Laplacian of the indicator

{{Short description|Limit of sequence of smooth functions}} [[File:Laplacian of the indicator v2.jpg|thumb|upright=1.5|right|An approximation of the negative indicator function of an ellipse in the plane (left), the derivative in the direction normal to the boundary (middle), and its Laplacian (right). In the limit, the right-most graph goes to the (negative) Laplacian of the indicator. Purely intuitively speaking, the right-most graph resembles an elliptic castle with a castle wall on the inside and a moat in front of it; in the limit, the wall and moat become infinitely high and deep (and narrow).]]

In [[potential theory]] (a branch of [[mathematics]]), the '''Laplacian of the indicator''' is obtained by letting the [[Laplace operator]] work on the [[indicator function]] of some [[Domain (mathematical analysis)|domain]] ''D''. It is a generalisation of the [[derivative (mathematics)|derivative]] (or "prime function") of the [[Dirac delta function]] to [[higher dimension]]s; it is non-zero only on the [[surface (mathematics)|surface]] of ''D''. It can be viewed as a ''surface delta prime function'', the derivative of a ''surface delta function'' (a generalization of the Dirac delta). The Laplacian of the indicator is also analogous to the [[second derivative]] of the [[Heaviside step function]] in one dimension.

The Laplacian of the indicator can be thought of as having infinitely positive and negative values when evaluated very near the boundary of the domain ''D''. Therefore, it is not strictly a [[function (mathematics)|function]] but a [[generalized function]] or [[measure (mathematics)|measure]]. Similarly to the derivative of the Dirac delta function in one dimension, the Laplacian of the indicator only makes sense as a [[mathematical object]] when it appears under an integral sign; i.e. it is a [[Distribution (mathematics)|distribution]] function. Just as in the formulation of distribution theory, it is in practice regarded as a [[limit of a sequence]] of [[smooth function]]s; one may meaningfully take the Laplacian of a [[bump function]], which is smooth by definition, and let the bump function approach the indicator in the [[limit (mathematics)|limit]].

==History== [[Paul Dirac]] introduced the [[Dirac delta function|Dirac {{mvar|δ}}-function]], as it has become known, as early as 1930.<ref>{{citation |last=Dirac|first=Paul | author-link=Paul Dirac |year=1958|title=Principles of quantum mechanics|edition=4th|publisher=Oxford at the Clarendon Press|isbn=((978-0-19-852011-5))}}</ref> The one-dimensional Dirac {{mvar|δ}}-function is non-zero only at a single point. Likewise, the multidimensional generalisation, as it is usually made, is non-zero only at a single point. In [[Cartesian coordinate system|Cartesian coordinates]], the ''d''-dimensional Dirac {{mvar|δ}}-function is a product of ''d'' one-dimensional {{mvar|δ}}-functions; one for each Cartesian coordinate (see e.g. [[Dirac delta function#Generalizations|generalizations of the Dirac delta function]]).

==Surface delta function== A generalisation of the Dirac delta is possible beyond a single point. The point zero, in one dimension, can be considered as the [[boundary (mathematics)|boundary]] of the positive [[halfline]]. The function '''1'''''x''>0 equals 1 on the positive halfline and zero otherwise, and is also known as the [[Heaviside step function]]. Formally, the Dirac {{mvar|δ}}-function and its derivative can be viewed as the first and second derivative of the Heaviside step function, i.e. [[∂]]''x'''''1'''''x''>0 and <math>\partial_x^2 \mathbf{1}_{x>0}</math>.

