There are apparently two parametrizations of this surface, one is simply using angles, another using the radius and an angle:
\begin{align}\sigma &\overset{\color{red}a}{=} \left(r\cos\theta, r\sin\theta, \sqrt{r^{\prime\,2}-r^2}\right)\\&\overset{\color{orange}b}{=} \left(r'\sin\theta\cos\varphi, r'\sin\theta\sin\varphi, \cos\varphi\right) \end{align} $$r' = \text{the radius of the sphere} \\r\in[0,r'],\quad\theta\in[0, \tau),\quad\varphi\in[0,\tau/4)$$
How can I show $\color{red}a$ and $\color{orange}b$ to be equivalent?
Currently they are not the same because there are some typo's. The first line characterises the surface of a half sphere of radius $r'$ and $z \geq 0$. Therefore it should read: $$ \sigma = (r \cos \theta,r \sin \theta,\sqrt{r'^2 - r^2}) = (r' \cos \theta \sin \phi,r' \sin \theta \sin \phi, r' \cos \phi) $$ Equivalence follows from realising $r = r' \sin \phi$ and hence $\sqrt{r'^2 - r^2} = r' \cos \phi$.