## Pre-requisite modules

## Co-requisite modules

## Prohibited combinations

Occurrence | Teaching cycle |
---|---|

A | Autumn Term 2021-22 to Summer Term 2021-22 |

The theory underpinning much of physics is expressed in terms of fields, continuous variables whose evolution is governed by partial differential equations (PDEs), e.g. electromagnetism: Maxwell's equations; quantum mechanics: Schrodinger's equation; fluid dynamics: Euler's equations or the Navier Stokes equations etc. Many important processes in continuum physics, for example those involving diffusion, heat flow and wave motion, are described by PDEs. Simpler problems can sometimes be solved analytically but, more normally, problems of interest require numerical solution. For the solution of such simulations to have meaning, results must replicate accurately the outcome of a real physical experiment or process. The main aims of the first part of this module are to introduce (i) the general properties of partial differential equations and (ii) the finite difference method for the solution of partial differential equations. These computational methods are powerful additions to the theoretician's toolkit. The theory developed in this part of the module is linked to computer experiments undertaken in an extended assignment.

The second part of the module extends calculus to functions of a complex variable, an elegant and useful topic in theoretical physics and beyond. In particular, it allows some integrations required in physics to be performed using a technique known as contour integration.

The third part of the module covers the calculus of variations, its applications to physics, shows the importance of Laplace's equation in physics and solves the equation in the common coordinate systems. In the course of solving Laplace's equation, Bessel's and Legendre's equations will be derived and the solutions of these equations explored. Examples of these equations from several branches of physics will be introduced.

The ideas in this module will be developed further through a number of computational related activities focusing on skills, e.g. peer reviewed and oral presentations, group seminars and individual articles.

Transferable skills have been embedded within the undergraduate programmes to align departmental teaching with the Employability Strategy and York Pedagogy, and to create a distinctive York graduate. But it is vital that students have an opportunity to reflect on the intellectual, practical and transferable skills gained during their degree, in order to appreciate how the education provided develops their employability.

It is important that students can evidence skills; for example the ability to work independently and/or in groups, tackle open-ended problems and communicate the outcomes succinctly in unfamiliar environments. It is also important that students appreciate how these skills and experiences developed both via participation with the programme and through engagement with other aspects of university life, and can map these to essential qualities required by potential employers and postgraduate programmes.

This module provides practical training and includes a team activity related to a programme specific open-ended physics problem, and two individual recorded presentations on a programme specific physics topic. It also prompts students to reflect on skills gained during their degree and to articulate how those skills have developed their employability by mapping these to potential career sectors. This is facilitated by workshops, the Physics Careers Event and the completion of a related pro-forma. Work culminates in the production of a CV and an application letter reflecting the skills and experiences which support application to a job sector or postgraduate programme.

Discuss the general properties of second order partial differential equations and classify these as elliptic, parabolic or hyperbolic

Write down finite differences for first and second order derivatives

Derive finite difference equations for simple linear PDEs (especially the diffusion and wave equations) using a variety of explicit and implicit schemes

Program the finite difference scheme

Test whether a function of a complex variable is analytic

Expand complex functions in Taylor or Laurent expansions

Determine the residue of the function with respect to a singularity

Evaluate contour integrals for a variety of functions and contours

Take the principal value of integrals with poles on the real axis

Make simple applications of contour integration to physics problems

Find the maxima and minima of many variable functions subject to constraints

Apply Euler-Lagrange methods to optimisation problems

Formulate some physics problems in terms of a least action principle.

Show how Laplace's equation arises for a general vector field at points outside the sources of the field

Given expressions for the Laplacian in cylindrical or spherical polar co-ordinates, obtain separable solutions to Laplace's equation

Recognise Bessel's equation, Legendre's equation and Legendre's associated equation and follow the process of solving these equations by a series solution

Sketch the graphs of Jn(x) and Yn(x)

Identify the recurrence relations and orthogonality relations for Bessel functions.

Use the orthogonality relations to determine coefficients in Bessel function expansion and apply this to boundary value problems

Write down the solution of Legendre's equation in terms of Legendre polynomials and associated Legendre functions

Explain the orthogonality and other properties of Legendre polynomials

Show Rodrigues formula can be used to generate Legendre polynomials

Show how the associated Legendre equation can be solved starting with solutions of Legendre's equation.

Apply the techniques developed above to the solution of physical problems.

