Express the unit vector $\mathbf{e}_{r}$ of spherical polar coordinates in terms of the orthonormal Cartesian basis vectors $\mathbf{i}, \mathbf{j}, \mathbf{k}$.

Express the equation for the paraboloid $z=x^{2}+y^{2}$ in (i) cylindrical polar coordinates $(\rho, \phi, z)$ and (ii) spherical polar coordinates $(r, \theta, \phi)$.

In spherical polar coordinates, a surface is defined by $r^{2} \cos 2 \theta=a$, where $a$ is a real non-zero constant. Find the corresponding equation for this surface in Cartesian coordinates and sketch the surfaces in the two cases $a>0$ and $a<0$.