Green's Identities - Green's Vector Identity

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. Although the second Green’s identity is always presented in vector analysis, only a scalar version is found on textbooks. Even in the specialized literature, a vector version is not easily found. 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. and . 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

$\sum\limits _{m}p_{m}\nabla q_{m} = \left(\mathbf{P}\cdot\nabla\right)\mathbf{Q}+\mathbf{P}\times\nabla\times\mathbf{Q},$

These results can be rigorously proven to be correct through evaluation fo the vector components. Therefore, the RHS can be written in vector form as

Putting together these two results, a theorem for vector fields analogous to Green’s theorem for scalar fields is obtained

$\color{OliveGreen} \mathbf{P}\cdot\nabla^{2}\mathbf{Q}-\mathbf{Q}\cdot\nabla^{2}\mathbf{P}=\nabla\cdot\left.$

The curl of a cross product can be written as ; Green’s vector identity can then be rewritten as

$\mathbf{P}\cdot\nabla^{2}\mathbf{Q}-\mathbf{Q}\cdot\nabla^{2}\mathbf{P}= \nabla\cdot\left.$

Since the divergence of a curl is zero, the third term vanishes and Green’s vector identity is

With a similar porcedure, 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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