Green's vector identity

The second derivative of two vector functions is related to the divergence of the vector functions with first order operators. Namely, $\mathbf {P} \cdot \nabla ^{2}\mathbf {Q} -\mathbf {Q} \cdot \nabla ^{2}\mathbf {P} =\nabla \cdot \left[\mathbf {P} \left(\nabla \cdot \mathbf {Q} \right)-\mathbf {Q} \left(\nabla \cdot \mathbf {P} \right)+\mathbf {P} \times \nabla \times \mathbf {Q} -\mathbf {Q} \times \nabla \times \mathbf {P} \right]$.

Introduction

Green’s second identity establishes a relationship between second and (the divergence of) first order derivatives of two scalar functions

$p_{m}\nabla ^{2}q_{m}-q_{m}\nabla ^{2}p_{m}=\nabla \cdot \left(p_{m}\nabla q_{m}-q_{m}\nabla p_{m}\right),$

where $p_{m}$ and $q_{m}$ are two arbitrary 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 $\mathbf {P} \cdot \left(\nabla \times \nabla \times \mathbf {Q} \right)-\mathbf {Q} \cdot \left(\nabla \times \nabla \times \mathbf {P} \right)=\nabla \cdot \left(\mathbf {Q} \times \nabla \times \mathbf {P} -\mathbf {P} \times \nabla \times \mathbf {Q} \right)$. This equation can be written in terms of the Laplacians using the well known identity $\nabla \times \nabla \times \mathbf {Q} =\nabla \left(\nabla \cdot \mathbf {Q} \right)-\nabla ^{2}\mathbf {Q}$,

$\mathbf {P} \cdot \nabla ^{2}\mathbf {Q} -\mathbf {Q} \cdot \nabla ^{2}\mathbf {P} +\mathbf {Q} \cdot \left[\nabla \left(\nabla \cdot \mathbf {P} \right)\right]-\mathbf {P} \cdot \left[\nabla \left(\nabla \cdot \mathbf {Q} \right)\right]=\nabla \cdot \left(\mathbf {P} \times \nabla \times \mathbf {Q} -\mathbf {Q} \times \nabla \times \mathbf {P} \right).$

However, the terms $\mathbf {Q} \cdot \left[\nabla \left(\nabla \cdot \mathbf {P} \right)\right]-\mathbf {P} \cdot \left[\nabla \left(\nabla \cdot \mathbf {Q} \right)\right]$, could not be readily written in terms of a divergence. The other approach introduces bi-vectors, this formulation requires a dyadic Green function .

Divergence of two vector fields

Consider that the scalar fields in ([eq:green]) are the Cartesian components of vector fields, i.e. $\mathbf {P} =\sum \limits _{m}p_{m}{\hat {\mathbf {e} }}_{m}$ and $\mathbf {Q} =\sum \limits _{m}q_{m}{\hat {\mathbf {e} }}_{m}$. Each component $m$ obeys an equation of the form of ([eq:green]); Summing up these equations, we obtain

$\sum \limits _{m}\left[p_{m}\nabla ^{2}q_{m}-q_{m}\nabla ^{2}p_{m}\right]=\sum \limits _{m}\left[\nabla \cdot \left(p_{m}\nabla q_{m}-q_{m}\nabla p_{m}\right)\right].$

The LHS according to the definition of the dot product may be written in vector form as

$\sum \limits _{m}\left[p_{m}\nabla ^{2}q_{m}-q_{m}\nabla ^{2}p_{m}\right]=\mathbf {P} \cdot \nabla ^{2}\mathbf {Q} -\mathbf {Q} \cdot \nabla ^{2}\mathbf {P} .$

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. $\sum \limits _{m}\left[\nabla \cdot \left(p_{m}\nabla q_{m}-q_{m}\nabla p_{m}\right)\right]=\nabla \cdot \left(\sum \limits _{m}p_{m}\nabla q_{m}-\sum \limits _{m}q_{m}\nabla p_{m}\right)$. Recall the vector identity for the gradient of a dot product

$\nabla \left(\mathbf {P} \cdot \mathbf {Q} \right)=\left(\mathbf {P} \cdot \nabla \right)\mathbf {Q} +\left(\mathbf {Q} \cdot \nabla \right)\mathbf {P} +\mathbf {P} \times \nabla \times \mathbf {Q} +\mathbf {Q} \times \nabla \times \mathbf {P}$

which, written out in vector components is given by $\nabla \left(\mathbf {P} \cdot \mathbf {Q} \right)=\nabla \sum \limits _{m}p_{m}q_{m}=\sum \limits _{m}p_{m}\nabla q_{m}+\sum \limits _{m}q_{m}\nabla p_{m}.$ 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 $p_{m}$’s) or the other ($q_{m}$’s) , the contribution to each term must be

