## Department of Mathematics and Statistics Programmes.

## Postgraduate Programme

## Undergraduate courses

#### Click on each course’s Heading for content

### **EMT 1201 – Precalculus**

This is the first course offered to first level students. It aims at solidifying basic mathematics concepts learnt in secondary school and helping students to transition to university level mathematics with ease. Topics covered are: Review of integer exponents, nth roots, factoring and fractional expressions, sets of real numbers, Absolute Value, Rectangular Coordinates, Graphs and Graphing Utilities, Equations of Lines, Symmetry and Graphs, Circles, Quadratic Equations, Inequalities, Functions, Graphs of Functions, Graphing Techniques, Inverse functions, Polynomial and Rational Functions, Exponential and Logarithmic Functions.

#### **Recommended textbooks**

1. Cohen D, Precalculus: A Problem-Oriented Approach, Thomson Brookes/Cole, 2005

### **EMT 1202 – Calculus I**

This course covers trigonometry and introduces students to calculus. Topics covered are Trigonometric Functions of Acute Angles, Algebra and the Trigonometric Functions, Right Triangle Applications, Trigonometric Functions of Angles, Trigonometric Identities, Radian Measure, Radian Measure and Geometry, Trigonometric Functions of Real Numbers, Graphs of the Sine and Cosine Functions, The Addition Formulas, Double-Angle Formulas, Product-to-Sum and Sum-to-Product Formulas, Trigonometric Equations, Inverse Trigonometric Functions, Tangent and Velocity Problems, The Limit of a Function, Calculating Limits using the Limit Laws, Continuity, Limits at Infinity – Horizontal Asymptotes, Derivatives and Rates of Change, The Derivative as a Function, Derivatives of Polynomial and Exponential Functions, Product and Quotient Rules, Derivatives of Trigonometric Functions, Hyperbolic Functions, Maximum and Minimum Values, The Mean Value Theorem, How Derivatives Affect the Shape of a Graph, 4.4 Indeterminate Forms and L’Hospital’s Rule

## **Recommended textbooks**

1. Cohen D, Precalculus: A Problem-Oriented Approach, Thomson Brookes/Cole, 2005

2. Stewart J, Calculus: Early Transcendentals, 6th Ed, Thomson Brookes/Cole, 2008.

#### **EMT 2301 – Calculus II**

Riemann Sums (Numerical integration, definite integral, properties of definite integrals, fundamental theorem of calculus, indefinite integrals and definite integrals, substitution rule), Applications of the definite integrals (area between two curves, volumes of a revolution, volumes of cylindrical shells, average value of a function), Techniques of integration (integration by parts, Trigonometric integrals, partial fractions, improper integrals), Further applications of integration (arc length, area of a surface of revolution), Differential equations (introduction, separable equations, linear first order equations), Parametric equations and polar equations (Calculus with parametric equations, arc length in parametric form, arc length in polar form).

#### **Recommended textbooks**

1. Stewart J, Calculus: Early Transcendentals, 6th Ed, Thomson Brookes/Cole, 2008.

2. Hughes-Hallett D, Gleason A.M, McCalum W.C., et.al., Calculus – Single Variable, 5th Ed., 2009

#### **EMT 2401 – Vector Calculus**

Vectors and the Geometry of Space (3-Dimensional coordinate systems, vectors, the dot product, the cross product, triple product, equations of lines and planes, functions and surfaces), Vector Functions (vector functions and space curves, derivatives and integrals of vector functions, Arc Length and Curvature), Partial Derivatives (Functions of Several Variables, Limits and Continuity, Partial Derivatives, Tangent Planes and Linear Approximations, The Chain Rule, Directional Derivatives and the Gradient Vector, Maximum and Minimum Values, Lagrange Multipliers), Multiple Integrals (Double Integrals over Rectangles, Iterated Integrals, Double Integrals over General Regions, Double Integrals in Polar Coordinates, Applications of Double Integrals, Surface Area, Triple Integrals, Triple Integrals in Cylindrical and Spherical Coordinates, Change of Variables in Multiple Integrals), Vector Calculus (Vector Fields, Line Integrals, The Fundamental Theorem of Line Integrals, Green’s Theorem, Curl and Divergence, Surface Integrals, Stoke’s Theorem, The Divergence Theorem)

