Heavyside's vector notation compressed Maxwell's twenty-some component equations into four. Thats elegant for teaching and teeshirts. However when I want to compute them on a computer grid, I use something closer to Maxwell's original equations.
The fully expanded coordinate version is based on an arbitrary choice of spacetime coordinates.
Ideally you'd use a computer language/environment with a rich enough basic formalism to let you write the 1 formula in a human-legible way and have your compiler expand out all of the coordinate version as appropriate for the particular problem for the sake of doing floating point arithmetic.