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PH30025: Mathematical methods

[Page last updated: 21 April 2022]

Academic Year: 2022/3
Owning Department/School: Department of Physics
Credits: 6 [equivalent to 12 CATS credits]
Notional Study Hours: 120
Level: Honours (FHEQ level 6)
Period:
Semester 2
Assessment Summary: EX 100%
Assessment Detail:
  • Examination (EX 100%)
Supplementary Assessment:
Like-for-like reassessment (where allowed by programme regulations)
Requisites: Before taking this module you must ( take PH20019 AND take PH20020 ) OR take PH20107
Learning Outcomes: After taking this unit the student should be able to:
* derive theorems of analytic functions and use them to evaluate integrals;
* use superposition methods for inhomogeneous ordinary differential equations;
* solve ordinary differential equations using Green functions;
* derive the Euler-Lagrange equation and solve problems using the calculus of variations.

Aims: The aim of this unit is to continue the development of students' mathematical knowledge and skills by introducing concepts and methods used in a mathematical description of the physical world.

Skills: Numeracy T/F A, Problem Solving T/F A.

Content: Functions of a complex variable (11 hours): Functions of z, multivalued functions, branch points and branch cuts. Differentiation, analytic functions, Cauchy-Riemann equations. Complex integration; Cauchy's theorem and integral. Taylor and Laurent expansions. Residue theorem, evaluation of real integrals. Kramers-Kronig relations.
Superposition methods (5 hours): Sturm-Liouville theory, eigenfunctions and eigenvalues, orthogonality of eigenfunctions. Solution of inhomogeneous equations. Green's functions: eigenfunction expansion; direct method; application to solution of inhomogeneous ordinary differential equations.
Calculus of variations (6 hours): Euler-Lagrange equation: derivation and examples. Variation subject to constraints: examples.

Programme availability:

PH30025 is Optional on the following programmes:

Department of Physics
  • USPH-AFB01 : BSc(Hons) Physics (Year 3)
  • USPH-AAB02 : BSc(Hons) Physics with Study year abroad (Year 4)
  • USPH-AKB02 : BSc(Hons) Physics with Year long work placement (Year 4)
  • USPH-AFB10 : BSc(Hons) Physics with Astrophysics (Year 3)
  • USPH-AAB10 : BSc(Hons) Physics with Astrophysics with Study year abroad (Year 4)
  • USPH-AKB10 : BSc(Hons) Physics with Astrophysics with Year long work placement (Year 4)
  • USPH-AFM02 : MPhys(Hons) Physics (Year 3)
  • USPH-AFM04 : MPhys(Hons) Physics with Research placement (Year 3)
  • USPH-AAM12 : MPhys(Hons) Physics with Study year abroad (Year 4)
  • USPH-AKM03 : MPhys(Hons) Physics with Professional Placement (Year 4)
  • USPH-AKM04 : MPhys(Hons) Physics with Professional and Research Placements (Year 4)
  • USPH-AFM10 : MPhys(Hons) Physics with Astrophysics (Year 3)
  • USPH-AFM11 : MPhys(Hons) Physics with Astrophysics with Research placement (Year 3)
  • USPH-AAM14 : MPhys(Hons) Physics with Astrophysics with Study year abroad (Year 4)
  • USPH-AKM10 : MPhys(Hons) Physics with Astrophysics with Professional Placement (Year 4)
  • USPH-AKM11 : MPhys(Hons) Physics with Astrophysics with Professional and Research Placements (Year 4)
  • USPH-AAM15 : MPhys(Hons) Physics with Astrophysics with Study year abroad and Research Placement (Year 4)
  • USPH-AAM13 : MPhys(Hons) Physics with Study year abroad and Research Placement (Year 4)
Programmes in Natural Sciences
  • UXXX-AFB01 : BSc(Hons) Natural Sciences (Biology with Physics stream) (Year 3)
  • UXXX-AAB02 : BSc(Hons) Natural Sciences (Biology with Physics stream) with Study year abroad (Year 4)
  • UXXX-AKB02 : BSc(Hons) Natural Sciences (Biology with Physics stream) with Year long work placement (Year 4)
  • UXXX-AFB01 : BSc(Hons) Natural Sciences (Chemistry with Physics stream) (Year 3)
  • UXXX-AAB02 : BSc(Hons) Natural Sciences (Chemistry with Physics stream) with Study year abroad (Year 4)
  • UXXX-AKB02 : BSc(Hons) Natural Sciences (Chemistry with Physics stream) with Year long work placement (Year 4)
  • UXXX-AFB01 : BSc(Hons) Natural Sciences (Physics with Biology stream) (Year 3)
  • UXXX-AAB02 : BSc(Hons) Natural Sciences (Physics with Biology stream) with Study year abroad (Year 4)
  • UXXX-AKB02 : BSc(Hons) Natural Sciences (Physics with Biology stream) with Year long work placement (Year 4)
  • UXXX-AFB01 : BSc(Hons) Natural Sciences (Physics with Chemistry stream) (Year 3)
  • UXXX-AAB02 : BSc(Hons) Natural Sciences (Physics with Chemistry stream) with Study year abroad (Year 4)
  • UXXX-AKB02 : BSc(Hons) Natural Sciences (Physics with Chemistry stream) with Year long work placement (Year 4)
  • UXXX-AFM01 : MSci(Hons) Natural Sciences (Biology with Physics stream) (Year 3)
  • UXXX-AAM02 : MSci(Hons) Natural Sciences (Biology with Physics stream) with Study year abroad (Year 4)
  • UXXX-AKM02 : MSci(Hons) Natural Sciences (Biology with Physics stream) with Professional Placement (Year 4)
  • UXXX-AAM02 : MSci(Hons) Natural Sciences (Chemistry with Physics stream) with Study year abroad (Year 4)

Notes:

  • This unit catalogue is applicable for the 2022/23 academic year only. Students continuing their studies into 2023/24 and beyond should not assume that this unit will be available in future years in the format displayed here for 2022/23.
  • Programmes 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.