Motto: 'Critical Thoughts for Critical Minds'

**Entry Requirements**

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.

**Duration of the programme**

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.

**Graduation Requirements**

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)

**BSc. Mathematics Courses **

**Copyright © 2016 - All Rights Reserved - Department of Mathematics (ABU Zaria)**

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

**Prerequisite – O/Level Mathematics**
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**

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

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

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

**Prerequisite – O/Level Mathematics**
Circular Measures: Trigonometric ratios of angles of any magnitude, inverse trigonometric functions.
Addition formulae: Sin (A+B), cos (A+B), tan (A+B) and their proofs. Multiple and half angles,
solutions of simple trigonometric equations. Factor formulae. Solution of triangles,
heights and distances (including three-dimensional problems)

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**

1. Mathematics for Fresh Undergraduates Vol. II, B.K. Jha, A.O. Ajibade, M.I. Yakubu and A.T. Imam, ABU press (2013)

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) :**

** Prerequisite – O/Level Mathematics. **
Functions of a real variable: Odd, even, periodic functions and their symmetries, graphs,
limits and continuity (Intuitive treatment only)

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**

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

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:**

** Prerequisite: O/Level Mathematics**
Introduction to computer systems. Components of computer systems and their functons. Windows operating
systems and its utilities. Hands-on exposure to Office application software (MS Office or Open Office):
Word processing, spreadsheets, presentation graphics and databases. Introduction to and use of Internet tools and technologies.

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**

1. S.B. Junaidu, A.F. Donfack-kana and A. Salisu, Fundamentals of information technology ABU press (2013)

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:**

** Prerequisite – O/Level Physics. **
Units and dimensions; Dimension methods for checking correctness of equations and for deriving simple relations.
Additions and subtraction of vectors, projectiles, Newton laws, conservation laws, Elastic collisions, work, energy and power.
Circular motion, simple harmonic motion, motion of rigid bodies, statics Gravitational potential, circular orbit, escape velocity.

**PHYS131 Heat and Properties of Matter:**

** Prerequisite – O/Level Physics. **
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) :**

** Prerequisite – O/Level Mathematics**
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) :**

** Prerequisite – O/Level Mathematics. **
Conics: Properties of parabola, ellipse, hyperbola, rectangular hyperbola, their Cartesian and parametric equations,
problems involving elimination of parameters, tangents and normals.

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**

1. Mathematics for Fresh Undergraduates Vol. II, B.K. Jha, A.O. Ajibade, M.I. Yakubu and A.T. Imam, ABU press (2013)

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) :**

** Prerequisite – O/Level Mathematics**
Vectors: Geometric representation of vectors in 1-3 dimensions, components, direction cosines. Addition, scalar multiplication,
linear independence and dependence of vectors. Scalar and vector products of vectors. Differentiation and integration of vectors w.r.t a scalar variable.

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**

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

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):**

** Prerequisite – O/Level Mathematics. **
Random experiment, Sample space, event space, definitions of probability, conditional probability, addition and multiplication theorems,
definition of random variable (discrete and continuous), mathematical expectations of a random variable, addition and multiplication
theorems of expectation, definition of moment, relationship between raw moments and central moments, the bi-variate frequency distribution,
fitting of curves by method of least squares, concepts of correlation and regression and their coefficients, the rank correlation coefficient.

**Text Books**

1. Statistics for Fresh Undergraduates, Yahaya A. and Nnamani C.N., ABU press (2013), Zaria.

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.

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

**Prerequisite – MATH105 or equivalence**
Applications of Calculus: Revision of different techniques of differentiation, successive differentiation, Leibniz’s theorem, Taylor’s and Maclaurin’s series. Tangents and normals to plane curves, curvature, Definite integrals. Methods of integration, reduction formulae, lengths of arc of a plane curve. Area enclosed by a plane curve. Improper integrals.

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 :**

1. Mathematical Methods, J. Heading, University Press, (1963).

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

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

**Prerequisite – MATH105 or equivalence**
Preliminaries: Properties of real numbers, algebraic and topological properties, identity theorem, density theorem for Q and R. Ordering and properties.

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:**

1. Introduction to Real Analysis, A. Olubunmo, Heinemann (1979).

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) :**

**Prerequisite – MATH101 or equivalence**
Logic and Methods of Proof: Sentential logic; statements, sentential connectives (negation, disjunction, conjunction, implication, ), truth tables, tautologies (or valid statement formulae), quantifiers. Methods of proof; indirect (or proof by contradiction) method, contrapositive proof, natural deduction.

