This course aims to provide a first approach to the subject of algebra, which is one of the basic pillars of modern mathematics. The focus of the course will be the study of certain structures called groups, rings, fields and some related structures. Abstract algebra gives to student a good mathematical maturity and enable learners to build mathematical thinking and skill. The topics to be covered are injective, subjective and objective mappings.  Product of mappings, inverse of a mapping. Binary operations on a set. Properties of binary operations (commutative, associative and distributive properties).  Identity element of a set and inverse of an element with respect to a binary operation. Relations on a set. Equivalence relations, equivalence classes. Partition of set induced by an equivalence relation on the set.  Partial and total order relations on a set. Well-ordered sets. Natural numbers; mathematical induction. Sum of the powers of natural numbers and allied series. Integers; divisors, primes, greatest common divisor, relatively prime integers, the division algorithm, congruencies, the algebra of residue classes.  Rational and irrational numbers. Least upper bound and greatest lower bound of a bounded set of real numbers. Algebraic structures with one or two binary operations. Definition, examples and simple properties of groups, rings, integral domains and fields.

The construction of mathematical models to address real-world problems has been one of the most important aspects of each of the branches of science. It is often the case that these mathematical models are formulated in terms of equations involving functions as well as their derivatives. Such equations are called differential equations. If only one independent variable is involved, often time, the equations are called ordinary differential equations. The course will demonstrate the usefulness of ordinary differential equations for modeling physical and other phenomena. Complementary mathematical approaches for their solution will be presented. The topics to be covered are vector algebra with applications to three-dimensional geometry. First order differential equations; applications to integral curves and orthogonal trajectories.  Ordinary linear differential equations with constant coefficients and equation reducible to this type. Simultaneous linear differential equations. Introduction to partial differential equations.

This course covers the fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, and the interchange of limit operations. It shows the utility of abstract concepts and teaches an understanding and construction of proofs. The topics to be covered include

limit of a sequence of real numbers, standard theorems on limits, bounded and monotonic sequences of real numbers, infinite series of real numbers, tests for convergence, power series, limit, continuity and differentiability of functions of one variable,  Rolle’s theorem, mean value theorems, Taylor’s theorem, definition and simple properties of the Riemann integral.

This course covers vector valued functions. It introduces students to the concept of change and motion and the manner in which quantities approach other quantities.  Topics include limits, continuity,  derivatives of vector functions, gradient, divergence, curl, formulae involving gradient, divergence, laplacian, orthogonal curvilinear coordinates,  line integrals, Green’s theorem in the plane, surface integrals. Other topics are the divergence theorem, improper integrals, Gamma functions, Beta functions, the Riemann Stieltjes Integral,  pointwise and uniform convergence of sequence and series, integration and differentiation term by term.

Limits, continuity and derivatives of vector functions; gradient, divergence and curl; formulae involving gradient, divergence, curl and laplacian and orthogonal curvilinear coordinates; line integrals; Green’s theorem in the plane; surface integrals; the divergence theorem; improper integrals; Gamma and Beta functions; The Riemann Stieltjes integral; pointwise and uniform convergence of sequence and series; integration and differentiation term by term.

This course is designed as a basic introductory course in the analysis of metric spaces. It is aimed at providing the abstract analysis components for the degree course of a student majoring in mathematics. This course affords students an opportunity to gain some familiarity with the axiomatic method in analysis. The topics to be covered are: metric spaces, open spheres, open sets, limit points, closed sets, interior, closure, boundary of a set, sequences in metric spaces, subsequences, upper and lower limits of real sequences,  continuous functions on metric spaces, uniform continuity, isometry, homomorphism, complete metric spaces, compact sets in a metric space, Heine-Borel theorem, connected set, and the inter-mediate value theorem.

This course is designed to offer a basic introduction to measure theory and Lebesgue’s integral. The topics to be covered are: countable and uncountable sets, countability of the rationals,  uncountability of the reals, measurable sets and functions, the Lebesgue’s integral where E is a measurable subset of the real line and f is measurable on E, the  spaces as metric spaces, Cauchy sequences in  spaces, completeness of spaces, the Riesz-Fischer theorem and Mean convergence in the space .