This identity is derived from the divergence theorem applied to the vector field F = ψ ∇φ and using the identity that ∇ ·(φX ) = ∇φ ·X + φ ∇·X: 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 ∆ ≡ ∇2 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.
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 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. 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.
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
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