Determining the direction of magnetic vector potential

[bfusco]

Homework Statement

Im doing this practice question and I am to determine the magnetic vector potential A for a cylindrical wire with a uniform current density J. i have already determined B both inside and outside the wire no problem.My issue is in the solution given my professor states that due to symmetry $\vec{A}=A\hat{z}$ then $\nabla \times A=\frac{-dA(s)}{ds}\hat{\phi}$

The part I don't understand is the statement "due to symmetry $\vec{A}=A\hat{z}$", how does he know that just by looking?

Is it because A is determined by the integral of the current, therefore A points in the same direction as the current?

 

[WannabeNewton]

In general, the direction of $\vec{A}$ will not necessarily be aligned with that of $\vec{j}$ (e.g. if the current density field points in different directions at different points, that is if the current doesn't all flow in one direction, then you can't immediately be certain that the direction of $\vec{A}$ aligns with that of $\vec{j}$ from point to point) but if $\vec{j}$ flows in one direction everywhere then what you said is basically true up to a sign.

Think about it: if $\vec{j}$ points in a single direction everywhere in space then this picks out a preferred axis in space right? If $\vec{A}$ didn't point along the same axis, how will it even know which way to point when there is no directional preference in the system off of that axis?