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

These results are rigorously proven to be correct in appendix [sec:Derivation-by-components] through evaluation of the vector components. Therefore, the RHS of eq. ([eq:comp vec green]) can be written in vector form as

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

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

$\mathbf {P} \cdot \nabla ^{2}\mathbf {Q} -\mathbf {Q} \cdot \nabla ^{2}\mathbf {P} =\nabla \cdot \left[\left(\mathbf {P} \cdot \nabla \right)\mathbf {Q} +\mathbf {P} \times \nabla \times \mathbf {Q} -\left(\mathbf {Q} \cdot \nabla \right)\mathbf {P} -\mathbf {Q} \times \nabla \times \mathbf {P} \right].$

Reassuringly, from the vector relationship ([eq:vec green]), we can go back to the scalar case as shown in appendix [sec:scalar-case]. The curl of a cross product can be written as $\nabla \times \left(\mathbf {P} \times \mathbf {Q} \right)=\left(\mathbf {Q} \cdot \nabla \right)\mathbf {P} -\left(\mathbf {P} \cdot \nabla \right)\mathbf {Q} +\mathbf {P} \left(\nabla \cdot \mathbf {Q} \right)-\mathbf {Q} \left(\nabla \cdot \mathbf {P} \right)$; Green’s vector identity can then be rewritten as

P²Q-Q²P=
[P(Q)-Q(P)-(PQ)+PQ-QP].

Since the divergence of a curl is zero, the third term vanishes and the identity can be written as

$\mathbf {P} \cdot \nabla ^{2}\mathbf {Q} -\mathbf {Q} \cdot \nabla ^{2}\mathbf {P} =\nabla \cdot \left[\mathbf {P} \left(\nabla \cdot \mathbf {Q} \right)-\mathbf {Q} \left(\nabla \cdot \mathbf {P} \right)+\mathbf {P} \times \nabla \times \mathbf {Q} -\mathbf {Q} \times \nabla \times \mathbf {P} \right].$

This result should prove useful when the divergence and curl of the fields can be established in terms of other quantities, as is the case in electromagnetism. There are several particular cases of interest of this expression: If the fields satisfy Helmholtz equation, the LHS of eq:vec green 2 is zero. Thus, a conserved quantity with zero divergence is obtained; If the fields are curl free so that they can be written in terms of the gradients of scalar functions $\alpha$ and $\beta$, expression eq:vec green 2 becomes

$\nabla \alpha \cdot \nabla ^{2}\left(\nabla \beta \right)-\nabla \beta \cdot \nabla ^{2}\left(\nabla \alpha \right)=\nabla \cdot \left[\nabla \alpha \left(\nabla ^{2}\beta \right)-\nabla \beta \left(\nabla ^{2}\alpha \right)\right].$

Another identity that may prove useful is obtained from the divergence of ([eq: grad prod]) $\nabla \cdot \nabla \left(\mathbf {P} \cdot \mathbf {Q} \right)=\nabla \cdot \left[\left(\mathbf {P} \cdot \nabla \right)\mathbf {Q} +\left(\mathbf {Q} \cdot \nabla \right)\mathbf {P} +\mathbf {P} \times \nabla \times \mathbf {Q} +\mathbf {Q} \times \nabla \times \mathbf {P} \right],$ invoking the Green’s vector identity ([eq:vec green]) derived above, the Laplacian of the dot product can be expressed in terms of the Laplacians of the factors

$\nabla ^{2}\left(\mathbf {P} \cdot \mathbf {Q} \right)=\mathbf {P} \cdot \nabla ^{2}\mathbf {Q} -\mathbf {Q} \cdot \nabla ^{2}\mathbf {P} +2\nabla \cdot \left[\left(\mathbf {Q} \cdot \nabla \right)\mathbf {P} +\mathbf {Q} \times \nabla \times \mathbf {P} \right].$

If the substitution of the vector identity ([eq:vec green]) is performed eliminating the terms $\left(\mathbf {Q} \cdot \nabla \right)\mathbf {P} +\mathbf {Q} \times \nabla \times \mathbf {P}$, the Laplacian of the dot product is

