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 theorem.
Green's first identity
This identity is derived from the divergence theorem applied to the vector field F = ψ∇φ: Let φ and ψ be scalar functions defined on some region U ⊂ Rd, and suppose that φ is twice continuously differentiable, and ψ is once continuously differentiable. Then
where ∆ is the Laplace operator, ∂U is the boundary of region U, n is the outward pointing unit normal of surface element dS and dS is the oriented surface element.
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,
Green's second identity
If φ and ψ are both twice continuously differentiable on U ⊂ R3, and ε is once continuously differentiable, one may choose F = ψε∇φ − φε∇ψ to obtain
For the special case of ε = 1 all across U ⊂ 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,
In particular, this demonstrates that the Laplacian is self-adjoint in the L2 inner product for functions vanishing on the boundary.
Green's third identity
For example, in R3, a solution has the form
Green's third identity states that if ψ is a function that is twice continuously differentiable on U, then
This form is used to construct solutions to Dirichlet boundary condition problems. To find solutions for Neumann boundary condition problems, the Green's function with vanishing normal gradient on the boundary is used instead.
It can be further verified that the above identity also applies when ψ is a solution to the Helmholtz equation or wave equation and G is the appropriate Green's function. In such a context, this identity is the mathematical expression of the Huygens Principle.
Green's identities hold on a Riemannian manifold, In this setting, the first two are
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 Δu = div(grad u) 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 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.
In vector diffraction theory, two versions of Green’s second identity are introduced.
This equation can be written in terms of the Laplacians,
However, the terms
could not be readily written in terms of a divergence.
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 through 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.
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