# How shall we show the surface integral approaches a limit (or does not blow up) at a field point near $S'$

Consider the electric field due to volume charge distribution in volume $$V'$$:

$$\mathbf{E}=\displaystyle \int_{V'} \rho' \dfrac{\mathbf{r}-\mathbf{r'}}{|\mathbf{r}-\mathbf{r'}|^3} dV'$$

The integrand blows up at a field point $$P \in V'$$. But the integral remains finite. This can be shown by choosing $$P$$ as the center of our Cartesian coordinate system and switching to spherical coordinate system.

Now consider the electric field due to surface charge distribution in surface $$S'$$:

$$\mathbf{E}=\displaystyle \int_{S'} \sigma' \dfrac{\mathbf{r}-\mathbf{r'}}{|\mathbf{r}-\mathbf{r'}|^3} dS'$$

At a field point $$P \in S'$$, it can be shown using Gaussian pillbox argument that there is a discontinuity of $$4\pi \sigma'$$ in the field $$\mathbf{E}$$ and hence the field is undefined at a point $$P \in S'$$.

Now as $$P$$ gets closer and closer to the surface, the integrand blows up. Now how shall we show that the integral doesn't blow up (or approaches a limit)?

Conditions:

1. $$\sigma'$$ is nowhere infinite
2. $$S'$$ is a closed and bounded surface (finite surface)
• Rephrasing your question: are you trying to prove that a function that is measurable almost everywhere can still have a well defined surface integral? If so, this is a well studied field of complex analysis and residuals theorems and the answer lies in there (totally independent of this application to the electric field). – gented May 29 '19 at 11:31
• @gented: Then how did people (eg: Poisson, Maxwell,etc) dealt with it before the birth of complex analysis? – N.G.Tyson May 29 '19 at 11:52
• Do you mean how they dealt with proving that it still converges or how they calculated it? For the former I don't know for sure but I believe they just didn't care :)...for the latter you can just practically integrate it by using the definition of surface integral and Dini's theorem (which reduces everything to standard Riemann integrals). – gented May 29 '19 at 11:55
• I see... So could it be that most of electromagnetism became rigorous after the introduction of these advanced topics (after Maxwell's death)? – N.G.Tyson May 29 '19 at 12:02
• Relevant: physics.stackexchange.com/q/450426 – jacob1729 May 29 '19 at 16:28