Let $S^2$ be the 2-sphere in $\mathbb{R}^3$, given with vector field $X=x \partial_x+ y \partial_y$ on the stereographic projection from north pole chart. What is the global extension of $X$ to $S^2$ as a vector field on $\mathbb{R}^3$?
Edit: The same vector field $X=x \partial_x+ y \partial_y$ on south pole chart(I checked that on overlap these define same vector fields) gives a globally defined vector field on sphere. I think this problem wants to ask how can $X$ be expressed in terms of $x,y,z$ in $\mathbb{R}^3$. (Since $x,y$ in $X$ is not the same $x,y$ in $\mathbb{R}^3$.)