FM 5001/5002 (Preparatory Sequence)

short description

• One-variable calculus (very brief review)
  • Derivatives
  • Integrals
  • Taylor approximations
• Linear algebra
  • Linear equations
  • Eigenvalues and eigenvectors
  • The spectral theorem
  • Cayley's theorem
  • Singular Value Decomposition and Principal Component Analysis
  • Cholesky Decomposition and Square-Root Matrices
• Multivariable calculus
  • Differential
    • the chain rule
    • Taylor approximations
    • Optimization
    • Lagrange multipliers and constrained optimization
  • Integral
    • Stokes' Theorem
    • planimeters
• Ordinary Differenital Equations
• Partial Differential Equations
  • The heat equation
• Probability Theory using ``Piecewise Constant Random Variables"
  • distributions
  • mean (expectation), variance and standard deviation
  • covariance and correlation
  • Bayes' Theorem
  • The Central Limit Theorem
    • via Fourier transforms
    • via Stirling's formula
  • ``Piecewise Constant Processes'' and Brownian motion approximation