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PH3081   Mathematics for Physicists

Academic year(s): 2019-2020

Key information

SCOTCAT credits : 15

ECTS credits : 7

Level : SCQF Level 9

Semester: 1

Planned timetable: 10.00 am Mon, Tue, Thu

The module aims to develop mathematical techniques that are required by a professional physicist or astronomer. There is particular emphasis on the special functions which arise as solutions of differential equations which occur frequently in physics, and on vector calculus. Analytic mathematical skills are complemented by the development of computer-based solutions. The emphasis throughout is on obtaining solutions to problems in physics and its applications. Specific topics to be covered will be Fourier transforms, the Dirac delta function, partial differential equations and their solution by separation of variables technique, series solution of second order ODEs, Hermite polynomials, Legendre polynomials and spherical harmonics. The vector calculus section covers the basic definitions of the grad, div, curl and Laplacian operators, their application to physics, and the form which they take in particular coordinate systems.

Relationship to other modules

Pre-requisite(s): Before taking this module you must pass PH2011 and pass PH2012 and ( pass MT2501 and pass MT2503 )

Anti-requisite(s): You cannot take this module if you take PH3082 or take MT3506

Learning and teaching methods and delivery

Weekly contact: 3 lectures plus fortnightly tutorials.

Scheduled learning hours: 36

Guided independent study hours: 114

Assessment pattern

As used by St Andrews: 2-hour Written Examination = 80%, Coursework = 20% (made up of Class Test = 15% and meaningful engagement with tutorial work = 5%)

As defined by QAA
Written examinations : 100%
Practical examinations : 0%
Coursework: 0%

Re-assessment: Oral Re-assessment, capped at grade 7

Personnel

Module coordinator: Dr C A Hooley
Module teaching staff: Dr C Hooley

Additional information from school

Aims & Objectives
  • A better working knowledge and understanding of the mathematics used in other honours

physics and astronomy modules.

  • An improved ability to formulate problems relating to physical phenomena in mathematical language, starting from intuitive ideas.
  • An ability to apply a range of mathematical techniques to the solution of such problems.
  • An ability to discriminate between alternative methods of solution as to which is the most suitable for the task at hand.
  • A significant enhancement of your problem solving ability as a practising physicist/astronomer.

 

Learning Outcomes

By the end of the semester, students are expected to be able to:

 

  • determine the components of a vector (or vector field) in Cartesian, cylindrical and spherical coordinates, write the fields in terms of appropriate unit vectors, and be able to translate an expression given in one coordinate system into either of the other two.

 

  • state the definition of the Dirac delta function; and apply that definition to derive basic identities and compute integrals.

 

  • represent functions as a discrete sum of orthonormal basis functions (e.g., sines/cosines, Hermite polynomials, Legendre polynomials, spherical harmonics, etc.), and apply the appropriate orthonormality relation to determine the series coefficients.

 

  • compute the Fourier transform (or inverse transform) of simple functions, exploit basic relationships between transform pairs and use the convolution theorem to solve ordinary differential equations and compute integrals.

 

  • solve second-order differential equations with non-constant coefficients using the method of Frobenius (power series solutions); this includes deriving recurrence relations, and determining closed-form expressions for the coefficients.

 

  • sketch the first few Hermite polynomials, and derive explicit expressions when given a recurrence relation or a generating formula. Students should also be explain the relationship between the degree of a Hermite polynomial and its symmetry.

 

  • state fundamental properties of solutions to Laplace's equation in a bounded region of space (e.g., uniqueness of the solution, existence of local maxima or minima, etc.).

 

  • describe the separation of variables technique, and use it to derive general solutions to second-order partial differential equations in Cartesian and spherical coordinate systems; then apply a given set of boundary conditions to determine a specific solution.

 

  • compute line, surface and volume integrals in Cartesian, cylindrical and spherical coordinates.

 

  • calculate the gradient or Laplacian of a scalar function, and the divergence or curl of a vector field, in Cartesian, cylindrical and spherical coordinates. Students should also be able to derive physical meaning from the resulting mathematical expressions, create and interpret visual representations of scalar and vector fields, and draw connections between them.

 

  • apply Stokes’ theorem and the divergence theorem to convert between differential and integral equations, as well as interpret physical meaning from the resulting expressions.

 

  • set up definite integrals of vector functions using Cartesian, cylindrical or spherical coordinates (as appropriate), and compute them for simple cases; this includes deriving expressions where all integration variables and limits are explicit (i.e., so that the integral could then be calculated by a computer).

 

Synopsis

Differential Equations

Dirac delta function

Fourier transforms

Series solutions, Hermite polynomials

Laplace’s equation in Cartesian and spherical coordinates

Legendre polynomials, spherical harmonics

 

Vector Calculus

Gradient, directional derivatives

Line and surface integrals

Divergence, divergence theorem

Curl, Stokes’ theorem

Helmholtz theorem, the Maxwell equations

Vector integration techniques

 

Additional information on continuous assessment etc.

Please note that the definitive comments on continuous assessment will be communicated within the module.  This section is intended to give an indication of the likely breakdown and timing of the continuous assessment. 

This module is part of the core JH programme, and as such there is a summary of deadlines etc on the School’s Students and Staff web pages.   There is a single class test in week five, contributing 15% to the module mark.   Students have compulsory tutorials every two weeks, with hand-in tutorial work counting for 5% of the module total.

 

Accreditation Matters

This module contains material that is or may be part of the IOP “Core of Physics”.  This includes

Three dimensional trigonometry

Vectors to the level of div, grad, and curl, divergence theorem and Stokes' theorem

Fourier series and transforms including the convolution theorem

 

Recommended Books

Please view University online record: http://resourcelists.st-andrews.ac.uk/modules/ph3081.html

 

General Information

Please also read the general information in the School's honours handbook that is available via st-andrews.ac.uk/physics/staff_students/timetables.php.