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MA42141: Differential and geometric analysis

[Page last updated: 22 May 2025]

Academic Year: 2025/26
Owning Department/School: Department of Mathematical Sciences
Credits: 5 [equivalent to 10 CATS credits]
Notional Study Hours: 100
Level: Masters UG & PG (FHEQ level 7)
Period:
Semester 1
Assessment Summary: EX 100%
Assessment Detail:
  • MA40254 Examination (EX 100%)
Supplementary Assessment:
Like-for-like reassessment (where allowed by programme regulations)
Requisites: Before taking this module you must take MA20219
Learning Outcomes: After taking this unit, students should be able to state and prove the principal theorems about the Frechet derivative, differential forms and the inverse function theorem, and apply them to simple examples.


Content: Review of the Frechet derivative, continuous differentiability and mean value inequality in finite dimensions, contraction mapping theorem.
Inverse and implicit function theorems for continuously differentiable maps, surfaces (submanifolds) defined by equations.
Differential forms: wedge product, exterior derivative, product rule, pullbacks and change of variables, closed and exact forms, Poincare lemma.
differentiable change of variable in n-dimensional integrals and integration of n-forms.
Further topics and applications which might include: matrix groups, Lagrange multiplier rule for constraints, integration of differential forms over simplicial k-chains and/or submanifolds with boundary, Stokes' theorem.

Skills: Numeracy T/F, A
Problem Solving T/F, A
Written Communication F (on problem sheets)

Aims: To develop the theory of continuously differentiable maps in finite dimensions, leading to a proof of a general inverse function theorem, and to introduce the theory of differential forms and its applications.

Course availability:

MA42141 is Optional on the following courses:

Department of Mathematical Sciences
  • USMA-AAB16 : BSc(Hons) Mathematical Sciences with Study year abroad (Year 4)
  • USMA-AKB16 : BSc(Hons) Mathematical Sciences with Year long work placement (Year 4)
  • USMA-AAB14 : BSc(Hons) Mathematics with Study year abroad (Year 4)
  • USMA-AKB14 : BSc(Hons) Mathematics with Year long work placement (Year 4)
  • USMA-AAB02 : BSc(Hons) Mathematics and Statistics with Study year abroad (Year 4)
  • USMA-AKB02 : BSc(Hons) Mathematics and Statistics with Year long work placement (Year 4)
  • USMA-AKM15 : MMath(Hons) Mathematics with Year long work placement (Year 4)

Notes:

  • This unit catalogue is applicable for the 2025/26 academic year only. Students continuing their studies into 2026/27 and beyond should not assume that this unit will be available in future years in the format displayed here for 2025/26.
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