The analogue of the step function in higher dimensions is the [[indicator function]], which can be written as '''1'''''x''∈''D'', where ''D'' is some domain. The indicator function is also known as the characteristic function. In analogy with the one-dimensional case, the following higher-dimensional generalisations of the Dirac {{mvar|δ}}-function and its derivative have been proposed:<ref name="Lange 2012">{{citation|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 |pages=1–49 | issue=11 |article-number=32 |bibcode=2012JHEP...11..032L |doi=10.1007/JHEP11(2012)032 |arxiv = 1302.0864 |s2cid=56188533}}</ref>

:<math>\begin{align} \delta(x) &\to -n_x\cdot\nabla_x\mathbf{1}_{x\in D}, \\ \delta'(x) &\to \nabla_x^2 \mathbf{1}_{x\in D}. \end{align}</math> Here ''n'' is the outward [[normal (geometry)|normal vector]]. Here the Dirac {{mvar|δ}}-function is generalised to a '''surface delta function''' on the boundary of some domain ''D'' in ''d'' ≥ 1 dimensions. This definition gives the usual one-dimensional case, when the domain is taken to be the positive halfline. It is zero except on the boundary of the domain ''D'' (where it is infinite), and it integrates to the total [[surface area]] enclosing ''D'', as shown [[#Normal derivative of the indicator|below]].

==Surface delta prime function== The one-dimensional Dirac delta prime function is generalised to a multidimensional ''surface delta prime function'' on the boundary of some domain ''D'' in ''d'' ≥ 1 dimensions. In one dimension and by taking ''D'' equal to the positive halfline, the usual one-dimensional {{mvar|δ'}}-function can be recovered.

Both the normal derivative of the indicator and the Laplacian of the indicator are supported by ''surfaces'' rather than ''points''. The generalisation is useful in e.g. quantum mechanics, as surface interactions can lead to [[boundary conditions]] in ''d > ''1, while point interactions cannot. Naturally, point and surface interactions coincide for ''d''=1. Both surface and point interactions have a long history in quantum mechanics, and there exists a sizeable literature on so-called surface delta potentials or delta-sphere interactions.<ref>{{citation|last1=Antoine|first1=J.P.|last2=Gesztesy|first2=F.|last3=Shabani|first3=J.|title=Exactly solvable models of sphere interactions in quantum mechanics|journal=Journal of Physics A: Mathematical and General |volume=20|number=12 |pages=3687–3712 |year=1999 |doi=10.1088/0305-4470/20/12/022|bibcode = 1987JPhA...20.3687A }}</ref> Surface delta functions use the one-dimensional Dirac {{mvar|δ}}-function, but as a function of the radial coordinate ''r'', e.g. δ(''r''−''R'') where ''R'' is the radius of the sphere.

Although seemingly ill-defined, derivatives of the indicator function can formally be defined using the [[Distribution (mathematics)|theory of distributions]] or [[generalized function]]s: one can obtain a well-defined prescription by postulating that the Laplacian of the indicator, for example, is defined by two [[integration by parts|integrations by parts]] when it appears under an integral sign. Alternatively, the indicator (and its derivatives) can be approximated using a [[bump function]] (and its derivatives). The limit, where the (smooth) bump function approaches the indicator function, must then be put outside of the integral.

==Proofs==

===Proof of the surface delta prime function===

This section will prove that the Laplacian of the indicator is a ''surface delta prime function''. The ''surface delta function'' will be considered below.

First, for a function ''f'' in the interval (''a'',''b''), recall the [[fundamental theorem of calculus]]

:<math> \int_a^b \frac{\partial f(x)}{\partial x}\,dx=\underset{x \nearrow b}\lim f(x)-\underset{x \searrow a}\lim f(x),</math>

assuming that ''f'' is locally integrable. Now for ''a'' < ''b'' it follows, by proceeding heuristically, that

:<math>\begin{align} \int_{-\infty}^{+\infty} \frac{\partial^2\mathbf{1}_{a<x<b}}{\partial x^2}\,f(x)\;dx&=\int_{-\infty}^{+\infty} \mathbf{1}_{a<x<b} \frac{\partial^2 f(x)}{\partial x^2}\;dx, \\ &=\displaystyle\int_a^b \frac{\partial^2 f(x)}{\partial x^2}\;dx, \\ &=\displaystyle\Big(\underset{ x \nearrow b}\lim -\underset{ x \searrow a}\lim\Big) \frac{\partial f(x)}{\partial x}. \end{align}</math>

