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Further information on which modules are specific to your programme.

Module description

This module consists of the content and assessment of all of PH3081 and the first part of PH3080. 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. In the other section of the module students are introduced to the Python language, and shown how this can be used to set up mathematical models of physical systems.

Intended learning outcomes

• To use Python to solve physical problems.
• 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 /physics-astronomy/students/ug/timetables-handbooks/