Green's identities
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 while using an extension of the product rule that : Let and be scalar functions defined on some region , and suppose that is twice continuously differentiable, and is once continuously differentiable. Using the product rule above, but letting , integrate over  . Then formula_1 where is the Laplace operator, is the boundary of region , is the outward pointing unit normal to the 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 , formula_2 Green's second identity. If and are both twice continuously differentiable on , and is once continuously differentiable, one may choose  to obtain formula_3 For the special case of all across , then, formula_4 In the equation above, is the directional derivative of in the direction of the outward pointing surface normal of the surface element , formula_5 Explicitly incorporating this definition in the Green's second identity with results in formula_6 In particular, this demonstrates that the Laplacian is a self-adjoint operator in the inner product for functions vanishing on the boundary so that the right hand side of the above identity is zero. 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: formula_7 For example, in , a solution has the form formula_8 Green's third identity states that if is a function that is twice continuously differentiable on , then formula_9 A simplification arises if is itself a harmonic function, i.e. a solution to the Laplace equation. Then and the identity simplifies to formula_10 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 (Dirichlet boundary condition), formula_11 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 formula_12 where and are smooth real-valued functions on , is the volume form compatible with the metric, formula_13 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 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 formula_14 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 formula_15 This equation can be written in terms of the Laplacians, formula_16 However, the terms formula_17 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., formula_18 Summing up the equation for each component, we obtain formula_19 The LHS according to the definition of the dot product may be written in vector form as formula_20 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., formula_21 Recall the vector identity for the gradient of a dot product, formula_22 which, written out in vector components is given by formula_23 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 formula_24’s) or the other (formula_25’s), the contribution to each term must be formula_26 formula_27 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 formula_28 Putting together these two results, a result analogous to Green's theorem for scalar fields is obtained, Theorem for vector fields: formula_29 The curl of a cross product can be written as formula_30  Green's vector identity can then be rewritten as formula_31 Since the divergence of a curl is zero, the third term vanishes to yield Green's vector identity: formula_32 With a similar procedure, the Laplacian of the dot product can be expressed in terms of the Laplacians of the factors formula_33 As a corollary, the awkward terms can now be written in terms of a divergence by comparison with the vector Green equation, formula_34 This result can be verified by expanding the divergence of a scalar times a vector on the RHS.