# Does Laplacian use first argument “theta” for polar angle or azimuthal angle?

Only example I could find in the help for Laplacian is this My question is, is $$\theta$$ meant to be the polar or the azimuthal angle? Using $$\theta$$ for polar is the Physics convention and is the more common one, but the other convention used in Mathworld is to use $$\theta$$ for azimuthal which is different from what Wikipedia uses here Of course, you could say, try both cases, and see which is which. I did, but Mathematica writes the Lapacian in complicated form, my eyes are glazed trying to decide which one it is supposed to be, and could not simplify to make it easier to read

 Laplacian[u[r, theta, phi], {r, theta, phi}, "Spherical"] Starring at the above for sometime, and comparing it to Wikipedia It seems Mathematica is using the same convention as Wikipedia and not as Mathworld. So "theta" is the polar angle.

My question is simple: Is the above correct? I just need confirmation.

It would be nice if Mathematica had this documented somewhere to reduce error.

You can use CoordinateChartData to get the expected standard names:

coords = CoordinateChartData["Spherical", "StandardCoordinateNames"]


{"r", "θ", "φ"}

and the corresponding ranges:

CoordinateChartData["Spherical","CoordinateRangeAssumptions"] @ coords


"r" > 0 && 0 < "θ" < π && -π < "φ" <= π

and the volume factor:

CoordinateChartData["Spherical", "VolumeFactor"] @ coords


"r"^2 Sin["θ"]

So, θ is the polar angle and φ is the azimuthal angle.

Carl's answer is very good. I'm offering this alternate answer for completeness. The documentation for CoordinateChartData contains a brief description of the different vcoordinates in the "Details" section. For example, it has this description for "Spherical":

spherical coordinates with poles along the $$z$$ axis and coordinates in the order radius, polar angle, azimuthal angle

For many coordinate systems, the ranges and/or "standard" names don't uniquely identify them so these descriptions are useful.

I'd also point out that we have to use this order for the coordinate system to be right-handed (unless you reverse the direction of the polar angle), which is pretty important if you're computing curls...