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PX440 - Mathematical Methods for Physicists III

  • Module code: PX440
  • Module name: Mathematical Methods for Physicists III
  • Department: Physics
  • Credit: 7.5

Content and teaching | Assessment | Availability

Module content and teaching

Principal aims

To help students develop mathematical skills and to cover material needed in 4th year physics modules

Principal learning outcomes

At the end of the module you should be able: To set up minimization problems and to derive and solve the corresponding Euler-Lagrange equations; To identify an analytic function and classify its singularities; To establish Cauchy's theorem from the identities of vector calculus; To use the calculus of residues to evaluate definite integrals.

Timetabled teaching activities

15 Lectures + 5 examples classes

Departmental link

Other essential notes

One third of this module is on the calculus of variations and two thirds on complex variables. The calculus of variations is concerned with the minimisation of integrals over sets of differentiable functions. Such integrals crop up in many contexts. For example, the ground state wavefunction of a quantum system minimises the expectation value of the energy. The classical equations of motion for both particles and fields can often be obtained by minimising what is called the action functional (which may be familiar if you took Hamiltonian Mechanics). Requiring functions of complex variables to be analytic (differentiable with respect to their complex argument in some domain) turns out to constrain such functions very strongly. As the module shows: only the constant function is differentiable everywhere, analytic functions are actually equal to their Taylor series and not just approximated by them, a function that is once differentiable is differentiable infinitely many times. Complex differentiable functions are clean, they are fun and they are important in physics. For example, response functions like the dielectric response function are analytic functions with the domain, in which the function is analytic, being related to causality.

Module assessment

Assessment group Assessment name Percentage
7.5 CATS (Module code: PX440-7.5)
B (Examination only) 1.5 hr examination (April) 100%

Module availability

This module is available on the following courses:

  • Undergraduate Physics (BSc MPhys) (F304) - Year 3
Optional Core