Prerequisites

Spheres in Cylindrical Coordinates

Surprisingly, it often turns out to be simpler to solve problems involving spheres by working in cylindrical coordinates. We indicate here one of the reasons for this.

The equation of a sphere of radius $a$ in cylindrical coordinates is 1) \begin{equation} r^2 + z^2 = a^2 \end{equation} so that \begin{equation} 2r\,dr + 2z\,dz = 0 \label{Spherecyl} \end{equation} Proceeding as for the paraboloid, we take \begin{eqnarray} d\rr_1 &=& r\,d\phi\,\phat \\ d\rr_2 &=& dr\,\rhat + dz\,\zhat  =  \left( -\frac{z}{r}\,\rhat + \zhat \right) dz \label{drdz} \end{eqnarray} where we now view $r$ as a function of $z$. We therefore have \begin{equation} d\SS = d\rr_1 \times d\rr_2 = (r\,\rhat + z\,\zhat) \,d\phi\,dz \end{equation} which results finally in \begin{equation} \dS = |d\SS| = a\,d\phi\,dz \end{equation} The surface element of a sphere is therefore the same as that of the cylinder of the same radius! Among other things, this means that projecting the Earth outward onto a cylinder is an equal-area projection, which is useful for cartographers.

1) Throughout this section, $r$ and $\rhat$ refer to the cylindrical radial coordinate.