Can I use the concept of dimensional analysis in problems of vector analysis? For example if I have Gauss' law: $\nabla D=\rho_v$ how can I get one side from the other dimensionally?
Same question goes for rotation and generally for operators.
Please explain downvotes if there's any! I'm not the kind to downvote back.
 A: Firstly, recall the limit definition of a derivative, namely
$$\frac{df}{dt} = \lim_{\delta t \to 0}\frac{f(t+\delta t)-f(t)}{\delta t}$$
if taken with respect to time. From this, it is immediately clear that $\frac{df}{dt}$ has dimensions of $f$ over dimensions of time, so $\frac{d}{dt}$ can be thought of as having dimensions $[T]^{-1}$.

Now, we have Gauss' law, $\nabla \cdot \vec E = \frac{\rho}{\epsilon_0}$. The divergence is explicitly,
$$\nabla \cdot \vec E = \frac{\partial E_x}{\partial x} + \frac{\partial E_y}{\partial y} + \frac{\partial E_z}{\partial z}$$
and so each term has dimensions of the electric field divided by dimensions of length. We know that the dimension of the permittivity of free space is,
$$[\epsilon_0] = [L]^{-3}[M]^{-1}[T]^2[C]^2$$
where $[C]$ is the dimension of charge. We also know $[E] = [M][L][T]^{-2}[C]^{-1}$ and $[\rho] = [C][L]^{-3}$. So we find unsurprisingly that the dimensions on both sides of Gauss' law are consistent. If we knew the right hand side depended only on $\rho$ and $\epsilon_0$, then using dimensional analysis we would find $\rho/\epsilon_0$ is the only consistent right hand side dimensionally, up to a constant.
