I have a question asking me to evaluate $\iint_\Sigma \mathbf{F} \cdot \mathbf{n}~dS$, where $\Sigma$ is the lower half of the ellipsoid $z = -2 \sqrt{1 - x^2 - y^2}$ with $\mathbf{n}$ directed upwards. However, I'm having trouble deriving the parametric equations for $\Sigma$. I know the general form of the parametric equations for an ellipsoid, but I'd like to be able to recover them given a temporary lapse of memory (say, on an exam).
If I attempt to derive an expression for $\rho$ in terms of $\phi$ and $\theta$ using spherical coordinates, I end up with a messy expression that doesn't conform to the expected answer. On the other hand, letting $x = a \sin \phi \cos \theta$, $y = b \sin \phi \sin \theta$ and $z = c \cos \phi$ and solving for $a$, $b$ and $c$ gives the expected answer. But this procedure doesn't seem correct, because these don't look like spherical coordinates. Perhaps someone can explain how to derive the parametric equations systematically?