User:Frode54/depot math

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Equation referencing numbering
$$ $$ \quad W_{i} = \frac {z_{i}} {M_{i}^{\varepsilon} \cdot MM} \quad \text{where} \quad MM = \sum_{j=1}^{N} \frac {z_{j}} {M_{j}^{\varepsilon} } $$

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a

$$ $$ \quad W_{i} = \frac {z_{i}} {M_{i}^{\varepsilon} \cdot MM} \quad \text{where} \quad MM = \sum_{j=1}^{N} \frac {z_{j}} {M_{j}^{\varepsilon} } $$

Her er en tekst med referanse ($$) og videre tekst.

b

Her er en tekst med referanse $$ og videre tekst.

c

Her er en tekst med referanse ($$) og videre tekst.

d

where Δf is the total change in f. Dividing ($$) by $$ d^3\bf{r}$$ $$ d^3\bf{p}$$ Δt and taking the limits Δt → 0 and Δf → 0, we have

The total differential of f is:

where ∇ is the gradient operator, · is the dot product,



\frac{\partial f}{\partial \mathbf{p}} = \mathbf{\hat{e}}_x\frac{\partial f}{\partial p_x} + \mathbf{\hat{e}}_y\frac{\partial f}{\partial p_y} + \mathbf{\hat{e}}_z \frac{\partial f}{\partial p_z}= \nabla_\mathbf{p}f $$

is a shorthand for the momentum analogue of ∇, and êx, êy, êz are Cartesian unit vectors.

Final statement
Dividing ($$) by dt and substituting into ($$) gives:


 * $$\frac{\partial f}{\partial t} + \frac{\mathbf{p}}{m} \cdot \nabla f + \mathbf{F} \cdot\frac{\partial f}{\partial \mathbf{p}} = \left(\frac{\partial f}{\partial t} \right)_\mathrm{coll}$$

In this context, F(r, t) is the force field acting on the particles in the fluid, and m is the mass of the particles. The term on the right hand side is added to describe the effect of collisions between particles; if it is zero then the particles do not collide. The collisionless Boltzmann equation is often called the Vlasov equation.

This equation is more useful than the principal one above, yet still incomplete, since f cannot be solved unless the collision term in f is known. This term cannot be found as easily or generally as the others – it is a statistical term representing the particle collisions, and requires knowledge of the statistics the particles obey, like the Maxwell–Boltzmann, Fermi–Dirac or Bose–Einstein distributions.