets, Vortices: Definitions of source and sinks in two and three dimensions. Velocity potentials due to (i) a three dimensional source (or sink), (ii) a three dimensional doublets. Complex potentials due to two dimensional sources and doublets. Complex potential due to a two-dimensional vortex. Concept of image of a simple source with regard to a plane.

Equations of Motion: Euler’s dynamical equations. Lagrange’s integration of Euler’s equations Bernoulli’s equation for (I) steady incompressible flows, and (ii) steady, compressible adiabatic flows. Irratational Motion in Two Dimensions: Introduction. Boundary conditions for a moving cylinder. Flow due to translation motion of a right circular cylinder. Flow of liquid past a circular cylinder. Force on a cylinder due to a uniform stream past it. D’Alembert’s paradox. Effect of a constant circulation about a circular cylinder placed in a uniform stream. Initial motion due to sudden movement of two co-axial cylinder.

**Text books:**

**MATH415 – Quantum Mechanics (3 Credits) :**

**Text books:**

**MATH417 – Biomathematics (3 Credit Units) :**

Mathematical Ecology: Mathematical models in ecology; Growth and decay of populations, isolated populations poisoned by their own metabolic products; Prey-predator models; Models for competition between the species. Differential equations of ecology. Stochastic models in ecology.

Mathematical genetics: Genetic matrices. Hardy-Weinberg law. Bayes’ theorem and its application in genetics. Mathematical theory of epidemics. Some simple epidemic models. Deterministic models (i) with removal and (ii) with no removal and migration. Stochastic models.

Mathematical models for the brain: Moculloch and Pitts models Stochastic models.

**MATH425 Graph Theory and Combinatorics (3 Credit Units) :**

Trees: Characteristics, centers and centroids, matroids.

Transversality: Eularian and Hamiltonian graph, line graphs and Transversality.

Enumeration of Graphs: Labeled graphs, polya’s enumeration theorem. Enumeration of graphs and trees.

Digraphs: Digraphs and connectedness. Directional duality and acyclic diagraphs. Digraphs and matrices. Tournaments.

**Text books:**

1. Graph Theory by Frank Harary, Addison Wesley, 1990.

**MATH402 – Functional Analysis (3 Credit Units) :**

Normed Spaces: Linear spaces, Norm function, Normed Linear boundedness principle. Open mapping and closed graph.

Hilbert Spaces: Definition and examples of Inner product spaces and Hilbert spaces, projection theorem, Riez representation theorem.

**Text books:**

2. Functional Analysis, W. Rudin, McGraw-Hill, (1974)

**MATH404 – General Topology II (3 Credit Units) :**

Metrizable spaces: Urysohn’s Lemma, Urysohn’s metrization theorem.

**Text books:**

2. Topology (A first course), James R. Munkres, Prentice-Hall, Inc (1975)

3. Topology, M. Eisenberg, Rinebort, (1974).

**MATH406 – Group Representations and Characters (3 Credit Units) :**

Character Theory: Definition and elementary properties of characters, class function, orthogonality relations, character relations of the first and second kind. Linear characters, irreducible characters. The character table, induced characters, lifted characters.

**Text books:**

2. Introduction to group characters, W. Ledermann.

3. Representation Theory of Finite Groups and Associative Algebras, C.W. Curtis & I. Reiner., John Wiley, (1962)

**MATH408 – Measure and Intergration (3 Credit Units) :**

Lebesgue Integral: Review of Riemann Integral. The Lebesgue integral of a bounded measurable function. Lebesgue theorems of boundeded monotone and dominated convergence. Egoroff’s theorem. Fatou’s Lemma. Extension of definition of Lebesgue integral to an unbounded measurable function. The analogues for infinite series. Integral over an unbounded set.

