The differential of inclusion from S^{2} to R^{3}
Here $z\circ i \circ \phi^{-1}=\sqrt{1-u^2-v^2}$ and so
$\frac{\delta}{\delta u} \sqrt{1-u^2-v^2}=$
$=\frac{-u}{\sqrt{1-u^2-v^2}}=-\frac{x}{z}$
so that
$i_*(\frac{\delta}{\delta u })=\frac{\delta}{\delta x}-\frac{x}{z}\frac{\delta}{\delta z}$
Similarly
$i_*(\frac{\delta}{\delta v })=\frac{\delta}{\delta y}-\frac{y}{z}\frac{\delta}{\delta z}$
Merry Christmas 🎄