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:**

1. Symbolic logic, C.I. Lewis and C.H. Longford, Prentice Hall (1975).

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) :**

**Prerequisite – MATH102 or equivalence**
Matrices: Definition, types of matrices, algebra of matrices, matrix
as a sum of symmetric and skew Symmetric matrices. Elementary
operations of matrices and echelon form, equivalence matrices.

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:**

1. Linear Algebra, S. lipschutz (Schaum’s Outline Series) Mc Graw-Hill (1987)

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

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

**Prerequisite – MATH105**
Accuracy in numerical calculations: errors and their sources, error accumulation in different operations.
Finite differences: difference operators and difference table.

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:**

1. Introduction to Numerical Analysis, Carl-Eric Froberg, Addison-Wesley publication, (1981).

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) :**

**Prerequisite – STAT102. **
Brief revision of various definitions of probability. Baye’s theorem, concepts of probability function, probability density function, cumulative probability density function and moment generating function. Univariate discrete probability distributions such as Bernoulli distribution, Binomial and Poisson distribution, type I and type II geometric distributions, negative binomial distribution, hypergeometric distribution, various properties of all these distributions, fitting of binomial, Poisson and geometric distributions.

**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) :**

**Prerequisite – MATH104 or **
Revision of first order ordinary differential equations. Derivation of differential equations from primitives.

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:**

1. Theory and Problems of Advanced Mathematics for Engineers and Scientists, M.R. Spiegel (Schaum’s Outline Series) Mc Graw-Hill (1974).

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) :**

**Prerequisites – MATH105 or **
Real Functions of one Variable: Limits of functions. Improper limits (limits at + and - Algebraic operations on limits of functions. Continuity of functions on sets and related results. Uniform continuity.

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 :**

1. A first course in Real Analysis, Protter , M.H. and Morrey, C.B, Springer-Verlag, (1977).

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

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

**Prerequisites – MATH102 or**
Group: Review of definition of a group, subgroups, criteria for subgroups, special subgroups, centralizer of an element, centre of a group, cosets, normal subgroups, quotient groups, conjugate elements and conjugacy relation, congruence relation modulo and subgroup H. Lagranges theorem. The class equation and applications.

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:**

1. Symbolic logic, C.I. Lewis and C.H. Longford, Prentice Hall (1975).

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) :**

**Prerequisite – MATH102**
Vector Spaces: Review of basic definitions and examples of vector spaces. Subspaces, linear dependence and independence. Bases, dimension of a vector space. Homomorphism and quotient space. Direct sum, Dual spaces.

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:**

1. Linear Algebra, S. Lipschutz (Schaum’s Outline Series) Mc Graw-Hill (1987)

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

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

**Prerequisite – STAT102**
Univariate continuous probability distributions such as Normal, Uniform, exponential, type I and type II beta and gamma distributions, various properties of these distributions, fitting of normal distribution. Concept of Bi-variate probability distribution, joint, marginal, conditional probability distribution, covariance and correlation of bi-variate r.v. sampling distribution and standard errors of statistics, distribution of functions of random variables using the techniques such as cumulative distribution function technique, moment generating function technique and transformation technique.

**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.

**COSC202: Fortran and Structured Programming:**

**Prerequisite – COSC101 or Equivalence**
Structured programming elements, structured design principle, abstraction, modularity, stepwise refinement, structured design techniques, teaching of structured programming language, FORTRAN: Characters, constant and variables, Arithmetic assignment statement. FORTRAN standard functions. READ and WRITE statement, Transfer of control, subscripted variables, DO statement. SUBPROGRAMMES: Arithmetic function, function, Subroutine and Subprogramme.

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

**Text Books:**

FORTRAN 90 for Engineers and Scientists, Larry Nyhoff and Sanford Leestma

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

**Prerequisites – MATH201**
Vector Fields: Revision of definitions and elementary results related to vectors; gradient, divergence and curl in different co-ordinate systems. Multiple integrals; areas and volumes, Surface and Line integrals; Stokes theorem, Divergence theorem. Green’s theorem.

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:**

1. Mathematical Methods, J. Heading, University Press (1963).

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) :**

**Prerequisite – MATH203**
Point Set Theory: Theorem of nested intervals, accumulation and isolated points, Bolzano – Weierstrass theorem, closed and open sets, interior, exterior and boundary points of a set and related theorems. Cantor’s decreasing set theorem. Lindeloff covering theorem and Heine-Borel theorem, Perfect sets, Cantor’s ternary set and some of its properties.

