Laplacian of the indicator
   HOME

TheInfoList



OR:

In mathematics, the Laplacian of the indicator of the domain ''D'' is a generalisation of the derivative of the
Dirac delta function In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
to higher dimensions, and is non-zero only on the ''surface'' of ''D''. It can be viewed as the ''surface delta prime function''. It is analogous to the second derivative of the
Heaviside step function The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive argume ...
in one dimension. It can be obtained by letting the
Laplace operator In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
work on the
indicator function In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\i ...
of some domain ''D''. 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''. From a mathematical viewpoint, it is not strictly a function but a
generalized function In mathematics, generalized functions are objects extending the notion of functions. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful in making discontinuous functions ...
or
measure Measure may refer to: * Measurement, the assignment of a number to a characteristic of an object or event Law * Ballot measure, proposed legislation in the United States * Church of England Measure, legislation of the Church of England * Mea ...
. 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 Distribution may refer to: Mathematics *Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations * Probability distribution, the probability of a particular value or value range of a vari ...
function. Just as in the formulation of distribution theory, it is in practice regarded as a limit of a sequence of smooth functions; one may meaningfully take the Laplacian of a
bump function In mathematics, a bump function (also called a test function) is a function f: \R^n \to \R on a Euclidean space \R^n which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supported. The set of all bump f ...
, which is smooth by definition, and let the bump function approach the indicator in the limit.


History

Paul Dirac Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. He was the Lucasian Professor of Mathematics at the Univer ...
introduced the Dirac -function, as it has become known, as early as 1930. The one-dimensional Dirac -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 coordinates, the ''d''-dimensional Dirac -function is a product of ''d'' one-dimensional -functions; one for each Cartesian coordinate (see e.g. generalizations of the Dirac delta function). However, a different generalisation is possible. The point zero, in one dimension, can be considered as the 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 The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive argume ...
. Formally, the Dirac -function and its derivative (i.e. the one-dimensional ''surface delta prime function'') can be viewed as the first and second derivative of the Heaviside step function, i.e. ∂''x''1''x''>0 and \partial_x^2 \mathbf_. The analogue of the step function in higher dimensions is the
indicator function In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\i ...
, 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 -function and its derivative have been proposed: :\begin \delta(x) &\to -n_x\cdot\nabla_x\mathbf_, \\ \delta'(x) &\to \nabla_x^2 \mathbf_. \end Here ''n'' is the outward
normal vector In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve at ...
. Here the Dirac -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 The surface area of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of arc ...
enclosing ''D'', as shown
below Below may refer to: *Earth *Ground (disambiguation) *Soil *Floor *Bottom (disambiguation) Bottom may refer to: Anatomy and sex * Bottom (BDSM), the partner in a BDSM who takes the passive, receiving, or obedient role, to that of the top or ...
. The one-dimensional Dirac -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 -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 mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to th ...
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. Surface delta functions use the one-dimensional Dirac -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
theory of distributions A theory is a rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or research. Theories may be s ...
or
generalized function In mathematics, generalized functions are objects extending the notion of functions. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful in making discontinuous functions ...
s: one can obtain a well-defined prescription by postulating that the Laplacian of the indicator, for example, is defined by two integrations by parts when it appears under an integral sign. Alternatively, the indicator (and its derivatives) can be approximated using a
bump function In mathematics, a bump function (also called a test function) is a function f: \R^n \to \R on a Euclidean space \R^n which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supported. The set of all bump f ...
(and its derivatives). The limit, where the (smooth) bump function approaches the indicator function, must then be put outside of the integral.


Dirac 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 The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, or ...
: \int_a^b \frac\,dx=\underset\lim f(x)-\underset\lim f(x), assuming that ''f'' is locally integrable. Now for ''a'' < ''b'' it follows, by proceeding heuristically, that :\begin \int_^ \frac\,f(x)\;dx&=\int_^ \mathbf_ \frac\;dx, \\ &=\displaystyle\int_a^b \frac\;dx, \\ &=\displaystyle\Big(\underset\lim -\underset\lim\Big) \frac. \end Here 1''a''<''x''<''b'' is the
indicator function In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\i ...
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 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: : \begin \int _\nabla_x^2\mathbf_\,f(x)\;dx&= \int _\mathbf_\,\nabla_x^2 f(x)\;dx,\\ &= \int _\,\nabla_x^2 f(x)\;dx,\\ &= \oint_\,\underset\lim n_\beta \cdot \nabla_x f(x)\;d\beta. \end 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 second identity In mathematics, Green's identities are a set of three identities in vector calculus relating the bulk with the boundary of a region on which differential operators act. They are named after the mathematician George Green, who discovered Green's ...
) 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 In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the ''flux'' of a vector field through a closed surface to the ''divergence'' of the field in the vol ...
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 -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 In potential theory, an area of mathematics, a double layer potential is a solution of Laplace's equation corresponding to the electrostatic or magnetic potential associated to a dipole distribution on a closed surface ''S'' in three-dimensions. ...
) 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.