Here '''1'''''a''<''x''<''b'' is the [[indicator function]] of the domain ''a'' < ''x'' < ''b''. The indicator equals one when the condition in its subscript is satisfied, and zero otherwise. In this calculation, two [[integration by parts|integrations by parts]] (combined with the fundamental theorem of calculus as shown above) show that the first equality holds; the boundary terms are zero when ''a'' and ''b'' are finite, or when ''f'' vanishes at infinity. The last equality shows a ''sum'' of outward normal derivatives, where the sum is over the boundary points ''a'' and ''b'', and where the signs follow from the outward direction (i.e. positive for ''b'' and negative for ''a''). Although derivatives of the indicator do not formally exist, following the usual rules of partial integration provides the 'correct' result. When considering a finite ''d''-dimensional domain ''D'', the sum over outward normal derivatives is expected to become an ''integral'', which can be confirmed as follows:

:<math> \begin{align} \int _{\mathbf{R}^d}\nabla_x^2\mathbf{1}_{x\in D}\,f(x)\;dx&= \int _{\mathbf{R}^d}\mathbf{1}_{x\in D}\,\nabla_x^2 f(x)\;dx,\\ &= \int _{D}\,\nabla_x^2 f(x)\;dx,\\ &= \oint_{\partial D}\,\underset{x \to \beta}\lim n_\beta \cdot \nabla_x f(x)\;d\beta. \end{align} </math>

where the limit is of x approaching surface β from inside domain ''D'', nβ is the [[unit vector]] normal to surface β, and ∇''x'' is now the multidimensional gradient operator. As before, the first equality follows by two integrations by parts (in higher dimensions this proceeds by [[Green's identities|Green's second identity]]) where the boundary terms disappear as long as the domain ''D'' is finite or if ''f'' vanishes at infinity; e.g. both '''1'''''x''∈''D'' and ∇''x'''''1'''''x''∈''D'' are zero when evaluated at the 'boundary' of '''R'''''d'' when the domain ''D'' is finite. The third equality follows by the [[divergence theorem]] and shows, again, a sum (or, in this case, an integral) of outward normal derivatives over all boundary locations. The divergence theorem is valid for piecewise smooth domains ''D'', and hence ''D'' needs to be piecewise smooth.

Thus the ''surface delta prime function'' (a.k.a. Dirac {{mvar|δ'}}-function) exists on a piecewise smooth surface, and is equivalent to the Laplacian of the indicator function of the domain ''D'' encompassed by that piecewise smooth surface. Naturally, the difference between a point and a surface disappears in one dimension.

In electrostatics, a surface dipole (or [[Double layer potential]]) can be modelled by the limiting distribution of the Laplacian of the indicator.

The calculation above derives from research on path integrals in quantum physics.<ref name="Lange 2012"/>

===Proof of the surface delta function {{anchor|Normal derivative of the indicator}}=== This section will prove that the (inward) normal derivative of the indicator is a ''surface delta function''.

For a finite domain ''D'' or when ''f'' vanishes at infinity, it follows by the [[divergence theorem]] that

:<math>\int _{\mathbf{R}^d}\nabla_x^2\left (\mathbf{1}_{x\in D}\,f(x)\right )\;dx= 0.</math>

By the [[product rule]], it follows that

:<math>\int _{\mathbf{R}^d}\,\nabla_x^2\mathbf{1}_{x\in D}\,f(x)\;dx+ \int_{\mathbf{R}^d}\mathbf{1}_{x\in D}\,\nabla_x^2 f(x)\;dx =-2 \int _{\mathbf{R}^d} \nabla_x \mathbf{1}_{x\in D}\cdot \nabla_x f(x)\;dx.</math>

Following from the analysis of the section [[#Proof of the surface delta prime function|above]], the two terms on the left-hand side are equal, and thus

:<math>\oint_{\partial D}\,\underset{\alpha \to \beta}\lim n_\beta \cdot \nabla_\alpha f(\alpha)\;d\beta =-\displaystyle \int _{\mathbf{R}^d}\nabla_x\mathbf{1}_{x\in D}\cdot \nabla_x f(x)\;dx.</math>

