Find a parametrization for a sphere $S$ of radius $2$ centered at the origin.

Solution The equation of the sphere, $x^{2}+y^{2}+z^{2}=4,$ in rectangular coordinates cannot be expressed explicitly as a function of two of its variables. However, in spherical coordinates, the sphere has the explicit equation $\rho =2.$ So if we use the spherical coordinates $\theta$ and $\phi$ as parameters, then parametric equations of the sphere are given by

and a parametrization of $S$ is $$\begin{equation*} \mathbf{r}\left( \theta ,\phi \right) =2\cos \theta \sin \phi \,\mathbf{i} +2\sin \theta \sin \phi \,\mathbf{j}+2\cos \phi \,\mathbf{k} \end{equation*}$$

where $0\leq \theta \leq 2\pi$ and $0\leq \phi \leq \pi .$