Recent content by DavideGenoa

Errata corrige: I think that another lemma (proved here) useful to prove the desired equality may be: if ##\rho\in C_c^k(\mathbb{R}^4)## and ##f:\mathbb{R^3}\mapsto \mathbb{R}## is locally summable (as ##\boldsymbol{z}\mapsto 1/\|\boldsymbol{y}\|## is), then for any , then we can differentiate...

I think that another lemma (proved here) useful to prove the desired equality may be: if ##\rho\in C_c^k(\mathbb{R}^4)## and ##f:\mathbb{R^3}\mapsto \mathbb{R}## is locally summable (as ##\boldsymbol{z}\mapsto 1/\|\boldsymbol{y}\|## is), then for any , then we can differentiate under the...

Hi, friends! Under particular conditions on ##\phi:\mathbb{R}^3\times\mathbb{R}\to\mathbb{R}## - I think, as said here, that it is sufficient that ##\phi\in C_c^1(\mathbb{R}^4)##: please correct me if I am wrong - the following equality holds$$\frac{\partial}{\partial r_k}\int_{\mathbb{R}^3}...

@stevendaryl and anyone else reading: Are my conclusions in the preceding post correct?

Dear @stevendaryl , I have tried to adapt the proof by Hawkeye18 to this problem, and, as far as I understand, it can used in an almost identical way, with ##\frac{\partial }{\partial r_k}\left(\frac{\phi(\boldsymbol{l},t-c^1\|\boldsymbol{r}-\boldsymbol{l}\|)}{\|\boldsymbol{r}-\boldsymbol{l}\|}...

Thank you very much! Yes, @stevendaryl, I meant: 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\}##...

Not at all. I have only studied Kolmogorov-Fomin's Элементы теории функций и функционального анализа, roughly corresponding to the English version Introductory Real Analysis, by myself, since I have not enrolled at university, at least not yet, but I study mathematics and physics, while working...

Thank you, @WWGD! I have begun to fragmentate the question by asking here why (and whether) we can differentiate under the integral sign the first time it is done...

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\}##, $$\frac{\partial}{\partial r_k}\int_V...

Thank you both for your answers! Since I still do not understand the steps necessary to prove the desired result shown in your links, nor the steps I have found in Griffiths' Introduction to Electrodynamics, § 10.2.1, I am going to ask about my problems with these last ones, which are the...

Dear friends, I have found a derivation of the fact that, under the assumptions made in physics on ##\rho## (to which we can give the physical interpretation of charge density) the function defined by $$V(\mathbf{x},t):=\frac{1}{4\pi\varepsilon_0}\int_{\mathbb{R}^3}...

Thank you so much again! I am not saying that ##\mathbf{A}## is time independent in general. I mean: if $$\mathbf{A}(\mathbf{x}):=\frac{\mu_0}{4\pi}\int \frac{\mathbf{J}(\mathbf{y})}{\|\mathbf{x}-\mathbf{y}\|}d\mu_{\mathbf{y}} $$ and ##\mathbf{J}\in C^2(\mathbb{R}^3)## is compactly supported, by...

Thank you so much! The problem is that I do not understand how the desired identities are derived in the .pdf. I mean, if ##\mathbf{A}## does not depend upon ##t##, provided, as I think we can do in physics, that ##\mathbf{J}## is compactly supported and of class ##C^2(\mathbb{R}^3)##, by using...

@vanhees 71, thank you very much for your hint, but I have never studied Green's function and I have no idee how what you say is derived and how to apply it to my problem, which precisely is finding the steps through which we see, by differentiating twice the components of ##\mathbf{A}##...

##\nabla V## should be ##\mathbf{E}=-\nabla V##, with the minus sign, in the original post.