= Two-patch vector potential for a radial monopole
{title2=$A_\pm=\frac{\pm1-\cos\theta}{r\sin\theta}e_\phi,\quad\nabla\times A_\pm=e_r/r^2$}
The radial <vector field> $B=e_r/r^2$ on punctured three-dimensional <Euclidean space> has flux $4\pi$ through every centered <sphere>. It has no <vector potential> smooth on an entire enclosing sphere: the <Stokes theorem> on a closed surface would force zero flux. Two overlapping patches suffice. In <spherical coordinates> take $A_+=(1-\cos\theta)e_\phi/(r\sin\theta)$, regular away from the negative $z$ axis, and $A_-=-(1+\cos\theta)e_\phi/(r\sin\theta)$, regular away from the positive $z$ axis. Their <curls> both equal $B$.
The Cartesian forms $A_+=(-y,x,0)/[r(r+z)]$ and $A_-=(y,-x,0)/[r(r-z)]$ show that the first is smooth at the north pole and the second at the south pole. On the overlap their difference is $A_+-A_-=2\nabla\phi$, where the <azimuth> $\phi$ is defined locally. Its multivaluedness around the axis explains the circulation difference $4\pi$. Splitting the sphere along a latitude, applying the <Stokes theorem> with $A_+$ on the north and $A_-$ on the south, and using their opposite boundary orientations recovers the total flux. A <coordinate singularity> in the tangential spherical basis does not by itself make a vanishing smooth vector field singular at a pole.
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