**Recommended textbooks:**

1. James Stewart (2008), Calculus – Early Transcendentals, 6th Edition, Thomson Brooke/Cole

2. Colley Susan Jane (2012), Vector Calculus 4th Ed., Pearson Education Inc

3. Harshbarger R.J. and Reynolds J.J. (1990) Calculus with Applications, D.C. Health and Company Lexington

4. Swokowski, E.W. and et al (1983) Calculus 6th Ed., PWS Publishing Company Boston

5. Anton, H. (1992) Calculus with Analytic Geometry, 4th Ed., John Wiley and Sons New York

#### **EMT 2402 – Introduction to Statistical Analysis**

Describing data with graphs, Describing data with numerical measures, Bivariate data, Probability and Probability Distributions, The Normal Probability Distribution, Sampling Distributions, Large sample estimation, large sample tests of hypotheses, Inference from small samples, Analysis of variance, Linear regression and correlation, Multiple regression analysis, Analysis of categorical data, Nonparametric statistics.

#### **Recommended textbook**

1. Mendehall and Beaver, Introduction to probability and statistics, 14th Ed, 2013

#### **EMT 3501 – Linear Algebra**

The axioms for the definition of a vector space and examples of vector spaces, Vector subspaces and examples, Free and generating sets in a vector space, Linear independence and a spanning set, Basis for a vector space and dimension of a vector space, Direct decomposition of a vector space, Quotient spaces of a vector space and their dimension, Linear mappings between vector spaces, Invariant subspaces, Semisimplicity, Isomorphism, homomorphism, endomorphism, image, kernel, The Homomorphism Theorems, Vector space of linear mappings, Dual spaces and dual mappings, Matrix representation of linear mappings, Change of basis, Equivalence of bases and matrices, rank of a matrix and its significance, Eigenvectors, eigenvalues and eigenspaces, Characteristic polynomial of an endomorphism and its associated companion matrix, The complexification of a vector space, Tensor product in a vector space, Euclidean spaces, scalar product, orthogonality, orthogonal decomposition, orthogonal complement, Linear mappings between Euclidean spaces, Adjoint of a linear map, trace and norm, Use of Maple Programming in solving simple problems in vector spaces.

#### **Recommended textbooks**

1. Spindler K., Abstract Algebra with Applications, Vol. 1 (Chapman & Hall / CRC Pure and Applied Mathematics) 1993

2. Abstract Algebra with Applications, Vol 2, by Karlheinz Spindler, (Dekker 2004)

3. Linear Algebra by Curtis, C.W. (Springer 1974)

4. Linear Algebra by Greub, W.H. (Springer 1967)

5. Finite Dimensional Vector Spaces by Halmos, P.R. (van Nostrand 1958)

6. Linear Algebra and its Applications by Strang, G. (Academic Press 1976)

7. A Survey of Modern Algebra by Birkhoff, G and Maclane, S. (MacMillian)

8. An Introduction to Abstract Algebra by Moore, J.T. (Academic Press)

9. Algebra Volume 1 by Cohn, P.M. (John Wiley and Sons)

#### **EMT 3502 – Mathematical Statistics I**

Probability distribution (review), chebyshev’s inequality, Multivariate distributions (joint, marginal and conditional distribution), Some special distributions (Gamma, chi-squared and normal distributions), Distribution of function of random variables (Transformations of variables, The Beta, t, and F- distributions, Distribution of order statistics, The distribution of chi and ns2/sigma2, Limiting distribution (convergence in distribution, convergence in probability, limiting moment – generating function, the Central limit Theorem), Regression modelling

