The stress tensor

For each surface on a unit cube, the stress on that surface can point in each of the three directions.

$F_x=\sum_i \partial \sigma_{i1} / \partial x$

The second moment of the particle distribution function, describing the flow of momentum in
the laboratory frame, is called the stress tensor,

$$\boldsymbol{P} = \int m \vec{v}\vec{v} f(\vec{r},\vec{v},t) d^3\vec{v} .$$

The third order moment of the particle distribution function measuring the energy flux density is denoted by 

$$\boldsymbol{Q(\vec{r},t)} = \int \frac{1}{2} m v^2 \vec{v} f(\vec{r},\vec{v},t) d^3\vec{v} .$$

The stress tensor measured in the rest-frame is called the pressure tensor, $\boldsymbol{P}$, whereas the energy flux density becomes the heat flux density, $\boldsymbol{q}$. We introduce the relative velocity,

\[ \vec{w} = \vec{v} - \vec{V} .

Then, we get

$$\boldsymbol{p(\vec{r},t)}= \int m \vec{w}\vec{w} f(\vec{r},\vec{v},t) d^3\vec{v} .$$

$$\boldsymbol{q(\vec{r},t)}= \int \frac{1}{2}m w^2 \vec{w} f(\vec{r},\vec{v},t) d^3\vec{v} .$$

The trace of the pressure tensor measures the ordinary (scalar) pressure,

$$p = \frac{1}{3}Tr(\boldsymbol{p}) =\frac{1}{3}(p_{xx}+p_{yy}+p_{zz}).$$

Note that the kinetic energy density is $3/2p$,

$$\frac{3}{2}p = \int \frac{1}{2} m w^2 f(\vec{r},\vec{v},t) d^3\vec{v} .$$

The moments measures in the two different frames are related.

$$\boldsymbol{P} = \boldsymbol{p} + mn\vec{V}\vec{V}$$

$$\boldsymbol{Q} = \boldsymbol{q}+\boldsymbol{p}\cdot \vec{V}+ \frac{3}{2}p\vec{V}+\frac{1}{2}mnV^2\vec{V}.$$

Reference:
https://physics.stackexchange.com/questions/152927/why-is-stress-a-tensor-quantity/152933
https://en.wikipedia.org/wiki/Cauchy_stress_tensor

https://farside.ph.utexas.edu/teaching/plasma/lectures1/node29.html