Generally for orthogonal coordinates the Laplace operator can be expressed:

Substituting in a few steps the actual scalar coefficients we get for n-dimensional spherical coordinates:

By using (1.19) and (1.20)

further

and get finally using (1.21) the Laplace operator in n dimensions:

What we need is to find out is all the h:as in the Laplace operator (2.4). Let us denote the purely angular part, the part that only depends on the angular coordinates:

Now we can write the Laplace-operator in n-dimensional spherical coordinates in the following form:

We define below a recursive angular differential operator. This operator has the property that it only includes angular coordinates for which . We can assume recursively that the angular coordinates for which already have been separated. We then derive a recursion formula that makes the separation of the angular coordinates possible in a logical sequence, starting from a known case with lower number of dimensions (say, from the three-dimensional spherical coordinates).

and from (1.16)

We have in effect defined two sets of recursive scalar coefficients:

A couple of results, to be used below

Define an angular differential operator of m angular variables:

To get the recursive formula, rewrite the sum-expression in a few steps:

using the result (2.15)

In the last term in (2.20) we can identify the angular operator, with dimensions one less than above

Using result (2.16) above for the recursive scalar coefficients, we finally come to the recursive formula for the angular differential operator:

We also know (from the two- or three-dimensional case) for the first angle coordinate (azimuth):

The total Lapalace operator for the n-dimensional case was, from formula (2.6)

Example: Spherical Laplace Operator in 4 dimensions