
Academic Year:  2012/3 
Owning Department/School:  Department of Mathematical Sciences 
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 MA10207 and 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 derivative to functions of several real 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 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. Skills: Numeracy T/F, A Problem Solving T/F, A Spoken and Written Communication F (in tutorials and on problem sheets) 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 and functions with singularities. Euclidean space, CauchySchwarz inequality, convergence, continuity, open and closed sets, BolzanoWeierstrass and Weierstrass theorems, norms including operator norm on matrices. Frechet derivative as best linear approximation, partial derivatives, directional derivative, Jacobi matrix, gradient, chain rule, mean value theorem, Lipschitz continuity. Hessian in n dimensions, higher derivatives, Taylor's theorem. In two dimensions: extrema, implicit function theorem, Lagrange multipliers. 
Programme availability: 
MA20218 is Compulsory on the following programmes:Department of Computer Science
