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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 $\mathbb{R}^3$.)

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    You've defined $X$ at every point of $\mathbb{S}^2$ except the north pole. What value does $X$ need to take in $T_{\mbox{north pole}}\mathbb{S}^2$ so that it is smooth there?2012-12-12
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    $\mathbb{R}^3$ is completely redundant here.2012-12-12
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    I made some edits to clarify things.2012-12-12
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    Dear Gobi,if I correctly interpret what you say, there is a change of sign in the expression of the field in the south pole chart: see my answer.2012-12-23

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