I know several parametrization of the sphere as $$(x,y,z)=(r\cos\theta \sin\varphi,r\sin\theta \sin\varphi, r\cos \varphi)$$
$$(x,y,z)=(r\sin\varphi, r\cos\theta \cos\varphi, r\cos \varphi\sin\theta )$$
where $\varphi\in [0,\pi]$ and $\theta \in [0,2\pi]$. The question is : does the jacobian (i.e. the determinant of the jacobian matrix) will depend on the parametrization or not ? i.e. will it be $r\sin^2 \varphi$ always or not ?