Editing Dirac delta function (section)

===Properties in ''n'' dimensions===
The delta distribution in an {{mvar|n}}-dimensional space satisfies the following scaling property instead,
<math display="block">\delta(\alpha\boldsymbol{x}) = |\alpha|^{-n}\delta(\boldsymbol{x}) ~,</math>
so that {{mvar|δ}} is a [[homogeneous function|homogeneous]] distribution of degree {{math|&minus;''n''}}.

Under any [[reflection (mathematics)|reflection]] or [[rotation (mathematics)|rotation]] {{mvar|ρ}}, the delta function is invariant,
<math display="block">\delta(\rho \boldsymbol{x}) = \delta(\boldsymbol{x})~.</math>

As in the one-variable case, it is possible to define the composition of {{mvar|δ}} with a [[Lipschitz function|bi-Lipschitz function]]<ref>Further refinement is possible, namely to [[submersion (mathematics)|submersions]], although these require a more involved change of variables formula.</ref> {{math|''g'': '''R'''<sup>''n''</sup> → '''R'''<sup>''n''</sup>}} uniquely so that the following holds
<math display="block">\int_{\R^n} \delta(g(\boldsymbol{x}))\, f(g(\boldsymbol{x}))\left|\det g'(\boldsymbol{x})\right| d\boldsymbol{x} = \int_{g(\R^n)} \delta(\boldsymbol{u}) f(\boldsymbol{u})\,d\boldsymbol{u}</math>
for all compactly supported functions {{mvar|f}}.

Using the [[coarea formula]] from [[geometric measure theory]], one can also define the composition of the delta function with a [[submersion (mathematics)|submersion]] from one Euclidean space to another one of different dimension; the result is a type of [[current (mathematics)|current]]. In the special case of a continuously differentiable function {{math|''g'' : '''R'''<sup>''n''</sup> → '''R'''}} such that the [[gradient]] of {{mvar|g}} is nowhere zero, the following identity holds{{sfn|Hörmander|1983|loc=§6.1}}
<math display="block">\int_{\R^n} f(\boldsymbol{x}) \, \delta(g(\boldsymbol{x})) \,d\boldsymbol{x} = \int_{g^{-1}(0)}\frac{f(\boldsymbol{x})}{|\boldsymbol{\nabla}g|}\,d\sigma(\boldsymbol{x}) </math>
where the integral on the right is over {{math|''g''<sup>−1</sup>(0)}}, the {{math|(''n'' − 1)}}-dimensional surface defined by {{math|1=''g''('''x''') = 0}} with respect to the [[Minkowski content]] measure.  This is known as a ''simple layer'' integral.

More generally, if {{mvar|S}} is a smooth hypersurface of {{math|'''R'''<sup>''n''</sup>}}, then we can associate to {{mvar|S}} the distribution that integrates any compactly supported smooth function {{mvar|g}} over {{mvar|S}}:
<math display="block">\delta_S[g] = \int_S g(\boldsymbol{s})\,d\sigma(\boldsymbol{s})</math>

where {{mvar|σ}} is the hypersurface measure associated to {{mvar|S}}.  This generalization is associated with the [[potential theory]] of [[simple layer potential]]s on {{mvar|S}}.  If {{mvar|D}} is a [[domain (mathematical analysis)|domain]] in {{math|'''R'''<sup>''n''</sup>}} with smooth boundary {{mvar|S}}, then {{math|''δ''<sub>''S''</sub>}} is equal to the [[normal derivative]] of the [[indicator function]] of {{mvar|D}} in the distribution sense,

<math display="block">-\int_{\R^n}g(\boldsymbol{x})\,\frac{\partial 1_D(\boldsymbol{x})}{\partial n}\,d\boldsymbol{x}=\int_S\,g(\boldsymbol{s})\, d\sigma(\boldsymbol{s}),</math>

where {{mvar|n}} is the outward normal.{{sfn|Lange|2012|loc=pp.29–30}}{{sfn|Gelfand|Shilov|1966–1968|p=212}} For a proof, see e.g. the article on the [[surface delta function]].

In three dimensions, the delta function is represented in spherical coordinates by:

<math display="block">\delta(\boldsymbol{r}-\boldsymbol{r}_0) = 
\begin{cases} 
    \displaystyle\frac{1}{r^2\sin\theta}\delta(r-r_0) \delta(\theta-\theta_0)\delta(\phi-\phi_0)& x_0,y_0,z_0 \ne 0 \\
    \displaystyle\frac{1}{2\pi r^2\sin\theta}\delta(r-r_0) \delta(\theta-\theta_0)& x_0=y_0=0,\ z_0 \ne 0 \\ 
    \displaystyle\frac{1}{4\pi r^2}\delta(r-r_0) & x_0=y_0=z_0 = 0  
\end{cases}</math>