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{{Main|Laplacian of the indicator}}
A particular indicator function is the [[Heaviside step function]]
<math display="block">H(x) := \mathbf{1}_{x > 0}</math>
The [[distributional derivative]] of the Heaviside step function is equal to the [[Dirac delta function]], i.e.
▲<math display=block>\tfrac{d H(x)}{dx}=\delta(x)</math>
and similarly the distributional derivative of <math display="block">G(x) := \mathbf{1}_{x < 0}</math> is
Thus the derivative of the Heaviside step function can be seen as the ''inward normal derivative'' at the ''boundary'' of the ___domain given by the positive half-line. In higher dimensions, the derivative naturally generalises to the inward normal derivative, while the Heaviside step function naturally generalises to the indicator function of some ___domain {{mvar|D}}. The surface of {{mvar|D}} will be denoted by {{mvar|S}}. Proceeding, it can be derived that the [[Laplacian of the indicator#Dirac surface delta function|inward normal derivative of the indicator]] gives rise to a 'surface delta function', which can be indicated by <math>\delta_S(\mathbf{x})
▲<math display=block>\tfrac{d G(x)}{dx}=-\delta(x)</math>
▲Thus the derivative of the Heaviside step function can be seen as the ''inward normal derivative'' at the ''boundary'' of the ___domain given by the positive half-line. In higher dimensions, the derivative naturally generalises to the inward normal derivative, while the Heaviside step function naturally generalises to the indicator function of some ___domain {{mvar|D}}. The surface of {{mvar|D}} will be denoted by {{mvar|S}}. Proceeding, it can be derived that the [[Laplacian of the indicator#Dirac surface delta function|inward normal derivative of the indicator]] gives rise to a 'surface delta function', which can be indicated by <math>\delta_S(\mathbf{x}):</math>
<math display=block>\delta_S(\mathbf{x}) = -\mathbf{n}_x \cdot \nabla_x\mathbf{1}_{\mathbf{x}\in D}</math>
where {{mvar|n}} is the outward [[Normal (geometry)|normal]] of the surface {{mvar|S}}. This 'surface delta function' has the following property:<ref>{{cite journal |last=Lange |first=Rutger-Jan |year=2012 |title=Potential theory, path integrals and the Laplacian of the indicator |journal=Journal of High Energy Physics |volume=2012 |issue=11 |pages=29–30 |arxiv=1302.0864 |bibcode=2012JHEP...11..032L |doi=10.1007/JHEP11(2012)032|s2cid=56188533 }}</ref>
<math display=block>-\int_{\R^n}f(\mathbf{x})\,\mathbf{n}_x\cdot\nabla_x\mathbf{1}_{\mathbf{x}\in D}\;d^{n}\mathbf{x} = \oint_{S}\,f(\mathbf{\beta})\;d^{n-1}\mathbf{\beta}.</math>
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