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