I am working with some kinetic theory. I have the distribution function $\Psi (\overrightarrow{r},\overrightarrow{p},t)$, Where $\overrightarrow{r}$ - radius vector, $\overrightarrow{p}$ - unit vector of orientation, $t$ - time.
But my question is pure mathematical. In the article I am trying to understand, there is an Integration-by-parts formula:
$$ \int_{S} \overrightarrow{p} \left(\nabla_p \cdot \left( \frac {d\overrightarrow{p}} {dt} \Psi\right) \right)d\overrightarrow{p}=-\int_{S} \frac {d\overrightarrow{p}} {dt} \Psi d\overrightarrow{p} $$ Here $S$ is the surface of the unit sphere.
I know, how to use the IBP formula when integration domain is some segment in 1D, area in 2D, or volume in 3D.
I don't understand, what is the "boundary" of the region here. Can anyone explain me the full formula, with a boundary term ( here, obviously, it is omitted, since it is zero).
Thanks
