Course Coordinator: 
Jonathan Miller
Mathematical Methods of Natural Sciences

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.

To develop expertise in application of advanced mathematical methods for natural scientists
Course Content: 
  1. Complex Analysis I: Introduction to complex analysis: analytic functions.
  2. Complex Analysis II: Cauchy Theorem and contour integration.
  3. Complex Analysis III: Numerical methods in complex analysis.
  4. Linear algebra I: Advanced eigenvalues and eigenvectors.
  5. Linear algebra II: Numerical methods.
  6. Ordinary differential/difference equations (ODDE) I: Properties and exact solutions.
  7. ODDE II: Approximate solutions.
  8. ODDE III: Numerical solution.
  9. Asymptotic expansion of sums and integrals I: elementary methods.
  10. Asymptotic expansion of sums and integrals II: steepest descents.
  11. Perturbation methods.
  12. Boundary layer theory.
  13. WKB theory.
  14. Vector fields, Stokes' theorem.
  15. Green's functions.
Course Type: 
Homework 60%, Midterm Exam 20%, Final Exam 20%
Text Book: 
Advanced Mathematical Methods for Scientists and Engineers, Bender and Orszag (1999) Springer
A Guided Tour of Mathematical Physics, Snieder. At: http://samizdat.mines.edu/snieder/
Mathematics for Physics: A Guided Tour for Graduate Students, Stone and Goldbart (2009) Cambridge.
Mathematical methods for Natural Scientists v. 1
Mathematical methods for Natural Scientists v. 2
Reference Book: 
Basic Training in Mathematics. R. Shankar. Plenum, 1995.
Geometrical methods of mathematical physics. B. Schutz. Cambridge, 1999.
Statistical Field Theory. G. Mussardo. Oxford, 2009.
Statistical Mechanics: Entropy, Order Parameters and Complexity J.P. Sethna. Oxford, 2008
Prior Knowledge: 

Calculus,  e.g. A104 Vector and Tensor Calculus or B28 Ordinary and Partial Differential Equations