We get:
$x = \sqrt{1 - y^2 - z^2}$
Now, we can define the surface patch $\sigma_1(u,v) = (\sqrt{1 - u^2 - v^2}, u, v)$, defined on the open set $U = \{(u,v) \in \mathbb{R}^2 \mid u^2 + v^2 < 1\}$.
Similarly, we can solve the equation for $y$ and $z$ in terms of
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