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:**

1. The elements of Real Analysis, Bartel R.G. John Wiley, (1964)

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

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

**Prerequisite – MATH206**
Structure of Integers: Euclidean algorithm, g.c.d, primes, relative
primes, unique factorization theorem
(or Fundamental theorem of arithmetic).

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:**

1. Rings, Field and Groups (Introduction to Abstract Algebra), R.B.J.T. Allenby, Edward Arnold, (1993).

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

**Prerequisite – MATH203**
Sequences and Series of Complex Numbers: Definition of sequences and series of complex numbers, Properties of convergence, uniform and absolute convergence of sequences of complex numbers. Algebraic operations on limits of sequences. Limit, Continuity and Differentiability of Complex Functions: Definition of complex function. Properties of continuous complex functions. Limit of a complex function and its properties. Continuity of a complex function. Differentiation of a complex function. Analytic and entire functions.

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:**

1. Theory and Problems of Complex Analysis, Spiegel, M.R. (Chaums’s Outline Series) Mc Graw-Hill, (1981).

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

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

**Prerequisite – MATH201**
Introduction: Velocity and acceleration of a particle along curve, Radial and Transverse components of velocity and acceleration, Velocity and acceleration in 3-dimensional motion. Angular velocity Relative velocity.

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:**

1. Classical Dynamics of particles and systems, Jerry B. Marion, Academic Press, (1970).

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

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

**Prerequisite – MATH201**
Methodology of Model building: Identification, formulation and solution of problems. Cause-effect diagrams. Modeling using graphs and proportionality: Modeling by interpolation using polynomials. Modeling using Least squares and Linear programming. Modeling deterministic behavior and probabilistic processes. Modeling using derivatives: applications using differential equations.

**Text books:**

1. A first course in Mathematical Modeling, F.R Giordano & M.D. Weir, Woodsworth, Inc. (1985).

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) :**

**Prerequisite MATH205**
Elements of First Order Logic, Cantorian Set Theory, Intuitive Notion of Cardinals and Paradoxes, Evaluation of Axiomatic Method, Properties of Axiomatic: Consistency, independence, and completeness. Examples.

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

**Text books:**

1. Intermediate Set Theory, Singh. D., and Drake, F.R., John Wiley, 1996.

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) :**

**Prerequisite-MATH205**
Division and Factorization properties for positive integer multiplicative arithmetical functions e.g Euler’s φ-function.The mobius function μ. Linear congruences, residue sets (mod m). Euler’s theorem. Fermat’s theorem. Chinese remainder theorem. The ring Z of residue classes (mod m).

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 :**

1. Number Theory. J. Hunter, Oliver and Boyd, 1964.

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

**Prerequisite – MATH209**
Solution of simultaneous equations and other linear system of equations: Eigenvalues and Eigenvectors. Numerical solutions of ordinary differential equations: Euler’s, Picard’s, Taylor’s and Runge-Kutta methods, prediction – corrector methods. Matrices and Determinants. Introduction to numerical solution of partial differential equations.

**Text books:**

1. Theory and Problems of Complex Analysis, Spiegel, M.R. (Schaums’s Outline Series) Mc Graw-Hill, (1981).

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

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

**Prerequisite – MATH201**
Sturm-Liouville Problem; orthogonal polynomials and functions. Gamma and Beta functions of complex variables.

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:**

1. Theory and Problems of Advanced Mathematics for Engineers and Scientists, M.R. Spiegel (Schaum’s Outline Series) Mc Graw-Hill (1974).

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) :**

**Prerequisite MATH303**
Topological Spaces: Definition and examples of topological spaces, open and closed sets, neighborhoods, limits (cluster) points, interior and closure of a set, boundary, coarser and finer topologies, Bases and Subbasis. Subspaces of Topological spaces. Product topology. Quotient topology. First and second countable spaces. Separable spaces. Separation axioms. Topology of metric spaces. Convergence of sequence in a topological space, pointwise and uniform convergence, limit of functions at given points. Limit of functions in first countable Hausdorff spaces.

