How can electric potential contain so much information?

[zezima1]

Question is pretty simple and arose when doing one of the usual integrals to find the electric field. This one concerned the field above the middle of a uniform line distribution of charge. If you wish to calculate the field you must take in mind that the horizontal components cancel. However if you just find the potential and then take the gradient you also get the correct field. How can the potential "know" that the horizontal parts cancel? It's quite amazing (too amazing for me) to be honest.

 

[gopher_p]

It's been a while since I took physics, so I may not have the best answer for you. But here goes ...

If I had to guess it boils down to one of (or perhaps a combination of)

1) The definition of electric potential. It has that information because it was designed to have that information.

2) The glory of the Fundamental Theorem of Calculus and its variants/generalizations. All of the FTCs (FTC, Green's Thm, Stokes' Thm, Divergence Thm, FTC for Line integrals, ...) essentially tell you that you can get information about one function's accumulated behavior on a set by looking at the behavior of its antiderivative/potential functions on the boundary of that set. Understanding why this works requires digging into the various proofs of those theorems. It really is one of the most amazing and beatiful results in all of mathematics.