2.2.2.1 Even Moments

Multiplication with weight $O_i$ and integration gives the equation for the unknown $M_i$. For the even moments this gives

$$\begin{gather*}\begin{split}& \partial_{x_3} \int v_3 u^i f(x_3,v_3,v_r) d(v_3,v... ... = & \quad \quad \quad \int Q(f) u^i d(v_3,v_r) . \end{split}\end{gather*}$$ (2.31)

We use partial integration on the second term which leads to

$$\partial_{v_3} u^i = \frac{\partial u^i}{\partial u} \frac{\parti... ...} = iu^{i-1}\frac{\partial (v_r^2 + v_3^2)}{\partial v_3} = iu^{i-1}(2v_3) .$$ (2.32)

For $i = 1,2$ and $O_{2i} = v^{2i} = u^i$ we get:

$$\partial_{x_3} M_{2i+1} - \frac{q}{m^{*}} E_{x_3} (2iM_{2i-1}) = \int Q(f) u^i d(v_3,v_r) .$$ (2.33)

For the special case of the zeroth moment we obtain from $O_0 = 1$

$$\partial_{x_3} M_{1} = 0 , %- \frac{\langle O_0 \rangle - \langle O_0 \rangle_0}{\tau_0}$$ (2.34)

which is the well-known continuity equation.