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A magnetic monopole carries an isolated magnetic charge, producing nonzero net magnetic flux through a closed enclosing surface. In the usual macroscopic Maxwell equations without magnetic charges, Gauss's law for magnetism instead gives $\nabla \cdot \mathbf{B}=0$. Cutting a conventional bar magnet produces smaller magnetic dipoles, rather than separating its north and south poles into isolated charges.

A model radial field $\mathbf{B}=g\hat{\mathbf{r}}/r^2$ away from the origin has flux $4\pi g$ through every enclosing sphere. It can be described locally by a magnetic vector potential, but not by one potential smooth over the whole enclosing sphere: the Stokes theorem would otherwise force its total flux to vanish. The two-patch vector potential for a radial monopole makes that obstruction explicit. A classical monopole model, a monopole solution in a field theory, and a monopole-like excitation in a material require their own physical interpretation and evidence.

On the sphere, constant $\theta$ gives a latitude circle of radius $a\sin \theta$ and height $a\cos \theta$. Constant $\varphi$ gives a meridian semicircle in a vertical plane through the $z$ axis. The orthonormal basis of spherical coordinates isHere ${\mathbf{e}}_r$ points outward, ${\mathbf{e}}_{\theta }$ along increasing polar angle, and ${\mathbf{e}}_{\varphi }$ along increasing azimuth. At either pole the radial unit vector is unambiguous, but the two tangential coordinate vectors depend on the undefined longitude. The figure shows their directions away from the poles.

This coordinate ambiguity does not make a vector field singular if its tangential coefficient vanishes suitably. In Cartesian coordinates,The first is smooth at the north pole and zero there; the second is smooth and zero at the south pole. Their actual singular axes are the opposite ones. These are the regular patches of a two-patch vector potential for a radial monopole.