#### **Recommended textbooks**

1. Hogg, R.V. and Crig, A.T. (1970), 3rd Ed Introduction to Math Statistics, Collier Macmillan Publisher (London)

2. Francis, A. (1979), first Ed, A – level statistics, Stanley Thormes (Publisher)Ltd.

3. Hoel, P.G. (1984), 5th Ed Introduction to Math Statistics, John Wiley & Sons Inc.

4. Mood, A.M, Graybill, F.A and Boes, D.C. (1974) 3rd Ed. Introduction to the theory of Statistics, MacGraw – Hill Book Company

#### **EMT 3601 – Ordinary Differential Equations**

Introduction to Differential Equations (Definitions and Terminology (ODEs and PDEs, Order of an equation, Linear and Non Linear Equations, Solution of a DE, Explicit and Implicit Solutions, Particular and General Solution), First Order Differential Equations (Separable variables, Exact Equations, Linear Equations, Solutions by Substitutions), Modelling With First Order Des (Growth and Decay, Population Models, Logistic Equations), Differential Equations of Higher Order (Initial Value and Boundary Value Problems, Homogeneous equations, Nonhomogenous equations, Reduction of order, Homogenous linear eqs with constant coefficients, Undetermined Coefficients (Superposition and Annihilator Approaches), Variation of Parameters, Cauchy-Euler Equation, Systems of Linear Equations), Modelling with Higher Order Des (Free undamped vibrations, Free damped vibrations), Series Solutions of Linear Equations (Review of power series; power series solutions, Solutions about ordinary points, Solutions about singular points)

#### **Recommended textbooks**

1. Zill, D.G., A First Course in Differential Equations (with Modelling Applications), 9th Ed, Brooks/Cole Cengage Learning, 2005

2. Boyce W.E. and DiPrima R.C., Elementay Differential Equations and Boundary Value Problems, 7th Ed., John Wiley & Sons Inc., 2001

#### **EMT 3602 – Real Analysis**

Review of Set Theory, Point Set topology (open sets, closed sets, continuity, connected nets, compact sets), Metric Spaces (basic topology in metric spaces, limits and continuity in metric spaces), Real numbers (as a completion of rationals, least upper bound, least upper bound, lim inf of real sequences), Functions of real variables (sequence and series of functions, uniform convergence, Taylor series), Integration theory (Riemann or Lebesgue), Measure theory on real numbers.

**Recommended textbooks**

1. Royden, H.L., Real Analysis, Macmillan Company: New York, 1985

2. Bryant, V., Metric Spaces, Iteration and Applications, CUP: Cambridge, 1985.

3. Russel, G.A., Real Analysis, A First Course, 2nd Edition, Addison Wesley, 2001

#### **EMT 4701 – Applied Mathematics**

Numerical methods for solving ODEs (Euler’s methods, Improve Euler method, Runge – Kutta methods, Bisection methods, Newton Raphson method, Fixed point iteration), Fourier Series, Partial differential equations (change of variables, Laplace’s equations, Heat and wave equations, Solution by use of Laplace transforms, solution by use of Fourier transforms), Calculus of variation (the Euler equation, Geodesics, The brachistochrone problem, Hamilton’s principle, Isoperimetric problems), Linear programming (systems of equations, simplex methods, Duality, Applications)

#### **EMT 4702 – Abstract Algebra**

Topics covered include Proofs, Sets and Equivalence Relations, Mathematical Induction, The Division Algorithm, Groups, Subgroups, Cyclic Groups, Permutation Groups, Cosets, Lagrange’s Theorem, Isomorphisms, Direct Products, Factor Groups, Normal Subgroups, Group Homomorphisms, Group Actions, Rings, Integral Domains and Fields, Ring Homorphisms and Ideals, Maximal and Prime Ideals, Polynomial Rings, Irreducible Polynomials, Vector Spaces