The gradient of the indicator vanishes everywhere, except near the boundary of ''D'', where it points in the normal direction. Therefore, only the component of ∇''x''''f''(''x'') in the normal direction is relevant. Suppose that, near the boundary, ∇''x''''f''(''x'') is equal to ''nxg''(''x''), where ''g'' is some other function. Then it follows that

:<math>\oint _{\partial D}\,g(\beta)\;d\beta=-\int_{\mathbf{R}^d}\,\nabla_x\mathbf{1}_{x\in D}\,\cdot\,n_x\,g(x)\;dx. </math>

The outward normal ''n''''x'' was originally only defined for ''x'' in the surface, but it can be defined to exist for all ''x''; for example by taking the outward normal of the boundary point nearest to ''x''.

The foregoing analysis shows that −''nx'' ⋅ ∇''x'''''1'''''x''∈''D'' can be regarded as the surface generalisation of the one-dimensional [[Dirac delta function]]. By setting the function ''g'' equal to one, it follows that the inward normal derivative of the indicator integrates to the [[surface area]] of ''D''.

In electrostatics, [[surface charge]] densities (or ''single boundary layers'') can be modelled using the surface delta function as above. The usual [[Dirac delta function]] be used in some cases, e.g. when the surface is spherical. In general, the surface delta function discussed here may be used to represent the surface [[charge density]] on a surface of any shape.

The calculation above derives from research on path integrals in quantum physics.<ref name="Lange 2012"/>

==Approximations by bump functions==

This section shows how derivatives of the indicator can be treated numerically under an integral sign.

In principle, the indicator cannot be differentiated numerically, since its derivative is either zero or infinite. But, for practical purposes, the indicator can be approximated by a [[bump function]], indicated by ''I''ε(''x'') and approaching the indicator for ε → 0. Several options are possible, but it is convenient to let the bump function be non-negative and approach the indicator ''from below'', i.e.

: <math>\begin{align} 0 \leq I_\varepsilon(x)& \leq \mathbf{1}_{{x}\in D}\quad \forall \varepsilon >0\\ \underset{\varepsilon \searrow 0}\lim\; I_\varepsilon(x)&=\mathbf{1}_{x\in D} \end{align}</math>

This ensures that the family of bump functions is identically zero outside of ''D''. This is convenient, since it is possible that the function ''f'' is only defined in the ''interior'' of ''D''. For ''f'' defined in ''D'', we thus obtain the following:

: <math>\begin{align} - \underset{\varepsilon \searrow 0}\lim \int _{\mathbf{R}^d}\,f(x)\, n_x \cdot \nabla_x I_{\varepsilon}(x)\;dx &= \oint _{\partial D}\,\underset{\alpha \to \beta}\lim f(\alpha)\;d\beta, \\ \underset{\varepsilon \searrow 0}\lim\,\int _{\mathbf{R}^d}\nabla_x^2 I_{\varepsilon}(x)\,f(x)\;dx&= \oint_{\partial D}\,\underset{\alpha \to \beta}\lim n_\beta \cdot \nabla_\alpha f(\alpha)\;d\beta, \end{align}</math>

where the interior coordinate α approaches the boundary coordinate β from the interior of ''D'', and where there is no requirement for ''f'' to exist outside of ''D''.

When ''f'' is defined on both sides of the boundary, and is furthermore differentiable across the boundary of ''D'', then it is less crucial how the bump function approaches the indicator.

== Discontinuous test functions ==

If the test function ''f'' is possibly discontinuous across the boundary, then distribution theory for discontinuous functions may be used to make sense of surface distributions, see e.g. section V in .<ref>{{citation |last=Lange|first=Rutger-Jan | year=2015|title=Distribution theory for Schrödinger's integral equation|journal=Journal of Mathematical Physics|volume=56|number=12|page=2015|article-number=122105 |doi=10.1063/1.4936302|arxiv=1401.7627|bibcode=2015JMP....56l2105L|s2cid=116896174 }}</ref> In practice, for the surface delta function this usually means averaging the value of ''f'' on both sides of the boundary of ''D'' before integrating over the boundary. Likewise, for the surface delta prime function it usually means averaging the outward normal derivative of ''f'' on both sides of the boundary of the domain ''D'' before integrating over the boundary.

