Nearly everybody uses $r$ and $\theta$ to denote polar coordinates. Most American calculus texts also utilize $\theta$ in spherical coordinates for the angle in the equatorial plane (the azimuth or longitude), $\phi$ for the angle from the positive $z$-axis (the zenith or colatitude), and $\rho$ for the radial coordinate. Virtually all other scientists and engineers — as well as mathematicians in many other countries — reverse the roles of $\theta$ and $\phi$ (and use some other letter, such as $R$, for the radial coordinate).

Why is this a problem? After all, the change in notation only affects students in particular fields, such as physics or electrical engineering. Furthermore, it's just a convention; surely these students have the maturity to deal with it. Based on our experience trying to implement this change during a second-year course in multivariable calculus, we feel that such sentiments underestimate the extent of the problem. Students find the complete interchange of the roles of $\theta$ and $\phi$ to be terribly confusing — and once confused, always confused.

Using different names for the radial coordinate, on the other hand, causes few problems. The use of $r$ for the spherical radial coordinate can be confused with the radial coordinate in polar or cylindrical coordinates, but computations requiring both at the same time are rare. While $\rho$ is not available to the physicist, as it is used to represent charge or mass density, students do not appear to be confused by the use of several different names for the spherical radial coordinate.

There is however a much more serious problem. Several of the most commonly used calculus texts list spherical coordinates in the order ($\rho$, $\theta$, $\phi$); the rest use ($\rho$, $\phi$, $\theta$). The first of these is left-handed! An orthogonal coordinate system is right-handed if the cross product of the first two coordinate directions points in the third coordinate direction. This is immaterial in the traditional mathematics treatment of vector calculus, but crucial to the way physicists and engineers treat the same material. These scientists often introduce basis vectors in the coordinate directions, analogous to $\{\ii,\jj,\kk\}$ for rectangular coordinates, and it is essential that these vectors form a right-handed system. This requires that the zenith be listed before the azimuth; with the standard mathematics convention, this is ($\rho$, $\phi$, $\theta$). Books which use the standard mathematics definitions of the angles but write ($\rho$, $\theta$, $\phi$) are doing their students a major disservice, although we reiterate that this is only an issue for material covered in subsequent courses.

Figure 7.1a: The definition of cylindrical coordinates.

There is a uniform standard for the use of spherical coordinates in applications, which is nowhere more apparent than in the definition of spherical harmonics. These special functions on the sphere are widely used, notably in the quantum mechanical description of electron orbitals, which in turn underlies much of chemistry. It can not be stated too strongly that everyone writes the spherical harmonics as $Y_{\ell m}(\theta,\phi)$, where $\theta$ is the zenith and $\phi$ the azimuth. There is simply no way to change this convention, which is embedded in generations of standard reference books.

One objection to this is that it is confusing to use the same label, $\theta$, for two different angles in polar and spherical coordinates. This objection can be easily resolved, even if the resolution may not be popular: Change the conventions for polar coordinates, that is, use $\phi$ rather than $\theta$.

We propose that these conventions be adopted by mathematicians, and we use them throughout these materials. Our conventions for both spherical and cylindrical coordinates are shown in Figure 7.1.

Figure 7.1b: The definition of spherical coordinates.