Editing Laplace operator (section)

==Definition==
The Laplace operator is a [[Second-order differential equation|second-order differential operator]] in the ''n''-dimensional [[Euclidean space]], defined as the [[divergence]] (<math>\nabla \cdot</math>) of the [[gradient]] (<math>\nabla f</math>). Thus if <math>f</math> is a [[derivative|twice-differentiable]] [[real-valued function]], then the Laplacian of <math>f</math> is the real-valued function defined by:
{{NumBlk||<math display="block">\Delta f = \nabla^2 f = \nabla \cdot \nabla f </math>|{{EqRef|1}}}}
where the latter notations derive from formally writing:
<math display="block">\nabla  = \left ( \frac{\partial }{\partial x_1} , \ldots , \frac{\partial }{\partial x_n} \right ).</math>
Explicitly, the Laplacian of {{math|''f''}} is thus the sum of all the ''unmixed'' second [[partial derivative]]s in the [[Cartesian coordinates]] {{math|''x<sub>i</sub>''}}:
{{NumBlk||<math display="block">\Delta f = \sum_{i=1}^n \frac {\partial^2 f}{\partial x^2_i}</math>|{{EqRef|2}}}}

As a second-order differential operator, the Laplace operator maps {{math|[[Continuously differentiable|''C{{i sup|k}}'']]}} functions to {{math|''C''{{i sup|''k''−2}}}} functions for {{math|''k'' ≥ 2}}. It is a linear operator {{math|Δ : ''C''{{i sup|''k''}}('''R'''<sup>''n''</sup>) → ''C''{{i sup|''k''−2}}('''R'''<sup>''n''</sup>)}}, or more generally, an operator {{math|Δ : ''C''{{i sup|''k''}}(Ω) → ''C''{{i sup|''k''−2}}(Ω)}} for any [[open set]] {{math|Ω ⊆ '''R'''<sup>''n''</sup>}}.

Alternatively, the Laplace operator can be defined as:

<math display="block">\nabla^2 f(\vec{x}) = \lim_{R \rightarrow 0} \frac{2n}{R^2} (f_{shell_R} - f(\vec{x})) = \lim_{R \rightarrow 0} \frac{2n}{A_{n-1} R^{1+n}} \int_{shell_R} f(\vec{r}) - f(\vec{x}) d r^{n-1} </math>

where <math>n</math> is the dimension of the space, <math>f_{shell_R}   </math> is the average value of <math>f</math> on the surface of an [[n-sphere]] of radius <math>R</math>, <math>\int_{shell_R} f(\vec{r}) d r^{n-1}</math> is the surface integral over an [[n-sphere]] of radius <math>R</math>, and <math>A_{n-1}</math> is the [[Unit sphere#Volume and area|hypervolume of the boundary of a unit n-sphere]].<ref>{{Cite journal |last=Styer |first=Daniel F. |date=2015-12-01 |title=The geometrical significance of the Laplacian |url=https://pubs.aip.org/aapt/ajp/article-abstract/83/12/992/1057202/The-geometrical-significance-of-the-Laplacian?redirectedFrom=fulltext |journal=American Journal of Physics |volume=83 |issue=12 |pages=992–997 |doi=10.1119/1.4935133 |bibcode=2015AmJPh..83..992S |issn=0002-9505 |archive-url=https://www2.oberlin.edu/physics/dstyer/Electrodynamics/Laplacian.pdf |archive-date=20 November 2015}}</ref>