Indicator function: Difference between revisions

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and similarly the distributional derivative of <math display="block">\ G(x) := \operatorname\mathbb{I}\!\bigl(\ x < 0\ \bigr)\ </math> is <math display=block>\ \frac{\ \mathrm{d}\ G(x)\ }{\ \mathrm{d}\ x\ } = -\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 inward [[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 \operatorname\mathbb{I}\!\bigl(\ \mathbf{x}\in D\ \bigr)\ </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>