Crthuky

Lets take for example the Laplacian. So I want to apply it in spherical coordinates, so I go the the associated documentation page
(https://reference.wolfram.com/language/ref/Laplacian.html?view=all)

Luckily, there is an example Laplacian[1, 1, 1, r, [Theta], [Phi], "Spherical"] // Expand. Yet still, I do not know whether [Theta] is the polar or azimuthal angle.

As far as I can tell nothing in the docs tells you the order of arguments. Is it radius, azimuth, polar angle or is it radius, azimuth, polar angle?

Anyway, I tried

??Laplacian
??"Spherical"

to no avail.

So my question is where do I find the order of arguments of default functions like this? (If not in the documentation).

I can't keep coming to stack exchange for every single function I use.
And trying all the permutations of the arguments until it works is rather tiring.

Is there a more in depth doc than the one I linked to? Also, what is the correct order of arguments in this case.

The "Details" section of that page refers to CoordinateChartData. Now this is a bit dense, but it contains everything you need. First of all, you can try to find out what kind of things you can find out about spherical coordinates:

In[9]:= CoordinateChartData["Spherical", "Properties"]

Out[9]= "AlternateCoordinateNames", "CoordinateRangeAssumptions",
"Dimension", "InverseMetric", "Metric", "ParameterRangeAssumptions",
"ScaleFactors", "StandardCoordinateNames", "StandardName",
"VolumeFactor"

Many functions in Mathematica have a "Properties" property that allows you to figure out what you can ask for. It's useful to keep that in mind.

Let's first find out what the standard names are for the coordinates:

In[10]:= CoordinateChartData["Spherical", "StandardCoordinateNames"]

Out[10]= "r", "θ", "φ"

There is also the "CoordinateRangeAssumptions" property which gives you the constraints on a given set of parameters, so let's use the parameter names we just got:

In[11]:= CoordinateChartData["Spherical", "CoordinateRangeAssumptions", %]

Out[11]= "r" > 0 && 0 < "θ" < π && -π < "φ" <= π

Now you know exactly which angle is which, since the polar angle is the one that ranges from 0 to π.

Another suggestion is to look at the references on the documentation page of Laplacian. For example, there is a linked tutorial about vector analysis which also mentions CoordinateChartData.

Alternatively, sometimes you just need to click around a bit among functions and symbols that seem related to what you need to know. For example, the linked guide about vector analysis lists the function ToSphericalCoordinates which has a helpful graphic in the Details section. Guides are quite useful for finding your way around since they tend to group functions and symbols by theme or application.