How to represent a time varying vector field

[ScienceVSmath]

Homework Statement

I'm solving for the magnetic potential vector field given a time varying electrical field. It might seem silly, but I don't really understand how to approach the problem since the field varies on both space and time. I'm given the motion of a charged particle. Can someone explain how exactly I would represent the time varying electrical, electric potential, and magnetic potential fields. I'm not really having an issue with the Physics, it's the math.

Homework Equations

E(x,t)=-∇Φ-(∂A/∂t)

The Attempt at a Solution

 

[BvU]

Hello Vs, welcome to PF

That is a bit of a generic question. Can be made as complicated as you want (relativistic ?). The issue is specifically adressed and worked out with the Lienard-Wiechert potential .

From Ohio state:

Differentiating the Lienard-Wiechert potentials for a moving point charge is,
as the textbook says, straightforward but tedious". If you want to know
the details, look in Jackson's Classical Electrodynamics, which specializes in
straightforward tediousness.⁵

​

⁵Sometimes, to be fair, it deviates into devious tediousness​

My Jackson (2nd ed, 1975) Ch 6 is tedious, indeed. Pages look as clean as new, so I must have avoided them :)
I would start with simple cases.