Coordinates notation in Spherical polar system

Question

There is a number of conventions for specifying coordinates in Spherical polar coordinate system: ($r$, $θ$, $φ$), ($r$, $φ$, $θ$), ($\rho$, $\theta$, $\phi$) and even ($r$, $\psi$, $θ$).

The article in citizendium states:

The notational convention introduced above (θ for the colatitude angle, φ for the azimuth angle) is used universally in physics. It is in accordance with advice of the International Standards Organization (ISO 31-11, which, however, advises the use of ρ instead of r ). In mathematics—especially in the older and the European literature—the convention is very widespread, too. To quote a few prestiguous mathematical books that apply it: Abramowitz and Stegun[1] (p. 332), Whittaker and Watson[2] (p. 391), Courant and Hilbert[3] (p.195), and Kline[4] (p. 527). Until the 1960s this convention was used universally, also in mathematical textbooks, see e.g. the 1959 edition of Spiegel[5] (p. 138).

Somewhere in the 1960s it became custom in American mathematical textbooks to use a notation in which φ and θ are interchanged, see e.g. Kay[6] (p. 24) and Apostol[7] (p. 419). This was done in order to not confuse students by changing the meaning of the Greek letter θ in the transition from 2D to 3D polar coordinates...

The article at mathworld.wolfram.com summarizes a number of conventions used by various authors.

The questions are:

1. What notation convention is more "standard" for physics and especially photometry and computer graphics?
2. Can I use ($r$, $θ$, $φ$) symbols to specify radial, azimuthal and polar coordinates?

Answer 1

In physics we usually use r to specify the radial component in spherical coordinates. We leave $\rho$ for cylindrical coordinates. How the two swapped, I don't know, but I feel comfortable using this formalism. :)

Now for the azimuthal angle. Typically in physics we use $\phi$ for the azimuthal angle. This is both in cylindrical and spherical coordinates. This allows an easy conversion from x and y to $\phi$ in both systems. I haven't looked at Arfken Weber lately, but from Jackson as well as Griffiths (which were burned into my brain my first year of grad school), I do feel that what I've stated is the norm.

To answer your second question more clearly, you may use your arrangement, but if I graded your homework and didn't realize clearly you used this style, it would take me much longer to grade your homework and I'd be confused for a bit. In physics, consistency is the key. This actually reminds me of a story about Richard Feynman, where he came up with his own style of writing certain trig functions. His friend said if he continued doing this, no one would understand what he was doing and to keep things consistent. This is true until he invented Feynman diagrams :)

I cannot state anything about computer graphics' style of spherical coordinates nor photometry.

Also what @Christian Blatter stated about the 0 of the polar angle at the north pole or on the equator is correct. This is typically at the north pole but can be rotated depending on the symmetry of the system.

Answer 2

This might be useful http://theoretical-physics.net/0.1/src/math/other.html

As Riccardo suggested as long as someone has consistent notation, a technical paper (with such notations) is not confusing to read.

Answer 3

The two important things with notations in 3D co ord systems are(a)a good diagram showing clearly what the various symbols stand for and obviously consistency of notations (in dissertations,papers,and classroom discussions). I have grown up with (r,theta) for plane polar,(rho,phi,z)fr cylindrical ,and (r,theta,phi) for spherical polar system. There are too many good books on the subject..admittedly Arfken and Weber is pretty good but I always recommend Thomas' Calculus for serious students.It has a mature discussion devoted to these co ord systems..

Answer 4

Usually you can use the notation $(\rho,\theta,\phi)$, but any notation is accepted. $(r,\theta,\phi)$ is ok. You can use whatever you want; the only problem is that when you choose a notation you must use it without change it in the rest of the paper.