Generalized functions and vector calculus theorems

Question

To apply the divergence theorem, for example, there are conditions on your function. Your function must be a function in the ordinary sense in the first place. But in Electrodynamics, sometimes our functions (charge density for example) happen to be generalized (Dirac delta) functions. Yet physicists apply these theorems safely (or perhaps blindly?). This leaves me with inconvenience, what to do?

Answer 1

If you have that the distribution $f$ can be decomposed into $f=g+u$ where $g$ is a distribution with compact support $C\subset\operatorname{int}S$ and $u$ is a regular function the integrals involved can be defined and the equality still holds:

$\int_S \nabla\cdot f\,dV = \langle\nabla\cdot g,\varphi\rangle + \int_S\nabla\cdot u\,dV$

where $\varphi$ is a test function that is $1$ on $C$, but still has support on $S$. Now since $g$ has support in $C$ and $\varphi$ is constant there we have $\langle\nabla\cdot g, \varphi\rangle = -\langle g, \nabla\varphi\rangle=0$ so

$\int_S \nabla\cdot f\,dV = \int_S\nabla\cdot u = \int_{\partial S}u\cdot dS = \int_{\partial S}f\cdot dS$