Dirac surface delta function

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 In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the ''flux'' of a vector field through a closed surface to the ''divergence'' of the field in the vol ...
that :\int _\nabla_x^2\left (\mathbf_\,f(x)\right )\;dx= 0. By the
product rule In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v + ...
, it follows that :\int _\,\nabla_x^2\mathbf_\,f(x)\;dx+ \int_\mathbf_\,\nabla_x^2 f(x)\;dx =-2 \int _ \nabla_x \mathbf_\cdot \nabla_x f(x)\;dx. Following from the analysis of the section above, the two terms on the left-hand side are equal, and thus :\oint_\,\underset\lim n_\beta \cdot \nabla_\alpha f(\alpha)\;d\beta =-\displaystyle \int _\nabla_x\mathbf_\cdot \nabla_x f(x)\;dx. 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 :\oint _\,g(\beta)\;d\beta=-\int_\,\nabla_x\mathbf_\,\cdot\,n_x\,g(x)\;dx. 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 In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
. By setting the function ''g'' equal to one, it follows that the inward normal derivative of the indicator integrates to the
surface area The surface area of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of arc ...
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 In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
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.


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 In mathematics, a bump function (also called a test function) is a function f: \R^n \to \R on a Euclidean space \R^n which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supported. The set of all bump f ...
, 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. : \begin 0 \leq I_\varepsilon(x)& \leq \mathbf_\quad \forall \varepsilon >0\\ \underset\lim\; I_\varepsilon(x)&=\mathbf_ \end 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: : \begin - \underset\lim \int _\,f(x)\, n_x \cdot \nabla_x I_(x)\;dx &= \oint _\,\underset\lim f(\alpha)\;d\beta, \\ \underset\lim\,\int _\nabla_x^2 I_(x)\,f(x)\;dx&= \oint_\,\underset\lim n_\beta \cdot \nabla_\alpha f(\alpha)\;d\beta, \end 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 . 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 Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, 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 Schrödinger equation with a Dirac delta potential. The one-dimensional Dirac delta ''prime'' potential, on the other hand, has caused controversy. The controversy was seemingly settled by an independent paper, although even this paper attracted later criticism. A lot more attention has been focused on the one-dimensional Dirac delta prime potential recently. 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, as well as the generalisation to a multidimensional surface. 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 -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.


Surface reconstruction

The divergence of the indicator and the Laplacian of the indicator (or of the
characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts: * The indicator function of a subset, that is the function ::\mathbf_A\colon X \to \, :which for a given subset ''A'' of ''X'', has value 1 at points ...
, as the indicator is also known) have been used as the sample information from which surfaces can be reconstructed.


See also

*
Distribution (mathematics) Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives d ...
*
Generalized function In mathematics, generalized functions are objects extending the notion of functions. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful in making discontinuous functions ...
*
Dirac delta function In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
*
Indicator function In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\i ...
*
Delta potential In quantum mechanics the delta potential is a potential well mathematically described by the Dirac delta function - a generalized function. Qualitatively, it corresponds to a potential which is zero everywhere, except at a single point, where it t ...
*
Potential theory In mathematics and mathematical physics, potential theory is the study of harmonic functions. The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely gravi ...
*
Electrostatics Electrostatics is a branch of physics that studies electric charges at rest (static electricity). Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for amber ...
*
Double layer potential In potential theory, an area of mathematics, a double layer potential is a solution of Laplace's equation corresponding to the electrostatic or magnetic potential associated to a dipole distribution on a closed surface ''S'' in three-dimensions. ...


References

{{Reflist, 30em Mathematics of infinitesimals Generalized functions Measure theory