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## MA20218: Analysis 2A

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, Cauchy-Schwarz inequality, convergence, continuity, open and closed sets, Bolzano-Weierstrass 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
• USCM-AFB20 : BSc (hons) Computer Science and Mathematics (Full-time) - Year 2
• USCM-AKB20 : BSc (hons) Computer Science and Mathematics (Full-time with Thick Sandwich Placement) - Year 2
• USCM-AAB20 : BSc (hons) Computer Science and Mathematics with Study Year Abroad (Full-time with Study Year Abroad) - Year 2
• USCM-AFM14 : MComp (hons) Computer Science and Mathematics (Full-time) - Year 2
• USCM-AKM14 : MComp (hons) Computer Science and Mathematics with Industrial Placement (Full-time with Thick Sandwich Placement) - Year 2
• USCM-AAM14 : MComp (hons) Computer Science and Mathematics with Study Year Abroad (Full-time with Study Year Abroad) - Year 2
Department of Mathematical Sciences
• USMA-AFB15 : BSc (hons) Mathematical Sciences (Full-time) - Year 2
• USMA-AKB16 : BSc (hons) Mathematical Sciences (Full-time with Thick Sandwich Placement) - Year 2
• USMA-AAB16 : BSc (hons) Mathematical Sciences with Study Year Abroad (Full-time with Study Year Abroad) - Year 2
• USMA-AFB13 : BSc (hons) Mathematics (Full-time) - Year 2
• USMA-AKB14 : BSc (hons) Mathematics (Full-time with Thick Sandwich Placement) - Year 2
• USMA-AFB01 : BSc (hons) Mathematics and Statistics (Full-time) - Year 2
• USMA-AKB02 : BSc (hons) Mathematics and Statistics (Full-time with Thick Sandwich Placement) - Year 2
• USMA-AAB02 : BSc (hons) Mathematics and Statistics with Study Year Abroad (Full-time with Study Year Abroad) - Year 2
• USMA-AAB14 : BSc (hons) Mathematics with Study Year Abroad (Full-time with Study Year Abroad) - Year 2
• USMA-AFB05 : BSc (hons) Statistics (Full-time) - Year 2
• USMA-AKB06 : BSc (hons) Statistics (Full-time with Thick Sandwich Placement) - Year 2
• USMA-AAB06 : BSc (hons) Statistics with Study Year Abroad (Full-time with Study Year Abroad) - Year 2
• USMA-AFM14 : MMath Mathematics (Full-time) - Year 2
• USMA-AAM15 : MMath Mathematics with Study Year Abroad (Full-time with Study Year Abroad) - Year 2
Department of Physics
• USXX-AFB03 : BSc (hons) Mathematics and Physics (Full-time) - Year 2
• USXX-AKB04 : BSc (hons) Mathematics and Physics with Placement (Full-time with Thick Sandwich Placement) - Year 2
• USXX-AAB04 : BSc (hons) Mathematics and Physics with Study Year Abroad (Full-time with Study Year Abroad) - Year 2
• USXX-AFM01 : MSci (hons) Mathematics and Physics (Full-time) - Year 2

Notes:
* This unit catalogue is applicable for the 2013/4 academic year only. Students continuing their studies into 2014/15 and beyond should not assume that this unit will be available in future years in the format displayed here for 2013/14.
* Programmes and units are subject to change at any time, in accordance with normal University procedures.
* Availability of units will be subject to constraints such as staff availability, minimum and maximum group sizes, and timetabling factors as well as a student's ability to meet any pre-requisite rules.