## Thomas' Calculus 13th Edition

$z=\sqrt{1-x^{2}-y^{2}}$
The whole sphere:$\quad x^{2}+y^{2}+z^{2}=1$ "Upper hemisphere" restricts the z-coordinates to $z\geq 0.$ Combined, $x^{2}+y^{2}+z^{2}=1,\quad z\geq 0.$ Express z in terms of x and y $z^{2}=1-x^{2}-y^{2},\quad z\geq 0.$ Take the square root (result is nonnegative, so we drop the condition on z) $z=\sqrt{1-x^{2}-y^{2}}$