Green's vector identity

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The second derivative of two vector functions is related to the divergence of the vector functions with first order operators. Namely,

see also wikipedia: Green's vector identity

Introduction [1]

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

where and 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 [2].

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 [3][4][5]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 [6][7].

Divergence of two vector fields

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

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

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

Reassuringly, from the vector relationship we can go back to the scalar case as shown in the scalar limit. 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 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 in the introduction 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.

derivation by components

In order to evaluate consider the first term in three dimensional Cartesian components that may be written as

The curl in the second term is The cross product is The second term is then

that expands to

Evaluate in the x direction

canceling out terms

Analogous results are obtained in the other directions so that

that may be written out in vector form as However, the terms can be rearranged as

and thus An equivalent procedure for gives

scalar case

If we take one component vectors, for example, , the vector relationship ([eq:vec green]) becomes Since ,

and . Therefore we recover Green’s second identity for the functions .

  1. M. Fernández-Guasti. Green's second identity for vector fields. ISRN Mathematical Physics, 2012:7, 2012. Article ID: 973968. [1]
  2. M. Fernández-Guasti. Complementary fields conservation equation derived from the scalar wave equation. J. Phys. A: Math. Gen., 37:4107–4121, 2004.
  3. A. E. H. Love. The Integration of the Equations of Propagation of Electric Waves. Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character, 197:pp. 1–45, 1901.
  4. J. A. Stratton and L. J. Chu. Diffraction Theory of Electromagnetic Waves. Phys. Rev., 56(1):99–107, Jul 1939.
  5. N. C. Bruce. Double scatter vector-wave Kirchhoff scattering from perfectly conducting surfaces with infinite slopes. Journal of Optics, 12(8):085701, 2010.
  6. W. Franz, On the Theory of Diffraction. Proceedings of the Physical Society. Section A, 63(9):925, 1950.
  7. Chen-To Tai. Kirchhoff theory: Scalar, vector, or dyadic? Antennas and Propagation, IEEE Transactions on, 20(1):114–115, jan 1972.