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Programme & Unit Catalogues

## MA20218: Analysis 2A

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 Rn: Euclidean inner product; convergence, continuity, open and closed sets; Bolzano-Weierstrass and Weierstrass theorems.
Frechet derivative as best linear approximation, partial derivative, directional derivative, Jacobi matrix, gradient, mean value theorem, Lipschitz continuity.
Hessian in Rn, higher derivatives, Taylor's theorem.
In two-dimensional space: extrema, implicit function theorem, Lagrange multipliers.
Programme availability:

#### MA20218 is Compulsory on the following programmes:

Department of Computer Science
• USCM-AFB13 : BSc (hons) Computer Science with Mathematics (Full-time) - Year 2
• USCM-AKB14 : BSc (hons) Computer Science with Mathematics (Full-time with Thick Sandwich Placement) - Year 2
• USCM-AAB14 : BSc (hons) Computer Science with 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

NB. Programmes and units are subject to change at any time, in accordance with normal University procedures.