Third year Mathematics Modules

Title	Abstract Algebra
Code	SMTH311	Department	Mathematical Sciences
Prerequisites	SMTH221, SMTH222	Co-requisites	none
Aim	To introduce students to the theories of groups, rings and fields.
Content	Theory of Groups: Fundamentals (Mappings, binary operations, relations). The integers. Groups. Subgroups. Cyclic groups. Isomorphisms. Homomorphisms. Finite permutation groups. Cayley’s theorem. Normal subgroups. Quotient groups. Some applications of the theory of groups Theory of Rings and Fields: Rings. Integral domains. Fields. Ideals. Quotient Rings. Ring homomorphism. The field of real numbers. Complex numbers. Quaternions. Polynomials over a ring.
Assessment	40% Continuous Assessment Mark 60% Formal end of module exam (3 hours)
DP Requirement	40% Continuous Assessment Mark 80% Attendance at lectures and tutorials.

Title	Real Analysis
Code	SMTH321	Department	Mathematical Sciences
Prerequisites	SMTH221, SMTH222	Co-requisites	none
Aim	To introduce students to the theory of functions of real variables and metric spaces.
Content	Real numbers and real functions. Topology of real line and plane. Compactness. Completeness. Countability. Cardinality. Order Metric and normed spaces. Metrics. Norms. Properties of metric and normed spaces. Riemann integral. Upper and lower Riemann integrals. Riemann integrability. Properties of the Riemann integral.
Assessment	40% Continuous Assessment Mark 60% Formal end of module exam (3 hours)
DP Requirement	40% Continuous Assessment Mark 80% Attendance at lectures and tutorials.

Title	Graph Theory
Code	SMTH322	Department	Mathematical Sciences
Prerequisites	SMTH221, SMTH222	Co-requisites	none
Aim	To explore proof techniques in graph theory and explore its applications in pure and applied mathematics.
Content	Introduction to Graph theory Types of graph, representation of graphs, Hamiltonian and Euler circuits Graph theorems, Vertex and edge colorings Practical applications of graphs Network problems. Mathematical applications Representation of an equation by means of a graph .Elementary aspects of category theory
Assessment	40% Continuous Assessment Mark 60% Formal end of module exam (3 hours)
DP Requirement	40% Continuous Assessment Mark 80% Attendance at lectures and tutorials.

Title	Complex analysis
Code	SMTH322	Department	Mathematical Sciences
Prerequisites	SMTH221, SMTH222	Co-requisites	none
Aim	To introduce students to the theory of functions of complex variables.
Content	Complex functions, their limits and continuity. Complex differentiation. CauchyRiemann equations. Complex integration. Cauchy’s theorem and formulas. Infinite series. The residue theorem and its application in evaluation of integrals and series. Conformal mapping.
Assessment	40% Continuous Assessment Mark 60% Formal end of module exam (3 hours)
DP Requirement	40% Continuous Assessment Mark 80% Attendance at lectures and tutorials.