Editing Laplace operator (section)

=== {{mvar|N}} dimensions ===
In arbitrary [[curvilinear coordinates]] in {{math|''N''}} dimensions ({{math|''ξ''<sup>1</sup>, ..., ''ξ<sup>N</sup>''}}), we can write the Laplacian in terms of the inverse [[metric tensor]], <math> g^{ij} </math>:
<math display="block">\Delta = \frac 1{\sqrt{\det g}}\frac{\partial}{\partial\xi^i} \left( \sqrt{\det g} \,g^{ij}  \frac{\partial}{\partial \xi^j}\right) ,</math>
from the [https://www.genealogy.math.ndsu.nodak.edu/id.php?id=59087 Voss]-[[Hermann Weyl|Weyl]] formula<ref>Archived at [https://ghostarchive.org/varchive/youtube/20211211/BD2AiFk651E Ghostarchive]{{cbignore}} and the [https://web.archive.org/web/20190220065415/https://www.youtube.com/watch?v=BD2AiFk651E&gl=US&hl=en Wayback Machine]{{cbignore}}: {{cite web | last1=Grinfeld | first1=Pavel | title=The Voss-Weyl Formula | website=[[YouTube]] | date=16 April 2014 | url=https://www.youtube.com/watch?v=BD2AiFk651E&list=PLlXfTHzgMRULkodlIEqfgTS-H1AY_bNtq&index=23 | access-date=9 January 2018 | language=en}}{{cbignore}}</ref> for the [[Divergence#General coordinates|divergence]].

In '''spherical coordinates in {{mvar|N}} dimensions''', with the parametrization {{math|1=''x'' = ''rθ'' ∈ '''R'''<sup>''N''</sup>}} with {{mvar|r}} representing a positive real radius and {{mvar|θ}} an element of the [[unit sphere]] {{math|[[N sphere|''S''<sup>''N''−1</sup>]]}},
<math display="block"> \Delta f = \frac{\partial^2 f}{\partial r^2} + \frac{N-1}{r} \frac{\partial f}{\partial r} + \frac{1}{r^2} \Delta_{S^{N-1}} f</math>
where {{math|Δ<sub>''S''<sup>''N''−1</sup></sub>}} is the [[Laplace–Beltrami operator]] on the {{math|(''N'' − 1)}}-sphere, known as the spherical Laplacian. The two radial derivative terms can be equivalently rewritten as:
<math display="block">\frac{1}{r^{N-1}} \frac{\partial}{\partial r} \left(r^{N-1} \frac{\partial f}{\partial r} \right).</math>

As a consequence, the spherical Laplacian of a function defined on {{math|''S''<sup>''N''−1</sup> ⊂ '''R'''<sup>''N''</sup>}} can be computed as the ordinary Laplacian of the function extended to {{math|'''R'''<sup>''N''</sup>∖{0}<nowiki/>}} so that it is constant along rays, i.e., [[homogeneous function|homogeneous]] of degree zero.