In
mathematics, Green's identities are a set of three identities in
vector calculus
Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subjec ...
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 theorem.
Green's first identity
This identity is derived from the
divergence theorem applied to the vector field while using an extension of 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 ...
that : Let and be scalar functions defined on some region , and suppose that is twice
continuously differentiable, and is once continuously differentiable. Using the product rule above, but letting , integrate over . Then
where is the
Laplace operator, is the boundary of region , is the outward pointing unit normal to the surface element and is the oriented surface element.
This theorem is a special case of the
divergence theorem, and is essentially the higher dimensional equivalent of
integration by parts
In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. ...
with and the gradient of replacing and .
Note that Green's first identity above is a special case of the more general identity derived from the
divergence theorem by substituting ,
Green's second identity
If and are both twice continuously differentiable on , and is once continuously differentiable, one may choose to obtain
For the special case of all across , then,
In the equation above, is the
directional derivative
In mathematics, the directional derivative of a multivariable differentiable (scalar) function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity ...
of in the direction of the outward pointing surface normal of the surface element ,
Explicitly incorporating this definition in the Green's second identity with results in
In particular, this demonstrates that the Laplacian is a
self-adjoint operator
In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map ''A'' (from ''V'' to its ...
in the inner product for functions vanishing on the boundary so that the right hand side of the above identity is zero.
Green's third identity
Green's third identity derives from the second identity by choosing , where the
Green's function
In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions.
This means that if \operatorname is the linear differenti ...
is taken to be a
fundamental solution
In mathematics, a fundamental solution for a linear partial differential operator is a formulation in the language of distribution theory of the older idea of a Green's function (although unlike Green's functions, fundamental solutions do not a ...
of the
Laplace operator, ∆. This means that:
For example, in , a solution has the form
Green's third identity states that if is a function that is twice continuously differentiable on , then
A simplification arises if is itself a
harmonic function
In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that is,
: \f ...
, i.e. a solution to the
Laplace equation
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as
\nabla^2\! f = 0 or \Delta f = 0,
where \Delta = \n ...
. Then and the identity simplifies to
The second term in the integral above can be eliminated if is chosen to be the
Green's function
In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions.
This means that if \operatorname is the linear differenti ...
that vanishes on the boundary of (
Dirichlet boundary condition
In the mathematical study of differential equations, the Dirichlet (or first-type) boundary condition is a type of boundary condition, named after Peter Gustav Lejeune Dirichlet (1805–1859). When imposed on an ordinary or a partial differential ...
),
This form is used to construct solutions to Dirichlet boundary condition problems. Solutions for
Neumann boundary condition
In mathematics, the Neumann (or second-type) boundary condition is a type of boundary condition, named after Carl Neumann.
When imposed on an ordinary or a partial differential equation, the condition specifies the values of the derivative appli ...
problems may also be simplified, though the
Divergence theorem applied to the differential equation defining Green's functions shows that the Green's function cannot integrate to zero on the boundary, and hence cannot vanish on the boundary. See
Green's functions for the Laplacian or for a detailed argument, with an alternative.
It can be further verified that the above identity also applies when is a solution to the
Helmholtz equation or
wave equation
The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and seism ...
and is the appropriate Green's function. In such a context, this identity is the mathematical expression of the
Huygens principle
Huygens (also Huijgens, Huigens, Huijgen/Huygen, or Huigen) is a Dutch patronymic surname, meaning "son of Hugo". Most references to "Huygens" are to the polymath Christiaan Huygens. Notable people with the surname include:
* Jan Huygen (1563–1 ...
, and leads to
Kirchhoff's diffraction formula Kirchhoff's diffraction formula (also Fresnel–Kirchhoff diffraction formula)
can be used to model the propagation of light in a wide range of configurations, either analytically or using numerical modelling. It gives an expression for the wave d ...
and other approximations.
On manifolds
Green's identities hold on a Riemannian manifold. In this setting, the first two are
where and are smooth real-valued functions on , is the volume form compatible with the metric,
is the induced volume form on the boundary of , is the outward oriented unit vector field normal to the boundary, and is the Laplacian.
Green's vector identity
Green's second identity establishes a relationship between second and (the divergence of) first order derivatives of two scalar functions. In differential form
where and are two arbitrary twice continuously differentiable scalar fields. This identity is of great importance in physics because continuity equations can thus be established for scalar fields such as mass or energy.
In vector diffraction theory, two versions of Green's second identity are introduced.
One variant invokes the divergence of a cross product and states a relationship in terms of the curl-curl of the field
This equation can be written in terms of the Laplacians,
However, the terms
could not be readily written in terms of a divergence.
The other approach introduces bi-vectors, this formulation requires a dyadic Green function. The derivation presented here avoids these problems.
Consider that the scalar fields in Green's second identity are the Cartesian components of vector fields, i.e.,
Summing up the equation for each component, we obtain
The LHS according to the definition of the dot product may be written in vector form as
The RHS is a bit more awkward to express in terms of vector operators. Due to the distributivity of the divergence operator over addition, the sum of the divergence is equal to the divergence of the sum, i.e.,
Recall the vector identity for the gradient of a dot product,
which, written out in vector components is given by
This result is similar to what we wish to evince in vector terms 'except' for the minus sign. Since the differential operators in each term act either over one vector (say
’s) or the other (
’s), the contribution to each term must be
These results can be rigorously proven to be correct throug
evaluation of the vector components Therefore, the RHS can be written in vector form as
Putting together these two results, a result analogous to Green's theorem for scalar fields is obtained,
Theorem for vector fields:
The
curl of a cross product can be written as
Green's vector identity can then be rewritten as
Since the divergence of a curl is zero, the third term vanishes to yield Green's vector identity:
With a similar procedure, the Laplacian of the dot product can be expressed in terms of the Laplacians of the factors
As a corollary, the awkward terms can now be written in terms of a divergence by comparison with the vector Green equation,
This result can be verified by expanding the divergence of a scalar times a vector on the RHS.
See also
*
Green's function
In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions.
This means that if \operatorname is the linear differenti ...
*
Kirchhoff integral theorem Kirchhoff's integral theorem (sometimes referred to as the Fresnel–Kirchhoff integral theorem) is a surface integral to obtain the value of the solution of the homogeneous scalar wave equation at an arbitrary point P in terms of the values of t ...
*
Lagrange's identity (boundary value problem)
In the study of ordinary differential equations and their associated boundary value problems, Lagrange's identity, named after Joseph Louis Lagrange, gives the boundary terms arising from integration by parts of a self-adjoint linear differential ...
References
External links
*
Green's Identities at Wolfram MathWorld
{{DEFAULTSORT:Green's Identities
Vector calculus
Mathematical identities