This course develops advanced mathematical techniques for application in the natural sciences. Particular emphasis will be placed on analytical and numerical, exact and approximate methods, for calculation of physical quantities. Examples and applications will be drawn from a variety of fields. The course will stress calculational approaches rather than rigorous proofs. There will be a heavy emphasis on analytic calculation skills, which will be developed via problem sets.
- Complex Analysis I: Introduction to complex analysis: analytic functions.
- Complex Analysis II: Cauchy Theorem and contour integration.
- Complex Analysis III: Numerical methods in complex analysis.
- Linear algebra I: Advanced eigenvalues and eigenvectors.
- Linear algebra II: Numerical methods.
- Ordinary differential/difference equations (ODDE) I: Properties and exact solutions.
- ODDE II: Approximate solutions.
- ODDE III: Numerical solution.
- Asymptotic expansion of sums and integrals I: elementary methods.
- Asymptotic expansion of sums and integrals II: steepest descents.
- Perturbation methods.
- Boundary layer theory.
- WKB theory.
- Vector fields, Stokes' theorem.
- Green's functions.
Calculus, e.g. A104 Vector and Tensor Calculus or B28 Ordinary and Partial Differential Equations