This identity is derived from the divergence theorem applied to the vector field and using the identity that : Let and be scalar functions defined on some region, and suppose that is twice continuously differentiable, and is once continuously differentiable. Then where is the Laplace operator, is the boundary of region, is the outward pointing unit normal of 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 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 of in the direction of the outward pointing normal to the surface element, In particular, this demonstrates that the Laplacian is self-adjoint in the 2inner product for functions vanishing on the boundary.
Green's third identity
Green's third identity derives from the second identity by choosing, where the Green's function is taken to be a fundamental solution 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, i.e. a solution to the Laplace equation. 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 that vanishes on the boundary of , 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 is a solution to the Helmholtz equation or wave equation and is the appropriate Green's function. In such a context, this identity is the mathematical expression of the Huygens principle, and leads to Kirchhoff's diffraction formula 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 second identity establishes a relationship between second and 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 or the other, the contribution to each term must be These results can be rigorously proven to be correct through . 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, 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 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.
