Mathematics II
Course Coordinator: 
Robert Sinclair

The students will be introduced to some more advanced mathematical topics, but without proofs. Linear algebra, vector fields, dynamical systems, stochastic differential equations and numerical methods for these will be covered. Vector fields will be discussed with a view to motivating fluid dynamics, meaning conservation of mass, compressibility and divergence will be discussed. Systems of differential equations and their solution using Euler’s and Heun’s methods will be introduced. Dynamical systems will include fixed points, their stability, and bifurcation. The meaning of stochastic differential equations and their solutions will be discussed.

An extension of the course Mathematics I for graduate biologists.
Detailed Syllabus: 
  1. Linear Algebra: Rotations in the plane and space. Matrix representation. Matrix multiplication.
  2. Linear Algebra: Solution of linear systems. Eigenproblems. Hardy-Weinberg equilibrium.
  3. Linear Algebra: Change of basis, discrete Fourier transform.
  4. Continuous flows: Vector fields, conservation of mass, compressibility and divergence.
  5. Exercises (individual)
  6. Systems of differential equations: Reduction to systems of first order. Euler’s method.
  7. Systems of differential equations: Reaction-diffusion equations. Heun’s method.
  8. Systems of differential equations: Hodgkin-Huxley equations.
  9. Dynamical Systems: Linear systems, fixed points.
  10. Dynamical Systems: Linearization of nonlinear systems.
  11. Dynamical Systems: Predator-prey systems. Bifurcation. Chaos.
  12. Stochastic differential equations: Euler-Maruyama method.
  13. Student presentations: Preparation.
  14. Student presentations: Preparation.
  15. Student presentations: Presentation.
Course Type: 
Weekly written exercises.