I've seen the derivation where:(adsbygoogle = window.adsbygoogle || []).push({});

## \frac {df}{dt} = \frac {\partial f}{\partial t} + \vec {v} \cdot \vec {\nabla} f + \vec {a} \cdot \vec \nabla_{\vec{v}} f ##

Although I was told this should more generally be written as:

## \frac {df}{dt} = \frac {\partial f}{\partial t} + \vec {\nabla} \cdot ({ {\vec {v}}} f) + \vec {\nabla}_{{\vec{v}}} \cdot ({{\vec {a}}} f) ##

Which follows straightforwardly from the first expression if velocity is independent of position and acceleration is independent of velocity. But would you happen to know where I can find the derivation for the latter expression more generally from first principles (as shown for the first expression)?

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# I General Boltzmann equation

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