$\nabla ^{2}\left(\mathbf {P} \cdot \mathbf {Q} \right)=-\left(\mathbf {P} \cdot \nabla ^{2}\mathbf {Q} -\mathbf {Q} \cdot \nabla ^{2}\mathbf {P} \right)+2\nabla \cdot \left[\left(\mathbf {P} \cdot \nabla \right)\mathbf {Q} +\mathbf {P} \times \nabla \times \mathbf {Q} \right].$

Conclusions

Green’s second identity relating the Laplacians with the divergence has been derived for vector fields. No use of bi-vectors or dyadics has been made as in some previous approaches. In diffraction theory, the vector identity was stated before in terms of the curl. However, this earlier formulation had the drawback that the Laplacian could not be invoked without involving extra terms. As a corollary, the awkward terms in eq:vec difrac ident can now be written in terms of a divergence by comparison with eq:vec green 2

$\mathbf {P} \cdot \left[\nabla \left(\nabla \cdot \mathbf {Q} \right)\right]-\mathbf {Q} \cdot \left[\nabla \left(\nabla \cdot \mathbf {P} \right)\right]=\nabla \cdot \left[\mathbf {P} \left(\nabla \cdot \mathbf {Q} \right)-\mathbf {Q} \left(\nabla \cdot \mathbf {P} \right)\right].$

This result can be verified by expanding the divergence of a scalar times a vector on the RHS.

The condition imposed by Helmholtz equation $\nabla ^{2}\mathbf {P} =-k^{2}\mathbf {P}$ can be readily incorporated in the present formulation of Green’s second identity. This result is particularly useful if the vector fields satisfy the wave equation.

Derivation by components

In order to evaluate $\left(\mathbf {P} \cdot \nabla \right)\mathbf {Q} +\mathbf {P} \times \nabla \times \mathbf {Q}$ consider the first term in three dimensional Cartesian components $\left(\mathbf {P} \cdot \nabla \right)\mathbf {Q} =\left(p_{x}{\frac {\partial }{\partial x}}+p_{y}{\frac {\partial }{\partial y}}+p_{z}{\frac {\partial }{\partial z}}\right)\left(q_{x}{\hat {\mathbf {e} }}_{x}+q_{y}{\hat {\mathbf {e} }}_{y}+q_{z}{\hat {\mathbf {e} }}_{z}\right)$ that may be written as

$\left(\mathbf {P} \cdot \nabla \right)\mathbf {Q} =\left(p_{x}{\frac {\partial q_{x}}{\partial x}}+p_{y}{\frac {\partial q_{x}}{\partial y}}+p_{z}{\frac {\partial q_{x}}{\partial z}}\right){\hat {\mathbf {e} }}_{x}+\left(p_{x}{\frac {\partial q_{y}}{\partial x}}+p_{y}{\frac {\partial q_{y}}{\partial y}}+p_{z}{\frac {\partial q_{y}}{\partial z}}\right){\hat {\mathbf {e} }}_{y}+\left(p_{x}{\frac {\partial q_{z}}{\partial x}}+p_{y}{\frac {\partial q_{z}}{\partial y}}+p_{z}{\frac {\partial q_{z}}{\partial z}}\right){\hat {\mathbf {e} }}_{z}.$

The curl in the second term is $^{R}\mathbf {Q} =\nabla \times \mathbf {Q} =\left({\frac {\partial q_{z}}{\partial y}}-{\frac {\partial q_{y}}{\partial z}}\right){\hat {\mathbf {e} }}_{x}+\left({\frac {\partial q_{x}}{\partial z}}-{\frac {\partial q_{z}}{\partial x}}\right){\hat {\mathbf {e} }}_{y}+\left({\frac {\partial q_{y}}{\partial x}}-{\frac {\partial q_{x}}{\partial y}}\right){\hat {\mathbf {e} }}_{z}.$ The cross product is $\mathbf {P} \times {}^{R}\mathbf {Q} =\left(p_{y}{}^{R}q_{z}-p_{z}{}^{R}q_{y}\right){\hat {\mathbf {e} }}_{x}+\left(p_{z}{}^{R}q_{x}-p_{x}{}^{R}q_{z}\right){\hat {\mathbf {e} }}_{y}+\left(p_{x}{}^{R}q_{y}-p_{y}{}^{R}q_{x}\right){\hat {\mathbf {e} }}_{z};$ The second term is then