**Text books:**

2. The Lebesgue Integral, Burkill, J.C., Cambridge University Press, (1975).

3. Theory of Functions, E.C. Titchmarsh, Oxford University Press, (1962).

4. Theory and Problems of Real Variables, Schaum’s Outline Series, McGraw-Hill, (1969).

**MATH412 – Partial Differential Equations (3 Credit Units) :**

**Text books:**

2. Partial Differential Equations, F. John, Springer - Verlag, New York, 1978.

**MATH414 – Hydrodynamics II (3 Credit Units) :**

Water Waves: Introduction. Mathematical representation of a wave motion. Preliminary definitions, Standing or stationary waves. Surface waves. Simple harmonic surface waves. Paths of particles below surface waves. Deep water surface waves. Paths of particles below stationary waves. Group velocity. Wavelength and wave velocity.

Viscous Flows and Boundary-Layer Theory: Viscosity. Stresses in fluid motion. Stress-strain relation for a Newtonian fluid, Navier-stokes equations. Equation of motion in cylindrical and spherical polar coordinates. Reynolds number Steady flow of viscous fluids between parallel plates. Hagen-Poiseulle flow. Couette flow. Flow in tubes of cross-section other than circular. Steady motion of fluid due to a slowly rotated sphere. Boundary-layer concept. Boundry-layer equations and flow along a flat plate. Boundary-layer thickness. Dependence of boundry-layer on Reynolds number. Some simple exact solutions of boundary-layer equation. Unsteady flow due to a suddenly accelerated plane wall-Stokes’s first problem.

**Text books:**

2. Fluid Dynamics, D.R. Rutherford.

**MATH416 – Complex Analysis II (3 Credit Units) :**

**Text books:**

2. Elements of complex variables, Gordon L. I. & Lasher S., Rinehart (1963).

**MATH422 – Differential Geometry (3 Credit Units) :**

**Text books:**

2. Differential Geometry, Loius Auslander, Harper’s Series, 1967

3. Differential Geometry of three dimensions, C.E. Weatherburn, Cambridge University Press, 1964.

**MATH424 – Electromagnetic Theory and Waves (3 Credit Units) :**

**Text books:**

**MATH428 – Analytical Dynamics II (3 Credits) :**

**Text books:**

2. Classical Dynamics by Goldstein, 1990.

**STAT412 – Operations Research (3 Credit Units) :**

**Text Books:**

Operations Research, Swaroop, Gupta, P.K. and Mohan, M., Sultan Chand and Sons, New Delhi, India.

**MATH161 – Mathematics for Agriculture I (2 Credit Units) :**

Surds: Surds of the form . Rationalization of the denominators.

Sets: Types of sets, empty set, subset of a set, complement of sets. Venn diagrams, inclusion-exclusion principle.

Trigonometry: sine, cosine and tangent of various angles and their reciprocals, trigonometric ratios of special angles.

Simultaneous and quadratic equations: Methods of solving simultaneous equations, Cramer’s rule, examples. Methods of solving Quadratic Equations: factorization, completing the square, quadratic formulae and graphical method. Relationship between roots & coefficients of quadratic equations.

**Text books:**

2. Set Theory and Related Topics. Seymour Lipschutz. McGraw-Hill Book Company, New York, 1964.

**MATH162 – Mathematics for Agriculture II (2 Credit Units) :**

Integration: Integration of simple algebraic functions.

Statistics: Method of data presentation. Measures of central tendency (mean, mode, median). Measures of variation (mean deviation, standard deviation). Probability.

**Text books:**

2. Statistics for Beginners, Second Edition, S.O. Adamu, and Tinuke L. Johnson, SAAL Publications, Ibadan, 1985.

**MATH241 – Calculus I (3 Credit Units) :**

Differential Calculus: Definition a derivative. Differentiability of a function of one variable.

Geometrical and physical interpretation of a derivative. Techniques of differentiation. Rolle’s and the Mean-Value Theorems. Taylor’s and Maclaurin’s series expansions.

Applications of Differentiation: Maxima and minima of function of a single variable, curve sketching in Cartesian co-ordinates. L’Hospitals’s rule of evaluation of limits of functions in the indeterminate forms. Tangents and normals, curvature and evolutes of plane curves. Leibnitz’s formula for finding the nth differential coefficient of a product of two functions.

