Polarisation current In a dielectric we know that:
$$\rho_{pol}=-\nabla\cdot\vec{P}.$$
At the same time we know that the polarisation charge density can be the source of the polarisation current density:
$$\vec{\jmath}_{pol}=\frac {\partial \rho_{pol}}{\partial t}.$$
This would imply that:
$$\vec{\jmath}_{pol}= \frac {\partial}{\partial t}\left(-\nabla\cdot\vec P\right).$$
But in wikipedia (https://en.wikipedia.org/wiki/Polarization_density) it is:
$$\vec{\jmath}_{pol}= \frac {\partial \vec P}{\partial t}.$$
Why is it like this and not like I wrote initially?
 A: Your expression for $\vec{\jmath}_{{\rm pol}}$ is incorrect.  The local charge conservation equation for a charge density $\rho$ and its corresponding current density $\vec{\jmath}$ takes the form
$$\vec{\nabla}\cdot\vec{\jmath}+\frac{\partial\rho}{\partial t}=0.$$
This condition is consistent with $\vec{\jmath}_{{\rm pol}}=\frac{\partial\vec{P}}{\partial t}$ and $\rho_{{\rm pol}}=-\vec{\nabla}\cdot\vec{P}$, since in that case
$$\vec{\nabla}\cdot\vec{\jmath}=-\frac{\partial\rho}{\partial t}=\frac{\partial}{\partial t}\left(\vec{\nabla}\cdot\vec{P}\right).$$
Your original expression for $\vec{\jmath}_{{\rm pol}}$ has the wrong units for a current density, but this is an easy mistake to make.  The units of current density, $\vec{\jmath}$, are not current per unit volume, but rather current per unit area.  The units of $\vec{P}$ are dipole moment (charge times distance) divided by volume, so charge per unit area.  Differentiating this to with respect to time, $\frac{\partial\vec{P}}{\partial t}$, gives charge per unit area per time, or (since current is charge divided by time) current per area—thus current density.
