- linear vs NL ODEs
- separation of variables for 1st order ODE  
- integrating factor for linear 1st order ODE  
- homogeneous and particular solns for linear inhomogenous nth order ODE  
- constant coefficient nth order ODEs
- equidimensional nth order ODEs
- the energy method for 2nd order special form NL ODEs
- conservation of energy for 1-D Newton
- turning points

- coupled ODEs
- Cartesian and polar unit vectors
- some unit vectors are functions of t
- particle orbits in Cartesian and Polar coordinates
- conservation of angular momentum for central forces
- conservation of energy for conservative forces
- constants of motion reduce order of ODEs
- fields: central and conservative force fields 
- sketch simple contours and simple force fields
- df = dr.grad(f)
- grad, div, curl in Cartesian
- divergence-free force fields
- force fields exhibiting curl
- Gauss' and Stokes Theorem
- equivalence of F=-grad(f), curl(F)=0, and Integral(F.dr)=0
- direct solution of orbits vs using constants of motion
- the Lorentz force

- column vectors, N-dimension
- column vectors, infinite-dimension f(x)
- definition of vector space
- inner product definition
- operators (matrices and differential operators)
- eigenproblems
- Hermitean operators
- eigenvectors of H operators => complete set
- Fourier's trick to find coefficients
- Fourier series (for different boundary conditions)
- periodic complete set from momentum op of QM
- Legendre polynomials 
- completeness in the mean




Solving Linear PDE's
====================
- appropriate coordinate system
- express grad, etc, in that coordinate system
- separation of variables
- separation constant
- convenient sign and form of constant (oscillatory vs evanescent solns)
- solution combos
- eliminate combos from boundary/initial conditions
- quantization of constant
- sum over all possible solutions
- Fourier's trick

Some other factors in eliminating/constraning combos:
- no blow up
- more than one boundary
- linear combination of combos to yield convenient forms
- periodic boundary conditions