Integral Calculus: Indefinite integrals. Techniques of integration, change of variables, integration by parts and reduction formulae. Integration of rational functions (standard integrals and method of partial fractions). The definite integral, interpretation and properties.

Applications of integration: Average value of a function, finding lengths of arcs, plane areas, volumes of solids of revolution, area of surface of revolution, pressure, etc.

**Text books:**

2. Advanced Calculus, W. Fulks, John Wiley and Sons, Inc. New York. 1961.

**MATH242 – Calculus II (2 Credit Units) :**

Improper integrals: Improper integrals of types I, II and III. Evaluation of improper integrals, convergence of Improper integrals (Convergence in the Cauchy principal value sense). Tests of convergence.

Partial Differentiation: Partial derivatives of functions of two or three variables. Total differentials and applications.

Ordinary Differential Equations: First order differential equations with variables separable. Exact equations and integrating factors. Linear first order equations and those reducible to linear form. The Bernoulli equation. Applications (Geometrical and physical situation).

**Text books:**

2. Advanced Engineering Mathematics, E. Kreyszig, Wiley (1987)

3. Introduction to Real Analysis, A Olubumo, Heinemann, (1979)

**MATH243 – Methods of Linear Algebra I (2 Credit Units) :**

Vector Algebra: Definition of vector and physical examples; addition, multiplication by scalar, scalar and vector products. Components, applications in geometry.

Vector Analysis: Cartesian and polar coordinates in two and three dimensions. Vector functions of a real variable, continuity and application to curves and surfaces in 3-dimensional space, equation of straight line, plane and sphere, tangents and normals to curves. Tangent plane and normal to a surface.

**Text books:**

2. Advanced Engineering Mathematics, RK Jain et el, Narosa, New Delhi (2002)

**MATH244 – Method of Linear Algebra II (3 Credit Units) :**

**Text books:**

2. Linear Algebra and Matrix Theory, ED Nerring, John Wiley (1967)

**MATH341 – Ordinary Differential Equations (3 Credit Units) :**

**Text books:**

2. Elementary Differential Equations with Linear Algebra, R.L. Finney and D.R. Osibeng, Addison – Wesley Publishing Co., London, 1976.

**MATH342 – Functions of Several Variables (2 Credit Units) :**

Series of Functions: Uniform Convergence. Differentiation and integration of series. Power series, radius of convergence. Expansions of elementary functions in Maclaurin and Taylor series. Use of series to find values of integrals and roots periodic functions.

Fourier Series: definition and evaluation of Fourier coefficients. Expansion in Fourier sine –cosine series of a periodic function of period 2. Fourier sine and cosine series for a function defined on the
interval (a,b). Complex exponential form of a Fourier series.

**MATH441 Complex Analysis (3credit Units) :**

**Text Book:**

2. Elements of Complex Variables, L.I. Gordon and S. Lasher, Holt, Rinechart and Winstone. Inc., New York, 1963

**MATH443 Numerical Analysis (3 Credit Units) :**

**Text Book:**

2. Numerical Analysis, P.R. Turner, Macmillan, London, 1994

**MATH541: Conformal Transformation and Applications (3 Credit Units) :**

Invariance of harmonic functions under conformal transformations. The Schwarz, Christoffel transformation. The use of the complex potential in electrostatics and hydrodynamics (two dimensional problems),Application of conformal transformation to solve potential problems in electrostatics and Hydrodynamic, Riemann surfaces.

**Text Book:**

2. Elements of Complex Variables, L.I. Gordon and S. Lasher, Holt, Rinechart and Winstone. Inc., New York, 1963

**MATH542 Partial Differential Equations (3 Credit Units) :**

**Text books:**

2. Partial Differemntial Equations, F. John, Springer – Verlag, New York, 1978.,

3. Further Engineering Mathematics, K.A Stroud,4th Edition, Palgrave, Macmillan.

4. Advanced Engineering Mathematics, K.A Stroud,4th Edition, Palgrave, Macmillan.