Given the two charts for the stereographic projection from the north and south poles respectively:
$$ \phi_N(x,y,z)= \frac{1}{1-z}(u,v)$$
$$ \phi_S(x,y,z)= \frac{1}{1+z}(u,v)$$
and the changes in coordinates:
$$ (u,v) = \alpha_{NS}(s,t)= \frac{1}{s^2+t^2}(s,t)$$
$$ (s,t) = \alpha_{SN}(u,v)= \frac{1}{u^2+v^2}(u,v)$$
how can we show that $\phi_{N,*}^{-1} \frac{\partial}{\partial u}$ prolongs itself to a vector field in $S^2$ with a unique zero?