Hello, friends! I know, thanks to @Hawkeye18 who proved this identity to me, that, if ##\phi:V\to\mathbb{R}## is a bounded measurable function defined on the bounded measurable domain ##V\subset\mathbb{R}^3##, then, for any ##k\in\{1,2,3\}##,(adsbygoogle = window.adsbygoogle || []).push({});

$$\frac{\partial}{\partial r_k}\int_V \frac{\phi(\boldsymbol{r})}{\|\boldsymbol{r}-\boldsymbol{r}\|} d\mu_{\boldsymbol{r}}= \int_V \frac{\partial}{\partial r_k}\left(\frac{\phi(\boldsymbol{r})}{\|\boldsymbol{r}-\boldsymbol{r}\|}\right) d\mu_{\boldsymbol{r}}.$$

An important application of this result is the proof that the magnetostatic vector potential ##\frac{\mu_0}{4\pi}\int_\mathbb{R}^3 \frac{\boldsymbol{J}(\boldsymbol{r})}{\|\boldsymbol{r}-\boldsymbol{r}\|} d\mu_{\boldsymbol{r}}## satisfies Ampère's law.

Since I am, fuitlessly until now, trying to prove to myself that the Lorenz gauge magnetodynamic retarded vector potential satisfies Maxwell's equations, I am wondering whether, given a constant ##c##, it is also true that $$\frac{\partial}{\partial r_k}\int_V \frac{\phi(\boldsymbol{r},t-c^{-1}\|\boldsymbol{r}-\boldsymbol{r}\|)}{\|\boldsymbol{r}-\boldsymbol{r}\|} d\mu_{\boldsymbol{r}}= \int_V \frac{\partial}{\partial r_k}\left(\frac{\phi(\boldsymbol{r},t-c^{-1}\|\boldsymbol{r}-\boldsymbol{r}\|)}{\|\boldsymbol{r}-\boldsymbol{r}\|}\right) d\mu_{\boldsymbol{r}}$$for, say ##\phi\in C^2(\mathbb{R}^4)## and ##\phi(-,t)## compactly supported as a function of the first vector variable.

If it is true, how could we prove it?

I ##\infty##-ly thank any answerer!

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# I Differentiation under the integral in retarded potentials

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