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This subject develops the mathematical methods of applied mathematics and mathematical physics with an emphasis on integral transform and related techniques. An introduction is given to the calculus of variations and the Euler-Lagrange equation. Advanced complex contour integration techniques are used to evaluate and invert Fourier and Laplace transforms. The general theory includes convolutions, Green’s functions and generalized functions. The methods of Laplace, stationary phase, steepest descents and Watson’s lemma are used to asymptotically approximate integrals. Throughout, the theory is set in the context of examples from applied mathematics and mathematical physics such as the brachistochrone problem, Fraunhofer diffraction, Dirac delta function, heat equation and diffusion.
Intended learning outcomes
After completing this subject students should:
- have learned how the calculus of variations, transform methods and associated asymptotic analysis apply in a variety of areas in applied mathematics and mathematical physics;
- appreciate the role of advanced contour integration techniques of complex analysis and to be able to use these techniques to calculate transform integrals;
- understand the basic concepts of asymptotic evaluation of integrals, know how to implement Laplace’s method, stationary phase and steepest descents and appreciate their applicability and limitations;
- be familiar with the basic properties of generalized functions and Green’s functions in applied mathematics and mathematical physics and their applications;
- have the ability to pursue further studies in these and related areas.
In addition to learning specific skills that will assist students in their future careers in science, they will have the opportunity to develop generic skills that will assist them in any future career path. These include:
- problem-solving skills: the ability to engage with unfamiliar problems and identify relevant solution strategies;
- analytical skills: the ability to construct and express logical arguments and to work in abstract or general terms to increase the clarity and efficiency of analysis;
- collaborative skills: the ability to work in a team;
- time-management skills: the ability to meet regular deadlines while balancing competing commitments.
Last updated: 3 November 2022