Course Coordinator: 
Jonathan Miller
Mathematical Methods of Natural Sciences

An exploration and practicum in advanced mathematical techniques for application in the natural sciences.  The emphasis is on calculating physical quantities using analytical and numerical, exact and approximate methods. Instruction stresses calculational approaches rather than rigorous proofs with substantial practice in analytic calculation skills acquired via problem sets.  Examples and applications are drawn from a variety of fields.

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: 
  • A Guided Tour of Mathematical Physics, Snieder. At: http://samizdat.mines.edu/snieder/
  • Advanced Mathematical Methods for Scientists and Engineers, Bender and Orszag (1999) Springer
  • Mathematical methods for Natural Scientists v. 1, Lev Kantorovich (2022) Springer
  • Mathematical methods for Natural Scientists v. 2, Lev Kantorovich (2022) Springer
  • Mathematics for Physics: A Guided Tour for Graduate Students, Stone and Goldbart (2009) Cambridge.
Reference Book: 
  • Basic Training in Mathematics. R. Shankar (1995) Plenum
  • Geometrical methods of mathematical physics. B. Schutz (1999) Cambridge
  • Statistical Field Theory. G. Mussardo (2009) Oxford
  • Statistical Mechanics: Entropy, Order Parameters and Complexity J.P. Sethna (2008) Oxford
Prior Knowledge: 
Calculus,  e.g. A104 Vector and Tensor Calculus, or A108 Partial Differential Equations, Linear algebra, e.g., B29 Linear Algebra