Suppose we have $S^2$ and a vector $\vec{A}$ pointing at a random direction. Let us divide the sphere into $S_N$ for $0 \leq \theta \leq \frac{\pi}{2}$ and $S_S$ for $\frac{\pi}{2} \leq \theta \leq \pi$. Then how can I show the following? $$ A_N \backsim \frac{1-\cos \theta}{\sin \theta} \hat{e}_{\phi} $$ $$ A_S \backsim \frac{1+\cos \theta}{\sin \theta} \hat{e}_{\phi} $$
How do I split an arbitrary radial vector as above to a north and southern part therefore? Note that $\phi$ is the other angle with $0 \leq \phi \leq 2\pi.$
The question arise from page 10 of this document.
I believe this is a misunderstanding. The text is not dividing a vector $\vec A$ into two components. It is describing how to consistently define a vector potential for the magnetic field of a magnetic monopole despite the fact that no single vector potential can describe this field. It does so by defining two different vector potentials on the two hemispheres and showing that they are compatible at the boundary. There is no vector $\vec A$ of which $\vec A_N$ and $\vec A_S$ form components.