The Dirac magnetic monopole is defined as \begin{align} \vec{B}=\frac{g\vec{r}}{r^3}\,, \end{align} where $g$ is the strength of the monopole and $\vec{r}$ a vector. It is possible to show that the magnetic field is generated by the following "north" and "south" vector potentials \begin{align} A^N=\frac{g\sin\theta}{r(\cos\theta+1)}\hat{\phi}\,,\quad A^S=-\frac{g\sin\theta}{r(\cos\theta+1)}\hat{\phi}\,, \end{align} respectively, which are defined everywhere except but $\theta=\pi,0$, respectively. These singularities are known as the Dirac string, and it means that we cannot define the magnetic field everywhere.
In this regard, I have two questions:
a) Could someone show me an illustrative draw or image where the Dirac string and the nature of the vector potentials (or the magnetic field) can be seen explicitly on it?
b) According to Nakahara,
"the singularity along the $z$-axis is called the Dirac string and reflects the poor choice of the coordinate system".
I know that with gauge transformations it is possible to move the Dirac string, but it is a nonavoidable singularity (just like $r=0$ in the Schwarzschild case), or a singularity of the coordinates (just like $r=2MG$ in the Schwarzschild case, which can be "eliminated" with Kruskal-Szekeres coordinates) like Nakahara suggest? I am confused.