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:**

1. General Topology, Kelley, J.L., Van Nostrand, (1970)

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) :**

**Prerequisite MATH305**
Arithmetic Structures of groups: Definition and example of p-groups. Sylow p-subgroup, Sylow’s theorems (proofs and applications). Determination of all groups of low order, up to order 15.

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:**

1. Topics in Algebra, I.N. Herstein, Blaisdell, (1964)

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

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

**Prerequisite – MATH303**
Uniform Convergence of Sequences and Series of Functions: Pointwise and uniform convergences, Cauchy’s general principle of uniform convergence, test for uniform convergence; Mn -test, Weistrass M test, Abel’s test, Dirichlet’s test. Uniform convergence and continuity, Dini’s theorem. Integrability of uniform limit of a
uniformly convergence series of integrable functions, term by term integration.

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:**

1. The elements of Real Analysis, Bartel R.G. John Wiley, (1964)

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

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

**Prerequisite – MATH301**
Kinematics of Fluids: Lagrangian and Eulerian methods of treating motion of fluids. Steady and unsteady flows. Streamlines. Resolution of fluid motion into translation, rotation and deformation. Irrotational motion. Velocity potential. Fluid acceleration in Eulerian method. Acceleration components in Cartesian cylindrical and spherical polar coordinates. The significance of the operator D/Dt. V. condition for a boundary surface.

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:**

1. Hydromechanics, Besant and Ramsey
2. Fluid Dynamics, D.R. Rutherford.

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

**Prerequisite- MATH309**
Experimental observations. Bohr’s model of the atom and classical quantization. Uncertainty and complementary principles. Hermitian operators. Eigenvalues and eigenvectors. The commutation relation [x,y]=ih/2π. Schrödinger equation. One-dimensional square-well potential, infinite barriers. Differential equation and operator methods for linear harmonic oscillator. 3-dimensional central potentials, hydrogen atoms. Differential and scattering cross-sections. Laboratory and centre of mass frames. Partial wave analysis of the scattering amplitude.

**Text books:**

1. Quantum Mechanics by L.I.Schiff, McGraw Hill, 1915

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

**Prerequisite – MATH311**
The role of Mathematics in Biology and Medicine: Introduction and examples of some models.

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) :**

**Prerequisite- MATH303**
Graphs: Varieties of graphs, degrees, Extrema graphs, intersection graphs, operations on graphs.

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) :**

**Prerequisite: MATH303**
Metric Spaces: Separability, Completeness and compactness, contraction mapping theorem. Arzela – Ascoli lemma. Stone-Weierstrass theorem.

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:**

1. Introduction to Functional Analysis, Taylor, A. E. John Wiley (1958)

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

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

**Prerequisite – MATH303**
Compactness: Lindeloff spaces, finite intersection property. Bolzano – Weierstrass property, sequential compactness, countable compactness, Relatively compact sets, compact subspaces of T2 – spaces, local compactness and compactification theorem. Continuous mappings of compact sets; images of compact sets , continuous mappings on compact sets ranged in Rn. Closed subspaces of compact spaces; compact subset of Rn.

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

**Text books:**

1. General Topology, Kelley, J.L., Van Nostrand, (1970)

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) :**

**Prerequisite – MATH305**
Introduction: Historical background. Types of representations, permutations, automophism and matrix (principle, linear and faithful), equivalence representations, G – submodules, G –homomorphisms. Reducible representation: Reducibility and G – submodules, irreducibility, Maschke’s theorem. Complete reducibility and direct sum of G – submodules. Canonical decomposition of representations. The regular representation. The Schur’s Lemma. The commulant algebra. Tensor products of matrices. The group algebra (KG). Decomposition of the regular representations. Number of in equivalence irreducible representations of a group is equal to the number of thedistinct conjugacy classes. Lifting process, induced representation.

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:**

1. Group Representations and Characters, V.E. Hill.

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) :**

**Prerequisite – MATH303**
The Lebesgue Measure on the real line: Outer measure, measurable sets, σ-algebra. Measurability of sets. Measurability of open sets, properties of Lebesgue measure and Lebesgue measurable sets, construction of a non-measurable sets. Cantor’s set. Measurable functions.

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:**

1. Measures and Integration, Munroe, M.E., Addison-Wesley, (1971).

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) :**

**Prerequisite - MATH202**
Basic concepts. Boundary and initial value problems.Theory and solutions of first and second order linear equations; wave, heat and Laplace equations in Cartesian and polar coordinates, classifications, characteristics, canonical forms. Cauchy problems. Elliptic equations; Laplace and Poisson equations, solution in cylindrical, polar and spherical coordinates. Hyperbolic and parabolic equations; Wave and diffusion equations. Green’s function, harmonic function, properties.

