Editing Laplace operator (section)

== Coordinate expressions ==

=== Two dimensions ===
The Laplace operator in two dimensions is given by:

In '''[[Cartesian coordinates]]''',
<math display="block">\Delta f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2}</math>
where {{mvar|x}} and {{mvar|y}} are the standard [[Cartesian coordinates]] of the {{math|''xy''}}-plane.

In '''[[polar coordinates]]''',
<math display="block">\begin{align}
\Delta f &= \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial f}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2 f}{\partial \theta^2} \\
&= \frac{\partial^2 f}{\partial r^2} + \frac{1}{r} \frac{\partial f}{\partial r} + \frac{1}{r^2} \frac{\partial^2 f}{\partial \theta^2},
\end{align}</math>
where {{mvar|r}} represents the radial distance and {{mvar|θ}} the angle.

===Three dimensions===
{{See also|Del in cylindrical and spherical coordinates}}
In three dimensions, it is common to work with the Laplacian in a variety of different coordinate systems.

In '''[[Cartesian coordinates]]''',
<math display="block">\Delta f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}.</math>

In '''[[cylindrical coordinates]]''',
<math display="block">\Delta f = \frac{1}{\rho} \frac{\partial}{\partial \rho} \left(\rho \frac{\partial f}{\partial \rho} \right) + \frac{1}{\rho^2} \frac{\partial^2 f}{\partial \varphi^2} + \frac{\partial^2 f}{\partial z^2 },</math>
where <math>\rho</math> represents the radial distance, {{math|''φ''}} the azimuth angle and {{math|''z''}} the height.

In '''[[spherical coordinates]]''':
<math display="block">\Delta f = \frac{1}{r^2} \frac{\partial}{\partial r} \left(r^2 \frac{\partial f}{\partial r} \right) + \frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta} \left(\sin \theta \frac{\partial f}{\partial \theta} \right) + \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 f}{\partial \varphi^2},</math>
or 
<math display="block">\Delta f = \frac{1}{r} \frac{\partial^2}{\partial r^2} (r f) + \frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta} \left(\sin \theta \frac{\partial f}{\partial \theta} \right) + \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 f}{\partial \varphi^2},</math>
by expanding the first and second term, these expressions read
<math display="block">\Delta f = \frac{\partial^2 f}{\partial r^2} + \frac{2}{r}\frac{\partial f}{\partial r}+\frac{1}{r^2 \sin \theta} \left(\cos \theta \frac{\partial f}{\partial \theta} + \sin \theta \frac{\partial^2 f}{\partial \theta^2} \right) + \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 f}{\partial \varphi^2},</math>
<!---**********PLEASE SEE THE DISCUSSION PAGE BEFORE CHANGING THIS.**********-->
where {{math|''φ''}} represents the [[azimuthal angle]] and {{math|''θ''}} the [[zenith angle]] or [[colatitude|co-latitude]]. In particular, the above is equivalent to

<math>\Delta f = \frac{\partial^2 f}{\partial r^2} + \frac{2}{r}\frac{\partial f}{\partial r} + \frac{1}{r^2}\Delta_{S^2} f ,</math>

where <math>\Delta_{S^2}f</math> is the [[Laplace–Beltrami operator|Laplace-Beltrami operator]] on the unit sphere. 
<!---**************************************************************-->

In general '''[[curvilinear coordinates]]''' ({{math|''ξ''<sup>1</sup>, ''ξ''<sup>2</sup>, ''ξ''<sup>3</sup>}}):
<math display="block">\Delta = \nabla \xi^m \cdot \nabla \xi^n \frac{\partial^2}{\partial \xi^m \, \partial \xi^n} + \nabla^2 \xi^m \frac{\partial}{\partial \xi^m } = g^{mn} \left(\frac{\partial^2}{\partial\xi^m \, \partial\xi^n} - \Gamma^{l}_{mn}\frac{\partial}{\partial\xi^l} \right),</math>

where [[Einstein summation convention|summation over the repeated indices is implied]],
{{math|''g<sup>mn</sup>''}} is the inverse [[metric tensor]] and  {{math|Γ''<sup>l</sup> <sub>mn</sub>''}} are the [[Christoffel symbols]] for the selected coordinates.

=== {{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.