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.
where ∆ ≡ ∇2 is the Laplace operator, ∂U is the boundary of region U, n is the outward pointing unit normal to the surface element dS and dS = ndS 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 with ψ and the gradient of φ replacing u and v. Note that Green's first identity above is a special case of the more general identity derived from the divergence theorem by substituting F = ψΓ,
In the equation above, ∂φ/∂n is the directional derivative of φ in the direction of the outward pointing surface normal n of the surface element dS,
Explicitly incorporating this definition in the Green's second identity with ε = 1 results in
In particular, this demonstrates that the Laplacian is a self-adjoint operator in the L2 inner product for functions vanishing on the boundary so that the right hand side of the above identity is zero.
Green's third identity derives from the second identity by choosing φ = G, where the Green's function G is taken to be a fundamental solution of the Laplace operator, ∆.
Green's third identity states that if ψ is a function that is twice continuously differentiable on U, then
A simplification arises if ψ is itself a harmonic function, i.e. a solution to the Laplace equation.
This form is used to construct solutions to Dirichlet boundary condition problems.
Solutions for Neumann boundary condition 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 [2] for a detailed argument, with an alternative.
In such a context, this identity is the mathematical expression of the Huygens principle, and leads to Kirchhoff's diffraction formula and other approximations.
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 the outward oriented unit vector field normal to the boundary, and Δu = div(grad u) is the Laplacian.
Green's second identity establishes a relationship between second and (the divergence of) first order derivatives of two scalar functions.
where pm and qm 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.
[3] In vector diffraction theory, two versions of Green's second identity are introduced.
One variant invokes the divergence of a cross product [4][5][6] and states a relationship in terms of the curl-curl of the field
The other approach introduces bi-vectors, this formulation requires a dyadic Green function.
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.
Recall the vector identity for the gradient of a dot product,
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
These results can be rigorously proven to be correct through evaluation of the vector components.
Putting together these two results, a result analogous to Green's theorem for scalar fields is obtained, Theorem for vector fields:
Since the divergence of a curl is zero, the third term vanishes to yield Green's vector identity:
This result can be verified by expanding the divergence of a scalar times a vector on the RHS.