==Applications==

===Quantum mechanics=== In [[quantum mechanics]], point interactions are well known and there is a large body of literature on the subject. A well-known example of a one-dimensional singular potential is the [[delta potential|Schrödinger equation with a Dirac delta potential]].<ref>{{citation|last1=Atkinson|first1=D.A.|last2=Crater|first2=H.W.|title=An exact treatment of the Dirac delta function potential in the Schrodinger equation|journal=American Journal of Physics|volume=43|issue=4|pages=301–304|year=1975 |bibcode = 1975AmJPh..43..301A |doi = 10.1119/1.9857 }}</ref><ref>{{citation|last=Manoukian|first=E.B.|title=Explicit derivation of the propagator for a Dirac delta potential|journal=Journal of Physics A: Mathematical and General|volume=22|number=1|pages=67–70|year=1999 |doi=10.1088/0305-4470/22/1/013|bibcode = 1989JPhA...22...67M }}</ref> The one-dimensional Dirac delta ''prime'' potential, on the other hand, has caused controversy.<ref>{{citation|last1=Albeverio|first1=S.|last2=Gesztesy|first2=F.|last3=Hoegh-Krohn|first3=R.|last4=Holden|first4=H.|title=Solvable models in quantum mechanics|year=1988|publisher=Springer-Verlag}}</ref><ref>{{citation|last=Zhao|first=B.H.|title=Comments on the Schrödinger Equation with delta'-interaction in one dimension|journal=Journal of Physics A: Mathematical and General|year=1992|volume=25|issue=10|page=617 | doi = 10.1088/0305-4470/25/10/003|bibcode = 1992JPhA...25L.617Z }}</ref><ref>{{citation|last1=Albeverio|first1=S.|last2=Gesztesy|first2=F.|last3=Holden|first3=H.|title=Comments on a recent note on the Schrodinger equation with a delta'-interaction|journal=Journal of Physics A: Mathematical and General|year=1993|volume=26|issue=15|pages=3903–3904 |doi=10.1088/0305-4470/26/15/037|bibcode = 1993JPhA...26.3903A }}</ref> The controversy was seemingly settled by an independent paper,<ref>{{citation|last=Griffiths|first=D.J.|title=Boundary conditions at the derivative of a delta function|journal=Journal of Physics A: Mathematical and General|year=1993|volume=26|issue=9|pages=2265–2267 |doi=10.1088/0305-4470/26/9/021|bibcode = 1993JPhA...26.2265G }}</ref> although even this paper attracted later criticism.<ref name="Lange 2012"/><ref>{{citation|last1=Coutinho|first1=F.A.B.|last2=Nogami|first2=Y.|last3=Perez|first3=J.F.|title=Generalized point interactions in one-dimensional quantum mechanics|journal=Journal of Physics A: Mathematical and General|year=1997|volume=30|issue=11|pages=3937–3945 |doi=10.1088/0305-4470/30/11/021|bibcode = 1997JPhA...30.3937C }}</ref>