#### **EMT 4801 – Introduction to Number Theory**

Divisibility (Divisors, Bezout’s Identity, Least common multiples, linear Diophantine equations), Prime numbers (Prime numbers and prime-power factorisations, distribution of primes, Fermat and Mersenne primes, Primality-testing and factorisation), Congruence (Modular arithmetic, linear congruences, simultaneous linear congruences, an extension of the Chinese Remainder Theorem), Congruences with a prime-power modulus (The arithmetic of Zp, Pseudoprimes and Carmichael numbers, solving congruences mod pe), Euler’s function (units, Euler’s function, applications of Euler’s function), The group of units (the group Un, Primitive roots, existence of primitive roots, applications of primitive roots, algebraic structure of Un, the universal exponent), Quadratic residues (quadratic congruences, the group of quadratic congruences, the Legendre symbol, quadratic reciprocity, quadratic residues for prime-power moduli, quadratic residues of arbitrary moduli)

#### **Recommended textbook**

1. Jones G.A. and Jones J.M., Elementary Number Theory, Springer-Verlag London Ltd, 1998

#### **EMT 4802 – Mathematical Statistics II**

Inference, Interval Estimation (random intervals: confidence intervals for means, difference of means and variances, Bayes estimates), Point Estimation (unbiasedness consistency and sufficiency: Rao – Blackwell Theorem, Rao-Cramer Inequality, Maximum likelihood estimation, – Decision functions and Bayes procedures), Nonparametric Methods (Confidence interval for distribution quantiles, The sign test, equality of two distributions, Mann-Whitney- Wilcoxon Test, Linear rank statistics)

#### **Recommended textbooks**

1. Hoel, P.G. (1984), Introduction to Mathematics Statistics, 5th Ed, John Wiley and Sons, New York.

2. Hogg, R.V. and Craig, A.T. (1970), Introduction to Mathematical Statistics, 3rd Ed, Collier Macmillan Publishers (London)

3. Mood, A.M.; Graybill, F. A. and Boes, D.C. (1974), Introduction to the Theory of Statistics, McGraw Hill Company, New York

4. Freund, J.E., Introduction to Mathematical Statistics

#### **Methodology courses**

The following mathematics methodology courses are offered to level 3 and level 4 Bachelor of Science (Education) students with Mathematics as a major or minor subject.

#### **EMM 3501 – Introduction to Mathematics education**

This course has been designed to ensure that students understand general issues involved in the teaching and learning of secondary school mathematics. By going through the history of mathematics, teaching strategies/approaches, challenges students meet in learning mathematics with syllabus familiarisation and assessment procedures, then this course will engage student teachers to realise the need to teach mathematics as a human endeavor.

#### **EMM 3501 – Introduction to Mathematics education**

This course has been designed to ensure that students understand general issues involved in the teaching and learning of secondary school mathematics. By going through the history of mathematics, teaching strategies/approaches, challenges students meet in learning mathematics with syllabus familiarization and assessment procedures, then this course will engage student teachers to realise the need to teach mathematics as a human endeavor.

#### **EMM 3601 – Mathematics Classroom Experiences**

The course will help students to experience the whole process of teaching mathematics. Students will go through rigorous practices of the concepts learned in order to gain experiences of teaching. Short and long term planning using the knowledge of Blooms Taxonomy will be the main focus of the course and student-teachers will conduct microteaching before the teaching practice.

#### **EMM 4701 – Curriculum Studies in Mathematics**

This course will help students to understand most of the issues related to mathematics curriculum and a review of the Teaching Practicum. Furthermore, general curriculum issues; nature of mathematics, contemporary issues in mathematics (mathematics for all; technology); ethno-mathematics and action research in mathematics will help student-teachers gain more experiences in mathematics teaching.

#### **EMM 4801 – Techniques in Mathematics Teaching**

This course is intended for teachers to understand their responsibility towards supporting students in mathematics learning. By going through topics like supporting slow learners in mathematics, concepts in homework assignment, proofs and logic in mathematics, mathematics literacy and research in mathematics it is believed that teachers will gain more experience for better teaching.