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In Reif's Fundamentals of Statistical and Thermal Physics, at page 499, chapter 13, he describes the Bolztmann equation in the absence of collisions and the distribution function $f(\textbf{r},\textbf{v},t)$ after the time interval $dt$ obeys the following relation:

$$f(\textbf{r}',\textbf{v}',t')=f(\textbf{r},\textbf{v},t)$$

Or $$f(\textbf{r}+\dot{\textbf{r}}dt,\textbf{v}+\dot{\textbf{v}}dt,t+dt)-f(\textbf{r},\textbf{v},t)=0$$

He then expresses the latter in terms of partial derivatives:

$$\left[\left(\frac{\partial{f}}{\partial{x}}\dot{x}+\frac{\partial{f}}{\partial{y}}\dot{y}+\frac{\partial{f}}{\partial{z}}\dot{z}\right)+\left(\frac{\partial{f}}{\partial{v_x}}\dot{v_x}+\frac{\partial{f}}{\partial{v_y}}\dot{v_y}+\frac{\partial{f}}{\partial{v_z}}\dot{v_z}\right)+\frac{\partial f}{\partial t}\right]dt=0$$

I'm having trouble moving from the second to the last equation. I apply the chain rule for each coordinate, but how do I apply it to the first term? Should I expand it first as in: $f(x+dx)=f(x)+\frac{df}{dx}dx$? I tried so, but can't get to the last equation.

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  • $\begingroup$ $ {}$ Bolztmann? $\endgroup$ – Qmechanic Jun 3 at 15:58
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Consider the function $f(\textbf{r},\textbf{v},t)$. We want to show that $f$ does not change. Calculate the total time derivative with the chain rule:

$$\frac{df(\textbf{r},\textbf{v},t)}{dt} dt = \left(\frac{df}{d\textbf{r}}\dot{\textbf{r}} + \frac{df}{d\textbf{v}}\dot{\textbf{v}} + \frac{\partial f}{\partial t}\right)$$

Since you are using Cartesian coordinates, $\textbf{r} = x \hat{x} + y \hat{y} + z \hat{z}$. The last equation follows since $f$ does not change.

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