Department of Mathematical Sciences, Unit Catalogue 2011/12 

Credits:  6 
Level:  Intermediate (FHEQ level 5) 
Period: 
Semester 1 
Assessment:  EX 100% 
Supplementary Assessment:  MA20218 Mandatory extra work (where allowed by programme regulations) 
Requisites:  While taking this unit you must take MA20216 and before taking this unit you must take MA10209 and take MA10210 
Description:  Aims: To define the Riemann integral for real functions of a single variable, and to prove its elementary properties rigorously. To extend the theory of continuity and the derivative to real functions of several variables. To engender a geometrical understanding of the multivariate derivative through the use of examples. Learning Outcomes: After taking this unit, students should be able to: * state definitions and theorems in real analysis and present proofs of the main theorems * construct their own proofs of simple unseen results and construct proofs of simple propositions * Present mathematical arguments in a precise, lucid and grammatical fashion. * Apply definitions and theorems to simple examples give a geometric interpretation of multivariate differentiation. * Give a geometric interpretation of multivariate differentiation. Skills: Numeracy T/F A Problem Solving T/F A Written and Spoken Communication F (in tutorials) A (written). Content: Riemann integration in R, fundamental theorem of calculus, substitution, integration by parts, interchanging integrals and limits, integration of power series, improper integrals: unbounded intervals, functions with singularities. Real normed vector spaces with special reference to Euclidean space R^{n}: Euclidean inner product; convergence, continuity, open and closed sets; BolzanoWeierstrass and Weierstrass theorems. Frechet derivative as best linear approximation, partial derivative, directional derivative, Jacobi matrix, gradient, mean value theorem, Lipschitz continuity. Hessian in R^{n}, higher derivatives, Taylor's theorem. In twodimensional space: extrema, implicit function theorem, Lagrange multipliers. 
Programme availability: 
MA20218 is Compulsory on the following programmes:Department of Computer Science
