# The Spherical Coordinate Convention

July 8, 2024

It has come to my attention (actually I think I noticed this a long time ago and then forgot…) that there is a correct answer as to the convention used for spherical coordinates.

The options are:

- The physicist’s convention, which has \(\theta\) as the polar (zenithal) angle, that is, the angle off of the north pole, which ranges in \((0, \pi)\), and \(\phi\) as the equatorial (azimuthal) angle, the angle off of a designated line of 0 longitude which is usually the \(+x\) axis, and which ranges in \((0, 2 \pi)\)
- The mathematician’s convention, which swaps them, and has \(\phi\) as the polar angle and \(\theta\) as the equatorial angle.

It turns out that there is a right answer. The mathematicians are right (for once).

The reason is simple. Just look at them:

\(\LARGE{\theta \, \, \, \phi}\)

They’re fricking *pictures* of which angle they are.

That’s right. I do not care that physics has always been doing it the other way, or even that all the reference books about spherical harmonics disagree (what a weird objection). There is only one right answer and it will be easy for everyone to remember. Please update your textbooks.