Green's identities
Encyclopedia
In mathematics
, Green's identities are a set of three identities in vector calculus. They are named after the mathematician George Green
, who discovered Green's theorem
.
applied to the vector field
: Let φ and ψ be scalar functions defined on some region U in R3, and suppose that φ is twice continuously differentiable, and ψ is once continuously differentiable. Then
where
is the Laplace operator
,
is the boundary of region U and n is the outward pointing unit normal of surface element dS. This theorem is essentially the higher dimensional equivalent of integration by parts with ψ and the gradient of φ replacing u and v.
For the special case of
all across U in R3 then:
In the equation above ∂φ / ∂n is the directional derivative of φ in the direction of the outward pointing normal n to the surface element dS:
, where G is a fundamental solution of the Laplace equation. This means that:
For example in
, the fundamental solution has the form:
Green's third identity states that if ψ is a function that is twice continuously differentiable on U, then
A further simplification arises if ψ is itself a harmonic function
, i.e. a solution to the Laplace equation. Then
and the identity simplifies to:
where u and v are smooth real-valued functions on M, dV is the volume form compatible with the metric,
is the induced volume form on the boundary of M, N is oriented unit vector field normal to the boundary, and 
is the Laplacian.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Green's identities are a set of three identities in vector calculus. They are named after the mathematician George Green
George Green
George Green was a British mathematical physicist who wrote An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism...
, who discovered Green's theorem
Green's theorem
In mathematics, Green's theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C...
.
Green's first identity
This identity is derived from the divergence theoremDivergence theorem
In vector calculus, the divergence theorem, also known as Gauss' theorem , Ostrogradsky's theorem , or Gauss–Ostrogradsky theorem is a result that relates the flow of a vector field through a surface to the behavior of the vector field inside the surface.More precisely, the divergence theorem...
applied to the vector field
: Let φ and ψ be scalar functions defined on some region U in R3, and suppose that φ is twice continuously differentiable, and ψ is once continuously differentiable. Then
where
is the Laplace operatorLaplace operator
In mathematics the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space. It is usually denoted by the symbols ∇·∇, ∇2 or Δ...
,
is the boundary of region U and n is the outward pointing unit normal of surface element dS. This theorem is essentially the higher dimensional equivalent of integration by parts with ψ and the gradient of φ replacing u and v.Green's second identity
If φ and ψ are both twice continuously differentiable on U in R3, and ε is once continuously differentiable:
For the special case of
all across U in R3 then:
In the equation above ∂φ / ∂n is the directional derivative of φ in the direction of the outward pointing normal n to the surface element dS:
Green's third identity
Green's third identity derives from the second identity by choosing, where G is a fundamental solution of the Laplace equation. This means that:For example in
, the fundamental solution has the form:Green's third identity states that if ψ is a function that is twice continuously differentiable on U, then
A further simplification arises if ψ is itself a harmonic function
Harmonic function
In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R which satisfies Laplace's equation, i.e....
, i.e. a solution to the Laplace equation. Then
and the identity simplifies to:
On manifolds
Green's identities hold on a Riemannian manifold, In this setting, the first two arewhere u and v are smooth real-valued functions on M, dV is the volume form compatible with the metric,
is the induced volume form on the boundary of M, N is oriented unit vector field normal to the boundary, and is the Laplacian.External links
- http://mathworld.wolfram.com/GreensIdentities.html Green's Identities at Wolfram MathWorld