# Can current density be written as $\frac{\partial}{\partial t}\nabla Q(\vec r)$?

A quick question. Given that charge density is the charge per unit length/area/volume, I feel that it should be possible to write current density using vector notation as $\frac{\partial}{\partial t}\nabla Q(\vec r)$ where Q is the charge. I haven't see this expression around however. Is this correct, or am I overlooking something?

No, that's not correct. For one, it doesn't have the correct units - if $Q$ is a charge, then $\partial_t\nabla Q$ is a charge per unit time per unit length, instead of per unit area. More importantly, it is perfectly possible to have a whole region (both in a line, as a surface, and in bulk space) where the charge density is zero and the current density is nonzero; to see examples, choose any current-carrying wire around you.
You can relate the current and charge densities through the continuity equation, $$\nabla\cdot\mathbf j+\frac{\partial \rho}{\partial t}=0,$$ with analogues in lower dimensions, but other than that there's no other universal ways to connect them.