When one talks about electrical charge distribution, one usually refers to the way in which charge is "spread" through space. For example one might say that charge is distributed continuously with a

• volume charge density $\rho~(C/m^3)$ in a volume,
• surface charge density $\sigma~(C/m^2)$ on a surface,
• linear charge density $\lambda~(C/m)$ on a curve.

Alternatively we could have a discrete charge distribution in which the total charge $q$ in a region of space is located at various points.

However, in mathematics there is also the concept of distributions which is a sort of a generalization of a function. Formally a distribution is a functional that acts on test functions. A famous example of this is the delta Dirac function $\delta$ which, in the case of a test function $\varphi$ is defined as $\delta(\varphi) = \varphi(0) $.

My question is: is there a connection between the concepts of "charge distribution" and "mathematical distribution (as described earlier)"? If so, what are the distributions and the test functions in the previous cases (where we have $\rho$, $\sigma$ and $\lambda$)?