**Text books:**

1. Partial Differential Equations, L. Bers, F. John and M. Schechter, John Wiley and sons, Inc., New York, 1964.

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

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

**Prerequisite – MATH301**
Irrotational Motion in Three dimensions: Motion symmetrical about an axis Stoke’s stream function. Values of Stoke’s stream function in case of simple source and of a doublet. Motion of a sphere through a liquid at rest and at infinity. Liquid streaming past a fixed sphere. Motion of liquid inside a rotating ellipsoidal shell.

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:**

1. Hydromechanics, Besant and Ramsey

2. Fluid Dynamics, D.R. Rutherford.

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

**Prerequisite – MATH307**
Taylor and Laurent series expansions. Isolated singularities and residue. The residue theorem and some of its consequences. Maximum modulus principle. Argument principle. Rouche theorem. The fundamental theorem of algebra. Principle of analytic continuation. Morera’s theorem. Cauchy-Liouville theorem. Conformal and bilinear mappings. Multiple-valued functions and Riemann surfaces.

**Text books:**

1. Theory and Problems of Complex Analysis, Spiegel, M.R. (Chaums’s Outline Series) Mc Graw-Hill, (1981).

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

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

**Prerequisite – MATH301**
Curves: Type of curves, Serret-Frenet formulae, natural equations, Local surfaces; fundamental forms, curves on surfaces, porallelism, normal forms, principal, Gauss and mean curvatures, special sufaces, Gauss and Codazzi-Mainardi equations, geodesics, isometries. Global surfaces; surfaces in Euclidean space, ovaloids, Gauss-Bonnet theorem, shortest connecting curves, convex surfaces.

**Text books:**

1. A course in Differential Geometry, Wilhelm Klingenberg, Springer-verlag. 1978.

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) :**

**Prerequisite – MATH301**
Electrostatics; conductor, surface charge, capacitors, uniqueness theorem, minimum energy theorem, Green’s reciprocal theorem, Dielectrics, Magnetostatics; two dimensional problems, steady currents, Fields, field energy, reciprocal theorem, circuits, magnetic media. Electrodynamics; Maxwell’s equations, energy, forces and momentum relations in the electromagnetic field. Electromagnetic waves; wave equation and place waves.

**Text books:**

1. An introduction to electromagnetic theory , P.C. Clemmow, Cambridge University Press, 1973.

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

**Prerequisite- MATH309**
Integrals: The meaning of integrals, Jacobi’s integral. Noetherian forms and moment integral.
Stability: Definition and variational equation, indirect and direct methods of Lyapunov. Applications.
Celestial Problems: Central force problem, Apsidal, The n-body problem, the 2-body problem. Impulsive motion. Fundamental EQUATION. Impulsive motion theorem, Lagrange’s equations of impulsive motion.

**Text books:**

1. Classical Dynamics of Particles Systems by Jerry B. Marion, 1988.

2. Classical Dynamics by Goldstein, 1990.

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

**Prerequisite – MATH311**
Classical methods of optimization, Maxima and minima, Lagranges’ multipliers. Linear programming: Convex sets and functions, simplex and revised simplex methods, duality theory, applications. Linear programming applications to diet problems, transportation problems, manufacturing problems, Network Analysis, etc.

**Text Books:**

Operations Research, Sharma, J.K., Macmillan India.

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

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

**Prerequisite – O/Level Mathematics**
Indices: Laws of indices, relationship between indices and logarithm, Laws of logarithm of any positive number to a given base, change of bases in logarithm.

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:**

1. Additional Mathematics for West Africa, A Godman, and F. Talbert, Longman Group UK Limited, Essex, England, 1984.

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

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

**Prerequisite – O/L Mathematics**
Differentiation: Differentiation from first principle. Differentiation of sum, product and quotient of functions. Differentiation of simple explicit and implicit algebraic functions.

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:**

1. Additional Mathematics for West Africa, A. Godman and F. Talbert, Longman Group UK Limited, Essex, England, 1984.

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

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

**Prerequisite – MATH105**
Sequences and Functions: Infinite sequences and their limits. A short recollection of elementary functions and their properties. Limits and continuity of functions of a single real variable.

