1. ### I Expressing the magnetic vector potential in terms of its curl

@vanhees71 thanks, but I just wanna find A in terms of ∇'Φ and B.
2. ### I Expressing the magnetic vector potential in terms of its curl

well now we're at (35). I'm trying to express the electric field in terms of the "magnetic displacement current".
3. ### I Expressing the magnetic vector potential in terms of its curl

hmm, sadly, I think there might've been a misunderstanding but anyway, equation 35: \mathbf{A}=\frac{1}{4 \pi c^{2}} \frac{\partial}{\partial t} \int \frac{\left[\boldsymbol{\nabla}^{\prime} \Phi+\partial \mathbf{A} / \partial t\right]}{R} d^{3} r^{\prime}+\boldsymbol{\nabla} \times \int...
4. ### I Expressing the magnetic vector potential in terms of its curl

(writing in progress...)
5. ### I Expressing the magnetic vector potential in terms of its curl

ok, regardless, (30) is correct regarding the retarded vector fields described in the OP right? Just to confirm.
6. ### I Expressing the magnetic vector potential in terms of its curl

thanks! But in terms of B, its the same right?
7. ### I Expressing the magnetic vector potential in terms of its curl

thank you for the detailed analysis If we start with the Lorenz gauge, we will eventually arrive at the same spot right? interesting, that's kinda tricky Don't worry, its been pretty useful to me for understanding electromagnetism. :) I'd just like to see one more step. Is it correct to...
8. ### I Instantaneous solutions to Maxwell's equations' potentials conversion?

thanks, so far instantaneous potential solutions have eluded me after searching. I have found some stuff, but they aren't solely in terms of J and ρ.
9. ### I Expressing the magnetic vector potential in terms of its curl

but in (31), it seems like the author's applying the curl to the numerator B only.
10. ### I Expressing the magnetic vector potential in terms of its curl

sorry, I'm confused, I thought the nabla applies to the numerator only.
11. ### I Instantaneous solutions to Maxwell's equations' potentials conversion?

thank you, so, based on the Coulomb gauge, do you know what are the magnetic and electric potential solutions in terms of the charge and current densities?
12. ### I Instantaneous solutions to Maxwell's equations' potentials conversion?

ok, is the general solution you are referring to (509) and (510)? And the linear differential equations (506) and (507)?
13. ### I Instantaneous solutions to Maxwell's equations' potentials conversion?

okay, but doesn't excluding instantaneous solutions require a mathematical proof regarding Maxwell's equations?
14. ### I Expressing the magnetic vector potential in terms of its curl

I see, thanks, that makes sense. I noticed that from (30) to (31), the curl operator applied to the fraction ends up being equivalent to applying it to the numerator only. Do you know why that's the case?
15. ### I Instantaneous solutions to Maxwell's equations' potentials conversion?

cos I was thinking, for example, some versions of the Biot-Savart law don't use retarded time but still satisfy Maxwell's equations. anyway, what's the proof against instantaneity?