# Why, exactly, can't two field lines cross? [duplicate]

My physics textbook says two electric field lines never intersect. Their explanation runs somewhat like this:

If two field lines crossed, there would be two different directions to the electric field at the intersection point, which is impossible by definition.

A similar explanation is provided as to why two stream lines never cross.

However, mathematically, two intersecting curves can have the same direction of tangent at their intersection point. For example, consider the $x-$axis and the curve $y=x^3$ is the $xy$ plane. Or any two curves of the form $y=x^{2k+1}$ with $k\in\mathbb N$, plotted for $-1<x<1$.

Is the explanation then wrong, considering these counterexamples? If not, why not? If yes, then what's the correct explanation?

Edit: The answers to this Phys.SE question seem to focus on the very explanation I am having trouble understanding; I don't see how this is a duplicate of that one. I'd rather describe this as a follow-up question to that one.

## marked as duplicate by anna v, Yashas, Qmechanic♦Apr 3 '17 at 6:26

• Possible duplicate of Why can two (or more) electric field lines never cross? – anna v Apr 3 '17 at 5:14
• If you draw what you are describing, you are drawing a source at x=0 . Sources are charges in experimental observations and as modeled by classical electrodynamics. electric field lines theoreticlly cross on the point where a point charge is. – anna v Apr 3 '17 at 5:17
• This is really a mathematical question, related to the existence and uniqueness theorem for ODEs. The answer depends on how pathological you allow your situations to be. – knzhou Apr 3 '17 at 5:25
• In a real physical setup, this wouldn't be allowed. By allowing field lines to merge into one, you increase the local electric flux density to infinity. – knzhou Apr 3 '17 at 5:26
• @knzhou I'm interested to know how the existence and uniqueness theorem for ODEs solves my problem; mind elaborating? And FWIW, I don't mind pathological situations. :) – Ankoganit Apr 3 '17 at 6:09