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:**

1. Concepts of Calculus, A.H. Lighstone, Happer and Row, Publishers London. 1965.

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

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

**Prerequisite- MATH105**
Infinite (number) series and their properties: test of convergence, complex series.

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:**

1. Mathematical Methods, J Heading, University Press (1963)

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) :**

**Prerequisite- MATH102**
Complex numbers: Addition, multiplication, division, Argand diagram, polar representation, Demoivre’s Theorem.

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:**

1. Vector Analysis, Murry R. Spiegel, Schaum’s Outline Series (1974)

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

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

**Prerequisite- MATH102**
Determinants and Matrices: Definition and properties of a determinant, its evaluation. Matrices, addition, multiplication by scalar, product, adjoint of a non-singular matrix, rank and its evaluation. Simultaneous linear equations: Consistency, linear dependence, solutions (including Cramer’s rule). Eigenvalues and special matrices, symmetric, skew- symmetric, orthogonal, etc.

**Text books:**

1. Linear Algebra, S Lipschultz (Schaum’s Outline Series) McGraw Hill (1987)

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

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

**Prerequisite – MATH241**
Exact equations, linear equations of first and second order with constant coefficients, geometrical interpretation, Isoclines. Statement of existence theorem. Linear second order equation with variable coefficients. Series solutions of Differential Equation with non singular points. Definition and properties of Bessel equation, Bessel function of the first kind. Definition and properties of the Legendre equation, Legendre polynomials. Fourier integrals and transforms. Laplace transform and its applications to the solution of Differential equations.

**Text books:**

1. Introduction to Ordinary Differential Equations, A.L. Rabensten, Academic Press, London, 1966.

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) :**

**Prerequisite – MATH241**
Total differential and partial differentiation, chain rule, partial implicit function, Gradient and its geometrical interpretations. DIV, and CURL of Vector field. Laplacian. Maxima and minima. Multiple integrals and their evaluation by change of variables, applications; volume, mass, centre of gravity, moment of inertia. Surface and line integrals, Green, Gauss and Stroke’s theorems.

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) :**

**Prerequisite – MATH341**
Complex functions of real variable. Elementary functions of complex variable, Differentiation of complex variable, Cauchy-Riemann equations, Analytic and harmonic functions, Integration of complex variables, Cauchy Theorem, Simple examples of expansion in Taylor and Laurent Series, Poles and residues (without integral evaluation).

**Text Book:**

1. Complex analysis 2nd Edition, L.V. Allfors, McGraw-Hill New York, 1966.

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) :**

**Prerequisite- MATH241**
Accuracy in numerical calculations, errors, significant figures, calculation of sinx, cosx and exponent by Taylor expansion. Interpolation; Newton’s forward, backward and central differences formulae. Numerical differentiation and integration; trapezoidal and Simpson rules, automatic selection of interval size, Newton-Cotes formulae. Solution of algebraic and transcendental equations; graphical, bisection interaction, Newton- Raphson solution of simultaneous equations; (Gauss and Gauss-Siedel), Eigenvalues and eigenvectors. Numerical solution of ordinary differentials equations. Methods of Euler, Picards Taylor and Runge-Kutta predictor-corrector. Method for solving ordinary differential equations. Introduction to partial differential equations.

**Text Book:**

1. Introduction to Numerical Analysis, P.A. Stark, Macmillan, London, 1970

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

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

**Prerequisite MATH441**
Mobius transformations and effect on circle and straight lines. Other elementary transformations e.g. w = z2; w=sin z w=z+ .

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:**

1. Complex analysis 2nd Edition, L.V. Allfors, McGraw-Hill New York, 1966.

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) :**

**Prerequisite – MATH341**
Partial Differential Equation and special functions. The Fourier integral. Orthogonal functions, Sturm- Liouville problem, Partial differential Equations: Basic concepts. Vibrating String and one dimensional wave equation, variable separable method, D’Alembert’s solution. One-dimensional heat flow; heat heat flow in an infinite bar; vibrating membrane. Two dimensional wave equation; solutions for a rectangular and a circular membrane. Laplace equation; potential solution of Laplace equation, Solution on cylindrical polar and sphecrial polar coordinates (with polar symmetry). Functions Denined by infinite integrals, Gamma, Beta Functions, Error functions, Frosnel integrals, sine and cosine integrals.

**Text books:**

1. Partial Differential Equations, L Bers, F.John and M Schechter, John Wiley and sons, Inc, New York, 1964.

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.

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