PH3081 Mathematics for Physicists

Academic year

2025 to 2026 Semester 1

Further information on which modules are specific to your programme.

Key module information

SCOTCAT credits

15

The Scottish Credit Accumulation and Transfer (SCOTCAT) system allows credits gained in Scotland to be transferred between institutions. The number of credits associated with a module gives an indication of the amount of learning effort required by the learner. European Credit Transfer System (ECTS) credits are half the value of SCOTCAT credits.

SCQF level

SCQF level 9

The Scottish Credit and Qualifications Framework (SCQF) provides an indication of the complexity of award qualifications and associated learning and operates on an ascending numeric scale from Levels 1-12 with SCQF Level 10 equating to a Scottish undergraduate Honours degree.

Module coordinator

Dr I Leonhardt

Dr I Leonhardt
This information is given as indicative. Staff involved in a module may change at short notice depending on availability and circumstances.

Module description

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-requisites

BEFORE TAKING THIS MODULE YOU MUST PASS PH2012 AND PASS MT2501 AND PASS MT2503

Anti-requisites

YOU CANNOT TAKE THIS MODULE IF YOU TAKE PH3082 OR TAKE MT3506

Assessment pattern

2-hour Written Examination = 80%, continual assessment = 20%

Re-assessment

Oral Re-assessment, capped at grade 7

Learning and teaching methods and delivery

Weekly contact

3x 1hr lectures x10 weeks, 1hr tutorial x 5 weeks.

Scheduled learning hours

35

The number of compulsory student:staff contact hours over the period of the module.

Guided independent study hours

115

The number of hours that students are expected to invest in independent study over the period of the module.

Intended learning outcomes

  • State the definition of the Dirac delta function and use the delta function to compute a variety of integrals.
  • Decompose simple functions into their Fourier components. Students should be able to represent specified 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.
  • Determine 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.
  • Solve various types of 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.
  • Calculate and interpret the gradient or Laplacian of a scalar function, or the divergence and curl of a vector field, in Cartesian, cylindrical and spherical coordinates. Compute line, surface and volume integrals in Cartesian, cylindrical and spherical coordinates. Apply Stokes’ theorem and the divergence theorem and interpret physical meaning from the resulting expressions. 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.
  • Describe the technique of separation of variables and use it to derive general solutions to various second-order partial differential equations in Cartesian, cylindrical and spherical coordinate systems. Apply a given set of boundary conditions to determine a specific solution to the differential equation.

Additional information from school

For guidance on AS and PH modules please consult the School Handbook, at https://www.st-andrews.ac.uk/physics-astronomy/students/ug/timetables-handbooks/