1 Introduction
 1.1 Gyrokinetic?
 1.2 Guiding-center coordinates: a simple example
2 Transform Vlasov equation from particle coordinates to guiding-center coordinates
 2.1 Guiding-center transformation
 2.2 Choosing velocity coordinates
 2.3 Summary of the phase-space coordinate transform
 2.4 Distribution function in terms of guiding-center variables
 2.5 Spatial gradient operator in guiding-center coordinates
 2.6 Velocity gradient operator in guiding-center coordinates
 2.7 Time derivatives in guiding-center coordinates
 2.8 Final form of Vlasov equation in guiding-center coordinates
3 δf form of Vlasov equation in guiding-center variables
 3.1 Electromagnetic field perturbation
 3.2 Distribution function perturbation
 3.3 Gyrokinetic ordering
 3.4 Equation for macroscopic distribution function F_g
 3.5 Equation for δF_g
 3.6 Equation for the non-adiabatic part δG
4 Gyrokinetic equation in forms amenable to numerical computation
 4.1 Eliminate ∂⟨δϕ⟩_α∕∂t term on the right-hand side of Eq. (136)
 4.2 Eliminate ∂⟨δv ⋅δA⟩_α∕∂t term on the right-hand side of GK equation
 4.3 Mixed-variable pullback method[4]
 4.4 Discretizing Laplacian operator
 4.5 Summary of distribution function split
 4.6 Velocity space moment of ⟨v ⋅ δA⟩_α
5 Parallel Ampere’s Law
6 Poisson’s equation and polarization density
 6.1 Discussion on cancellation scheme
7 Polarization density expressed as local Fourier expansion
 7.1 Gyro-averaging of δΦ in guiding-center coordinates
 7.2 Gryo-angle integration in particle coordinates
 7.3 Pade approximation
8 Polarization density matrix obtained by numerically integrating in phase space using grid
 8.1 Direct evaluation of the double gyrophase integration
 8.2 Performing the parallel velocity integration
 8.3 Toroidal DFT of polarization density in field-aligned coordinates
 8.4 Using MC integration, to be continued, not necessary
A Adiabatic electron response
 A.1 Poisson’s equation with adiabatic electron response
B Characteristic curves of Frieman-Chen nonlinear gyrokinetic equation
 B.1 Time evolution equation for v_∥
C From (δΦ,δA) to (δE,δB)
 C.1 Expression of δB_⊥ in terms of δA
 C.2 Expression of δB_∥ in terms of δA
 C.3 Expressing the perturbed drift in terms of δE and δB
 C.4 Expressing the coefficient before ∂F₀∕∂𝜀 in terms of δE and δB
D Coordinate system and grid in TEK code
E Split-weight scheme for electrons in GEM code
F Implementation of gyrokinetics in particle-in-cell (PIC) codes
 F.1 Monte-Carlo evaluation of distribution function moment at grid-points
 F.2 Monte-Carlo sampling of 6D guiding-center phase-space
 F.3 Distribution function transform**check
 F.4 Moments of distribution function expressed as integration over guiding-center variables
G Diamagnetic flow **check**
H Transform gyrokinetic equation from (X,μ,𝜀,α) to (X,μ,v_∥,α) coordinates
 H.1 Recover equation in W. Deng’s 2011 NF paper
I Drift-kinetic limit
 I.1 Linear case
 I.2 Transform from (X,μ,𝜀) to (X,μ,v_∥) coordinates
 I.3 Parallel momentum equation
 I.4 Special case in uniform magnetic field
 I.5 Electron perpendicular flow
J Derivation of Eq. (123), not finished
K Modern view of gyrokinetic equation
L About this document
References