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MA22032: Analysis 2A

[Page last updated: 09 August 2024]

Academic Year: 2024/25
Owning Department/School: Department of Mathematical Sciences
Credits: 10 [equivalent to 20 CATS credits]
Notional Study Hours: 200
Level: Intermediate (FHEQ level 5)
Period:
Semester 1
Assessment Summary: EXCB 100%
Assessment Detail:
  • Examination (EXCB 100%)
Supplementary Assessment:
Like-for-like reassessment (where allowed by programme regulations)
Requisites: Before taking this module you must take MA12004
Learning Outcomes: After taking this unit, you should be able to:
  • demonstrate a good knowledge and understanding of Riemann integration, sequences and series of functions, normed spaces, and metric spaces;
  • use a range of techniques to integrate functions;
  • analyse the convergence properties of sequences and series of functions;
  • analyse the properties of normed and metric spaces;
  • apply the theory to simple examples.



Synopsis: This unit will familiarise you with some of the key ideas of mathematical analysis. You will develop your understanding of integration and explore a variety of tools and techniques to analyse functions.

Content: Integration on closed bounded intervals: Riemann sums, linearity, integrability of continuous functions, fundamental theorem of calculus, substitution, integration by parts. Integration for open and unbounded intervals, functions with singularities. Sequences of functions, uniform convergence. Integrals and limits, differentiating under the integral. Complex differentiation, real and complex power series, Weierstrass M-test, differentiation and integration of power series. Real and complex normed vector spaces, L2 and uniform norm, operator norm. Metric spaces, sequences, convergence, completeness. Open, closed and bounded sets, neighbourhoods; limits and continuity, characterisations via sequences and open sets; Lipschitz maps and uniform continuity, Contraction mapping theorem. Example: existence and uniqueness of solutions of ODEs.

Course availability:

MA22032 is Optional on the following courses:

Department of Mathematical Sciences
  • USMA-AFB30 : BSc(Hons) Mathematics (Year 2)
  • USMA-AFB32 : BSc(Hons) Mathematics and Statistics (Year 2)
  • USMA-AKB32 : BSc(Hons) Mathematics and Statistics with professional placement (Year 2)
  • USMA-AKB32 : BSc(Hons) Mathematics and Statistics with study abroad (Year 2)
  • USMA-AKB30 : BSc(Hons) Mathematics with professional placement (Year 2)
  • USMA-AKB30 : BSc(Hons) Mathematics with study abroad (Year 2)
  • USMA-AFM30 : MMath(Hons) Mathematics (Year 2)
  • USMA-AKM30 : MMath(Hons) Mathematics with professional placement (Year 2)
  • USMA-AKM31 : MMath(Hons) Mathematics with study abroad (Year 2)

Notes:

  • This unit catalogue is applicable for the 2024/25 academic year only. Students continuing their studies into 2025/26 and beyond should not assume that this unit will be available in future years in the format displayed here for 2024/25.
  • Courses and units are subject to change 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.
  • Find out more about these and other important University terms and conditions here.