# Partial derivatives in Boltzmann equation

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.

• ${}$ Bolztmann? – Qmechanic Jun 3 at 15:58

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.