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where Δ''f'' is the ''total'' change in ''f''. Dividing ({{EquationNote|1}}) by <math> d^3\bf{r}</math> <math> d^3\bf{p}</math> Δ''t'' and taking the limits Δ''t'' → 0 and Δ''f'' → 0, we have{{NumBlk|2=<math>\frac{d f}{d t} = \left(\frac{\partial f}{\partial t} \right)_\mathrm{coll}</math>|3={{EquationRef|2}}}}
where Δ''f'' is the ''total'' change in ''f''. Dividing ({{EquationNote|1}}) by <math> d^3\bf{r}</math> <math> d^3\bf{p}</math> Δ''t'' and taking the limits Δ''t'' → 0 and Δ''f'' → 0, we have{{NumBlk|2=<math>\frac{d f}{d t} = \left(\frac{\partial f}{\partial t} \right)_\mathrm{coll}</math>|3={{EquationRef|2}}}}
The total [[differential of a function|differential]] of ''f'' is:
The total [[differential of a function|differential]] of ''f'' is:
{{NumBlk|:|<math>\begin{align}
d f & = \frac{\partial f}{\partial t} \, dt
+\left(\frac{\partial f}{\partial x} \, dx
+\frac{\partial f}{\partial y} \, dy
+\frac{\partial f}{\partial z} \, dz
\right)
+\left(\frac{\partial f}{\partial p_x} \, dp_x
+\frac{\partial f}{\partial p_y} \, dp_y
+\frac{\partial f}{\partial p_z} \, dp_z
\right)\\[5pt]
& = \frac{\partial f}{\partial t}dt +\nabla f \cdot d\mathbf{r} + \frac{\partial f}{\partial \mathbf{p}}\cdot d\mathbf{p} \\[5pt]
& = \frac{\partial f}{\partial t}dt +\nabla f \cdot \frac{\mathbf{p}}{m}dt + \frac{\partial f}{\partial \mathbf{p}}\cdot \mathbf{F} \, dt
\end{align}</math>|{{EquationRef|3}}}}
where ∇ is the [[gradient]] operator, '''·''' is the [[dot product]],
where ∇ is the [[gradient]] operator, '''·''' is the [[dot product]],
:<math>
:<math>
</math>
</math>
is a shorthand for the momentum analogue of ∇, and '''ê'''<sub>''x''</sub>, '''ê'''<sub>''y''</sub>, '''ê'''<sub>''z''</sub> are [[cartesian coordinates|Cartesian]] [[unit vector]]s.
is a shorthand for the momentum analogue of ∇, and '''ê'''<sub>''x''</sub>, '''ê'''<sub>''y''</sub>, '''ê'''<sub>''z''</sub> are [[cartesian coordinates|Cartesian]] [[unit vector]]s.
===Final statement===
===Final statement===
20世纪90年代物理宇宙学，完全协变方法被用于研究宇宙微波背景辐射。更一般地说，对早期宇宙过程的研究往往试图考虑量子力学和广义相对论的影响。
20世纪90年代物理宇宙学，完全协变方法被用于研究宇宙微波背景辐射。更一般地说，对早期宇宙过程的研究往往试图考虑量子力学和广义相对论的影响。
== 方程求解 ==
== 方程求解 ==
this analytical approach provides insight, but is not generally usable in practical problems.
this analytical approach provides insight, but is not generally usable in practical problems.