Let:
$\displaystyle f=\int_V \dfrac{x-x'}{|\mathbf{r}-\mathbf{r'}|^3}\ dV'$
where $V'$ is a finite volume in space
$\mathbf{r}=(x,y,z)$ are coordinates of all space
$\mathbf{r'}=(x',y',z')$ are coordinates of $V'$
$|\mathbf{r}-\mathbf{r'}|=[(x-x')^2+(y-y')^2+(z-z')^2]^{1/2}$
How to prove that:
$\lim\limits_{\Delta x \to 0} \dfrac{f(x+\Delta x,y,z)-f(x,y,z)}{\Delta x}$ exist
$\text{ }$
MY TRY:
I am not sure whether this method would work. If it doesn't please suggest another method to reach my goal.
\begin{align} &\lim\limits_{\Delta x \to 0} \dfrac{f(x+\Delta x,y,z)-f(x,y,z)}{\Delta x}\\ =&\lim\limits_{\Delta x \to 0}\dfrac{\displaystyle\int_{V'} \dfrac{(x+\Delta x)-x'}{|\mathbf{r}(x+\Delta x,y,z)-\mathbf{r'}|^3}\ dV' - \int_{V'} \dfrac{x-x'}{|\mathbf{r}(x,y,z)-\mathbf{r'}|^3}\ dV'}{\Delta x}\\ =&\lim\limits_{\Delta x \to 0}\displaystyle\int_{V'} \dfrac{\left( \dfrac{(x+\Delta x)-x'}{|\mathbf{r}(x+\Delta x,y,z)-\mathbf{r'}|^3} -\dfrac{x-x'}{|\mathbf{r}(x,y,z)-\mathbf{r'}|^3} \right)}{\Delta x}dV' \end{align}
Now if only I could take the limit inside the integral (with respect to $V′$),I can proceed to show the limit exists.
If we can't do that and this method doesn't work, please suggest another method to show that the limit exists.
Assume that $C$ is a $3$-dimensional body with a smooth boundary and ${\rm vol}\ C<\infty$. Define $f(\textbf{r}) =\int_C\ \frac{x-x'}{|\textbf{r}-\textbf{r}'|^3}\ d{\rm vol}\ (\textbf{r}' ) $ Prove that $f$ is finite.
Proof : $\int_{B_\epsilon (0)}\ \frac{1}{|{\bf r}|^2} \ d{\rm vol}\ ({\bf r}) \leq C\epsilon$ for some $l>0$ and note that $ \frac{|x-x'|}{|\textbf{r}-\textbf{r}'|^3}\leq \frac{1}{| \textbf{r}-\textbf{r}'|^2}$
So $$ |f( \textbf{r} )| \leq \int_{B_\epsilon ({\bf r} )}\ \frac{|x-x'|}{|\textbf{r}-\textbf{r}'|^3}\ d{\rm vol}\ (\textbf{r}') +\int_{C-B_\epsilon(\textbf{r}) }\ \frac{1}{|\textbf{r}- \textbf{r}'|^2}\ d{\rm vol}\ (\textbf{r} ') $$
$$ \leq l\epsilon +\int_{C-B_\epsilon(\textbf{r}) }\ \frac{1}{\epsilon^2} \ d{\rm vol}\ (\textbf{r} ') \leq l\epsilon + \frac{1}{\epsilon^2}{\rm vol}\ C $$