What would be the derivative of the divergence with respect to time?

Question

Is this true?
${\frac{d\nabla}{dt} = 0}$

And if it is, then would this be different: ${\frac{d\nabla}{dt} . E = 0}$

Answer 1

If your coordinate system is stationary then $[\partial_{t}, \nabla]=0$. I.e. $\partial_{t}\nabla\cdot\bf{E}=\nabla\cdot(\partial_{t}\bf{E})$. If you are in the comooving coordiantes, like you always find yourself to be in hydrodynamics, then the above is not tru anymore.