Editing Laplace operator (section)

===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>
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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. 
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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.