A lot more attention has been focused on the one-dimensional Dirac delta prime potential recently.<ref>{{citation|last1=Kostenko|first1=A.|last2=Malamud|first2=M.|title=Spectral Theory of Semibounded Schrödinger Operators with δ′-Interactions|journal=Annales Henri Poincaré|volume=15|issue=3|year=2012 |page=617 |arxiv=1212.1691|bibcode = 2012arXiv1212.1691K|doi=10.1007/s00023-013-0245-9|s2cid=119727685}}</ref><ref>{{citation|last1=Brasche|first1=J.F.|last2=Nizhnik|first2=L.|title=One-dimensional Schrödinger operators with δ′-interactions on a set of Lebesgue measure zero |journal=Operators and Matrices |volume=7 |issue=4 |page=887 |year=2012|arxiv=1112.2545|bibcode = 2011arXiv1112.2545B |doi=10.7153/oam-07-49|s2cid=67790330}}</ref><ref>{{citation|last1=Carreau|first1=M.|last2=Farhi|first2=E.|last3=Gutmann|first3=S.|title=Functional integral for a free particle in a box|journal=Physical Review D|volume=42|number=4|pages=1194–1202|year=1990 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A point on the one-dimensional line can be considered both as a point and as surface; as a point marks the boundary between two regions. Two generalisations of the Dirac delta-function to higher dimensions have thus been made: the generalisation to a multidimensional point,<ref>{{citation|last1=Scarlatti|first1=S.|last2=Teta|first2=A.|title=Derivation of the time-dependent propagator for the three-dimensional Schrodinger equation with one point interaction|journal=Journal of Physics A: Mathematical and General|volume=23|number=19|pages=L1033|year=1990 |doi=10.1088/0305-4470/23/19/003|bibcode = 1990JPhA...23L1033S }}</ref><ref>{{citation|last=Grosche|first=C.|title=Path integrals for two-and three-dimensional δ-function perturbations|journal=Annalen der Physik|volume=506|number=4|pages=283–312|year=1994 |arxiv = hep-th/9308082 |bibcode = 1994AnP...506..283G |doi = 10.1002/andp.19945060406 |s2cid=119436723}}</ref> as well as the generalisation to a multidimensional surface.<ref name="Lange 2012"/><ref>{{citation|last1=Moszkowski|first1=S.A.|title=Derivation of the surface delta interaction|journal=Physical Review C|volume=19|number=6|pages=2344–2348|year=1997 |doi=10.1103/physrevc.19.2344|bibcode = 1979PhRvC..19.2344M }}</ref><ref>{{citation|last1=Antoine|first1=J.P.|last2=Gesztesy|first2=F.|last3=Shabani|first3=J.|title=Exactly solvable models of sphere interactions in quantum mechanics|journal=Journal of Physics A: Mathematical and General|volume=20|number=12|pages=3687–3712|year=1999 |doi=10.1088/0305-4470/20/12/022|bibcode = 1987JPhA...20.3687A }}</ref><ref>{{citation|last1=Shabani|first1=J.|last2=Vyabandi|first2=A.|title=Exactly solvable models of delta-sphere interactions in relativistic quantum mechanics|journal=Journal of Mathematical Physics|volume=43|issue=12|page=6064|year=2002 |bibcode = 2002JMP....43.6064S |doi = 10.1063/1.1518785 }} </ref><ref>{{citation|last1=Hounkonnou|first1=M.N.|author1-link=Mahouton Norbert Hounkonnou|last2=Hounkpe|first2=M.|last3=Shabani|first3=J.|title=Exactly solvable models of δ′-sphere interactions in nonrelativistic quantum mechanics|journal=Journal of Mathematical Physics|volume=40|number=9|pages=4255–4273|year=1999 |bibcode = 1999JMP....40.4255H |doi = 10.1063/1.532964 }}</ref>

The former generalisations are known as point interactions, whereas the latter are known under different names, e.g. "delta-sphere interactions" and "surface delta interactions". The latter generalisations may use derivatives of the indicator, as explained here, or the one-dimensional Dirac {{mvar|δ}}-function as a function of the radial coordinate ''r''.