$\mathbf {P} \times \nabla \times \mathbf {Q} =\left(p_{y}\left({\frac {\partial q_{y}}{\partial x}}-{\frac {\partial q_{x}}{\partial y}}\right)-p_{z}\left({\frac {\partial q_{x}}{\partial z}}-{\frac {\partial q_{z}}{\partial x}}\right)\right){\hat {\mathbf {e} }}_{x}+\left(p_{z}\left({\frac {\partial q_{z}}{\partial y}}-{\frac {\partial q_{y}}{\partial z}}\right)-p_{x}\left({\frac {\partial q_{y}}{\partial x}}-{\frac {\partial q_{x}}{\partial y}}\right)\right){\hat {\mathbf {e} }}_{y}+\left(p_{x}\left({\frac {\partial q_{x}}{\partial z}}-{\frac {\partial q_{z}}{\partial x}}\right)-p_{y}\left({\frac {\partial q_{z}}{\partial y}}-{\frac {\partial q_{y}}{\partial z}}\right)\right){\hat {\mathbf {e} }}_{z}$

that expands to

$\mathbf {P} \times \nabla \times \mathbf {Q} =\left(p_{y}{\frac {\partial q_{y}}{\partial x}}-p_{y}{\frac {\partial q_{x}}{\partial y}}-p_{z}{\frac {\partial q_{x}}{\partial z}}+p_{z}{\frac {\partial q_{z}}{\partial x}}\right){\hat {\mathbf {e} }}_{x}+\left(p_{z}{\frac {\partial q_{z}}{\partial y}}-p_{z}{\frac {\partial q_{y}}{\partial z}}-p_{x}{\frac {\partial q_{y}}{\partial x}}+p_{x}{\frac {\partial q_{x}}{\partial y}}\right){\hat {\mathbf {e} }}_{y}+\left(p_{x}{\frac {\partial q_{x}}{\partial z}}-p_{x}{\frac {\partial q_{z}}{\partial x}}-p_{y}{\frac {\partial q_{z}}{\partial y}}+p_{y}{\frac {\partial q_{y}}{\partial z}}\right){\hat {\mathbf {e} }}_{z}.$

Evaluate $\left(\mathbf {P} \cdot \nabla \right)\mathbf {Q} +\mathbf {P} \times \nabla \times \mathbf {Q}$ in the x direction

$\left[\left(\mathbf {P} \cdot \nabla \right)\mathbf {Q} +\mathbf {P} \times \nabla \times \mathbf {Q} \right]_{x}=\left(p_{x}{\frac {\partial q_{x}}{\partial x}}+p_{y}{\frac {\partial q_{x}}{\partial y}}+p_{z}{\frac {\partial q_{x}}{\partial z}}+p_{y}{\frac {\partial q_{y}}{\partial x}}-p_{y}{\frac {\partial q_{x}}{\partial y}}-p_{z}{\frac {\partial q_{x}}{\partial z}}+p_{z}{\frac {\partial q_{z}}{\partial x}}\right){\hat {\mathbf {e} }}_{x},$

canceling out terms $\left[\left(\mathbf {P} \cdot \nabla \right)\mathbf {Q} +\mathbf {P} \times \nabla \times \mathbf {Q} \right]_{x}=\left(p_{x}{\frac {\partial q_{x}}{\partial x}}+p_{y}{\frac {\partial q_{y}}{\partial x}}+p_{z}{\frac {\partial q_{z}}{\partial x}}\right){\hat {\mathbf {e} }}_{x}.$

Analogous results are obtained in the other directions so that

$\left(\mathbf {P} \cdot \nabla \right)\mathbf {Q} +\mathbf {P} \times \nabla \times \mathbf {Q} =\left(p_{x}{\frac {\partial q_{x}}{\partial x}}+p_{y}{\frac {\partial q_{y}}{\partial x}}+p_{z}{\frac {\partial q_{z}}{\partial x}}\right){\hat {\mathbf {e} }}_{x}+\left(p_{x}{\frac {\partial q_{x}}{\partial y}}+p_{y}{\frac {\partial q_{y}}{\partial y}}+p_{z}{\frac {\partial q_{z}}{\partial y}}\right){\hat {\mathbf {e} }}_{y}+\left(p_{x}{\frac {\partial q_{x}}{\partial z}}+p_{y}{\frac {\partial q_{y}}{\partial z}}+p_{z}{\frac {\partial q_{z}}{\partial z}}\right){\hat {\mathbf {e} }}_{z}.$

that may be written out in vector form as $\left(\mathbf {P} \cdot \nabla \right)\mathbf {Q} +\mathbf {P} \times \nabla \times \mathbf {Q} =\mathbf {P} \cdot {\frac {\partial \mathbf {Q} }{\partial x}}+\mathbf {P} \cdot {\frac {\partial \mathbf {Q} }{\partial y}}+\mathbf {P} \cdot {\frac {\partial \mathbf {Q} }{\partial z}}.$ However, the terms can be rearranged as

