# Can we find the following improper integral in any coordinate system using the usual methods of integration?

Consider the integral (expression for electric field at a point inside the charge distribution):

$$\mathbf{E}=\lim\limits_{\delta \to 0} \int_{V'-\delta} \rho'\ \dfrac{\mathbf{r}-\mathbf{r'}}{|\mathbf{r}-\mathbf{r'}|^3} dV'$$

where:

$$\mathbf{r'}$$ is coordinates of source points

$$\mathbf{r}$$ is coordinates of field points

$$V'$$ is the volume occupied by the charge

$$\delta$$ is a (small spherical) volume around the singular point $$\mathbf{r}=\mathbf{r'}$$

$$\rho'$$ is the charge density and is continuous throughout the volume $$V'-\delta$$

Now, since the function $$\left( \rho'\ \dfrac{\mathbf{r}-\mathbf{r'}}{|\mathbf{r}-\mathbf{r'}|^3} \right)$$ is continuous in the whole (open) domain, is it always possible to find this integral in any coordinate system using the usual methods of integration (i.e. finding anti-derivative and applying the appropriate limits)?

Or not (due to some other reasons like the domain being open)?

Actually it looks pretty straight forward. Use a spherical coordinate system with origin at the "source point". Then $$\rho= r- r'$$ and $$d\rho= dr$$. The problem becomes $$\lim_{\delta\to 0}\int_{\phi= 0}^\pi\int_{\theta= 0}^{2\pi}\int_{\rho= \epsilon}^\infty p(r,\theta,\phi)\frac{\rho}{\rho^3}\rho^2 sin(\theta)d\theta d\phi d\rho= \lim_{\delta\to 0}\int_{\phi= 0}^\pi\int_{\theta= 0}^{2\pi}\int_{\rho= \epsilon}^\infty p(r,\theta,\phi) sin(\theta)d\theta d\phi d\rho$$.
(Since I was using "$$\rho$$" as a coordinate, I changed your function "$$\rho$$" to "p".)
• I know we can use spherical coordinates (with origin at field point) to determine the integral: $$\mathbf{E}=\lim\limits_{\delta \to 0} \int_{V'-\delta} \rho'\ \dfrac{\mathbf{r}-\mathbf{r'}}{|\mathbf{r}-\mathbf{r'}|^3} dV'$$.
• But what I am asking is that "since the function $$\left( \rho'\ \dfrac{\mathbf{r}-\mathbf{r'}}{|\mathbf{r}-\mathbf{r'}|^3} \right)$$ is continuous in the whole (open) domain, can we use any coordinate system (with origin anywhere) to determine the integral $$\mathbf{E}=\lim\limits_{\delta \to 0} \int_{V'-\delta} \rho'\ \dfrac{\mathbf{r}-\mathbf{r'}}{|\mathbf{r}-\mathbf{r'}|^3} dV'$$ using the usual methods of integration (i.e. finding anti-derivative and applying the appropriate limits)"?