===Fluid dynamics=== The Laplacian of the indicator has been used in [[fluid dynamics]], e.g. to model the interfaces between different media.<ref>{{citation|last=Che|first=J.H.|title=Numerical simulations of complex multiphase flows: electrohydrodynamics and solidification of droplets|publisher=University of Michigan|year=1999|page=37}}</ref><ref>{{citation|last=Juric|first=D.|title=Computations of phase change|journal=PhD Thesis|year=1996|page=150|url=http://alum.wpi.edu/~damir.juric/CompBoilingFlows_ThesisJuric_1996.pdf|access-date=2012-12-29|archive-date=2014-02-22|archive-url=https://web.archive.org/web/20140222191408/http://alum.wpi.edu/~damir.juric/CompBoilingFlows_ThesisJuric_1996.pdf}}</ref><ref>{{citation|last1=Unverdi|first1=S.O.|last2=Tryggvason|first2=G.|title=A front-tracking method for viscous, incompressible, multi-fluid flows|journal=Journal of Computational Physics|volume=100|number=1|pages=29��30|year=1992 |doi=10.1016/0021-9991(92)90307-K|bibcode = 1992JCoPh.100...25U |url=https://deepblue.lib.umich.edu/bitstream/2027.42/30059/1/0000428.pdf|hdl=2027.42/30059|hdl-access=free}}</ref><ref>{{cite conference|last1=Goz|first1=M.F.|last2=Bunner|first2=B.|last3=Sommerfeld|first3=M.|last4=Tryggvason|first4=G.|title=Direct numerical simulation of bubble swarms with a parallel front-tracking method|conference=High Performance Scientific and Engineering Computing: Proceedings of the 3rd International FORTWIHR Conference on HPSEC, Erlangen, March 12–14, 2001|volume=21|pages=97–106|year=2002 |doi=10.1007/978-3-642-55919-8_11|series=Lecture Notes in Computational Science and Engineering|isbn=978-3-540-42946-3}}</ref><ref>{{citation|last1=Juric|first1=D.|last2=Tryggvason|first2=G.|title=A front-tracking method for dendritic solidification|journal=Journal of Computational Physics|volume=123|number=1|pages=127–148|year=1996 |bibcode = 1996JCoPh.123..127J |doi = 10.1006/jcph.1996.0011 |citeseerx=10.1.1.17.8419}}</ref><ref>{{citation|last1=Uddin|first1=E.|last2=Sung|first2=H.J.|title=Simulation of flow-flexible body interactions with large deformation|journal=International Journal for Numerical Methods in Fluids|year=2011 |doi=10.1002/fld.2731|volume=70|issue=9|pages=1089–1102|bibcode=2012IJNMF..70.1089U|s2cid=121032029 }}</ref>

===Molecular dynamics===

In [[molecular dynamics]], the divergence of the indicator can be shown to occur in the context of applying the [[control volume]] to the molecular [[Cauchy stress tensor]], which provides a bridge between [[continuum mechanics]] and molecular dynamics <ref>{{cite journal| last1 = Smith| first1 = E. R.| last2 = Heyes| first2 = D. M.| last3 = Dini| first3 = D.| last4 = Zaki| first4 = T. A.| title = Control-volume representation of molecular dynamics| journal = Physical Review E| volume = 85| issue = 5| article-number = 056705| year = 2012| publisher = American Physical Society| doi = 10.1103/PhysRevE.85.056705| arxiv = 1203.2453}}</ref>

===Surface reconstruction=== The divergence of the indicator and the Laplacian of the indicator (or of the [[Indicator function|characteristic function]], as the indicator is also known) have been used as the sample information from which surfaces can be reconstructed.<ref>{{cite conference |last=Kazhdan|first=M.|title=Reconstruction of solid models from oriented point sets|conference=Proceedings of the third Eurographics symposium on Geometry processing|year=2005|page=73|url=http://www.cs.jhu.edu/~misha/MyPapers/SGP05.pdf}} </ref><ref>{{cite book|last1=Kazhdan|first1=M.|last2=Bolitho|first2=M.|last3=Hoppe|first3=H|year=2006|title=Proceedings of the fourth Eurographics symposium on Geometry processing|pages=1–3–4|url=http://faculty.cs.tamu.edu/schaefer/teaching/689_Fall2006/poissonrecon.pdf}}</ref>

==See also==

* {{annotated link|Delta potential}} * {{annotated link|Dirac delta function}} * {{annotated link|Distribution (mathematics)}} * {{annotated link|Double layer potential}} * {{annotated link|Electrostatics}} * {{annotated link|Generalized function}} * {{annotated link|Indicator function}} * {{annotated link|Potential theory}}

==References== {{Reflist|30em}}

[[Category:Mathematics of infinitesimals]] [[Category:Generalized functions]] [[Category:Measure theory]] [[Category:Schwartz distributions]]