Demonstrate an ability to research and survey literature of a topic in computational physics.

Demonstrate an ability to communicate effectively through a short formal oral presentation

Demonstrate an ability to work together in groups towards a common goal

Demonstrate an ability to write an abstract and an article based on information collected and synthesized, in an appropriate style and format

**Syllabus**

__Part 1: Continuum Physics__

*Overview of Continuum Physics and PDEs*

- Prototype PDEs: Laplace’s equation, diffusion equation, wave equation
- Hyperbolic, parabolic and elliptic PDEs
- Initial and boundary value problems

*Finite difference method*

- Finite difference representation of first, second order derivatives
- Application to the diffusion equation
- Simple explicit (Euler type) method
- Stability and convergence criteria
- Consistency, order and truncation error
- Explicit finite difference schemes for parabolic equations: first order method, leapfrog method
- Hyperbolic PDEs: the advective equation
- Lax method and Courant-Friedrichs- Levy stability criterion
- Methods for general hyperbolic PDEs: Lax-Wendroff method, leapfrog method
- Explicit and implicit methods for the wave equation

__Part 2: Complex Variable Techniques__

- Cauchy-Riemann relations
- Cauchy's theorem
- Laurent expansion
- Cauchy residue theorem
- Contour integrals
- Physical examples of contour integration
- Poles on the real axis.

__Part 3: Variational Techniques, Differential Equations, and Special Functions__

- Lagrange multipliers
- Euler-Lagrange equation
- Principle of least action
- General vector field - scalar and vector potential and Laplace's equation.
- Application of the method of Frobenius to solve Laplace's equation.
- Bessel's equation and the properties of Bessel functions of the first and second kind.
- Legendre's equation and Legendre polynomials.
- The associated Legendre equation and associated Legendre polynomials.

**Skills ****Content (+training workshops and feedback sessions)**

- individual recorded 10 minute presentation on programme specific topic in physics with peer-assessment through VLE
- repeat of above building on feedback; presentations assessed by staff
- team activity: assessment centre exercise (formative)
- team activity: student groups prepare 10 minute presentation to cohort, reviewing a specific job sector, company profiles, essential skills required and how they map to the undergraduate programme, typical application process/timing
- team activity: ‘thinking like a physicist’ answering an open-ended complex problem on a programme specific theme culminating in production of a group solution document
- completion of an individual pro-forma (linked to attendance at Physics Careers Event) which asks a student to list potential careers sectors, identify key competencies, differentiate between occupations based on those aspects which are considered most relevant by the student
- production of a CV aligned to a potential sector
- production of a draft application letter aligned to a potential sector

Task | Length | % of module mark |
---|---|---|

Essay/courseworkApplication letter |
N/A | 5 |

Essay/courseworkCV and pro-forma |
N/A | 5 |

Essay/courseworkContinuum Physics Assignment |
N/A | 25 |

Essay/courseworkPeer review of presentation 1 |
N/A | 4 |

Essay/courseworkTeam exercise written report |
N/A | 5 |

Online ExamComputational and Mathematical Techniques for Theoretical |
N/A | 50 |

Oral presentation/seminar/examPresentation 2 |
N/A | 6 |

None

The Continuum Physics assignment (22%) is set in November, and submitted in January. It is open-book, anonymous. Students study and use a provided code (in Python) to write a report.

Task | Length | % of module mark |
---|---|---|

Essay/courseworkApplication letter |
N/A | 5 |

Essay/courseworkCV and pro-forma |
N/A | 5 |

Essay/courseworkContinuum Physics Assignment |
N/A | 25 |

Essay/courseworkPeer review of presentation 1 |
N/A | 4 |

Essay/courseworkTeam exercise written report |
N/A | 5 |

Online ExamComputational and Mathematical Techniques for Theoretical |
N/A | 50 |

Oral presentation/seminar/examPresentation 2 |
N/A | 6 |

Our policy on how you receive feedback for formative and summative purposes is contained in our Department Handbook.

Computational and Mathematical Techniques

Kreyszig E: Advanced Engineering Mathematics (John Wiley) ***

Arfken G: Mathematical methods for physicists (Academic Press/Elsevier Science & Technology)**

Skills/general

Warburton N: The basics of essay writing (Taylor & Francis/Routledge)

Levin P: Write great essays! 2nd edition (McGraw Hill) 2009