1. ## Sphere/hemisphere

Sketch $\displaystyle \sqrt{13 - x^2 - y^2}$

Attempt:

$\displaystyle z^2 = 13 - x^2 - y^2$

$\displaystyle z^2 = -x^2 - y^2 + 13$

$\displaystyle x^2 + y^2 + z^2 = 13$

Now I dont understand why this is half of a sphere (which opens up downwards) and not a whole sphere?

2. If this was $\displaystyle \displaystyle z = \sqrt{13 - x^2 - y^2}$, this is really $\displaystyle \displaystyle z = +\sqrt{13 - x^2- y^2}$, which is the upper half of the sphere.

To be a full sphere it would need to be $\displaystyle \displaystyle z = \pm \sqrt{13 - x^2 - y^2}$.