$\left(\mathbf {P} \cdot \nabla \right)\mathbf {Q} +\mathbf {P} \times \nabla \times \mathbf {Q} =\left(p_{x}{\frac {\partial q_{x}}{\partial x}}{\hat {\mathbf {e} }}_{x}+p_{x}{\frac {\partial q_{x}}{\partial y}}{\hat {\mathbf {e} }}_{y}+p_{x}{\frac {\partial q_{x}}{\partial z}}{\hat {\mathbf {e} }}_{z}\right)+\left(p_{y}{\frac {\partial q_{y}}{\partial x}}{\hat {\mathbf {e} }}_{x}+p_{y}{\frac {\partial q_{y}}{\partial y}}{\hat {\mathbf {e} }}_{y}+p_{y}{\frac {\partial q_{y}}{\partial z}}{\hat {\mathbf {e} }}_{z}\right)+\left(p_{z}{\frac {\partial q_{z}}{\partial x}}{\hat {\mathbf {e} }}_{x}+p_{z}{\frac {\partial q_{z}}{\partial y}}{\hat {\mathbf {e} }}_{y}+p_{z}{\frac {\partial q_{z}}{\partial z}}{\hat {\mathbf {e} }}_{z}\right),$

and thus $\left(\mathbf {P} \cdot \nabla \right)\mathbf {Q} +\mathbf {P} \times \nabla \times \mathbf {Q} =\left(p_{x}\nabla q_{x}+p_{y}\nabla q_{y}+p_{z}\nabla q_{z}\right).$ An equivalent procedure for $\left(\mathbf {Q} \cdot \nabla \right)\mathbf {P} +\mathbf {Q} \times \nabla \times \mathbf {P}$ gives $\left(\mathbf {Q} \cdot \nabla \right)\mathbf {P} +\mathbf {Q} \times \nabla \times \mathbf {P} =q_{x}\nabla p_{x}+q_{y}\nabla p_{y}+q_{z}\nabla p_{z}.$

[sec:scalar-case]scalar case

If we take one component vectors, for example, $\mathbf {P} \rightarrow p_{x}{\hat {\mathbf {e} }}_{x},\mathbf {Q} \rightarrow q_{x}{\hat {\mathbf {e} }}_{x}$, the vector relationship ([eq:vec green]) becomes $p_{x}\nabla ^{2}q_{x}-q_{x}\nabla ^{2}p_{x}=\nabla \cdot \left[p_{x}{\frac {\partial q_{x}}{\partial x}}{\hat {\mathbf {e} }}_{x}-q_{x}{\frac {\partial p_{x}}{\partial x}}{\hat {\mathbf {e} }}_{x}+\mathbf {P} \times \nabla \times \mathbf {Q} -\mathbf {Q} \times \nabla \times \mathbf {P} \right].$ Since $\nabla \times \mathbf {Q} ={\frac {\partial q_{x}}{\partial z}}{\hat {\mathbf {e} }}_{y}-{\frac {\partial q_{x}}{\partial y}}{\hat {\mathbf {e} }}_{z}$,

$\mathbf {P} \times \nabla \times \mathbf {Q} =\left(p_{x}{\frac {\partial q_{x}}{\partial y}}\right){\hat {\mathbf {e} }}_{y}+\left(p_{x}{\frac {\partial q_{x}}{\partial z}}\right){\hat {\mathbf {e} }}_{z}$

and $\mathbf {Q} \times \nabla \times \mathbf {P} =\left(q_{x}{\frac {\partial p_{x}}{\partial y}}\right){\hat {\mathbf {e} }}_{y}+\left(q_{x}{\frac {\partial p_{x}}{\partial z}}\right){\hat {\mathbf {e} }}_{z}$. Therefore $p_{x}\nabla ^{2}q_{x}-q_{x}\nabla ^{2}p_{x}=\nabla \cdot \left[p_{x}\nabla q_{x}-q_{x}\nabla p_{x}\right],$ we recover Green’s second identity for the functions $p_{x},\,q_{x}$.

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