The continuity equations can be derived by multiplying Boltzmann's
equation (2.3) by 
, i=0, ..., 4 and integrating over
. We note from
equation (2.7) that the integral of 
is zero
and that
The results in equation (2.8) can be seen by integrating by parts
and by noting that 
since the integrals
defined in equation (2.5) must
converge [12]. We also note that 
,
and t are independent variables and so
commutes with 
and 
. We are therefor considering
Consider first the case i=0. Multiplying by m and using equation (2.5) this gives the continuity equation
When i = 1, 2, and 3 we get
where
. Expressing the molecular velocity
in terms of the fluid velocity
and the peculiar velocity
we can write
The first term in equation (2.12) is simply 
and the second term is the pressure tensor 
[13]. Thus we have the Euler equation
Similarly, when i =4, we get the conservation of energy equation [12, 13],
where 
is the heat flux [12],
