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 for a detailed argument, with an alternative. 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 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 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