Since the divergence of a curl is zero, the third term vanishes and the identity can be written as
With a similar porcedure, the Laplacian of the dot product can be expressed in terms of the Laplacians of the factors
corollary
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 .
- ↑ Fernandezguasti04a M. Fernández-Guasti. Complementary fields conservation equation derived from the scalar wave equation. J. Phys. A: Math. Gen., 37:4107–4121, 2004.
- ↑ love1901 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.
- ↑ stratton1939 J. A. Stratton and L. J. Chu. Diffraction Theory of Electromagnetic Waves. Phys. Rev., 56(1):99–107, Jul 1939.
- ↑ bruce2010 N. C. Bruce. Double scatter vector-wave Kirchhoff scattering from perfectly conducting surfaces with infinite slopes. Journal of Optics, 12(8):085701, 2010.
- ↑ franz1950 W Franz. On the Theory of Diffraction. Proceedings of the Physical Society. Section A, 63(9):925, 1950.
- ↑ tai1972 Chen-To Tai. Kirchhoff theory: Scalar, vector, or dyadic? Antennas and Propagation, IEEE Transactions on, 20(1):114–115, jan 1972.
- ↑ M. Fernández-Guasti. Green's second identity for vector fields. ISRN Mathematical Physics, 2012:7, 2012. Article ID: 973968. [1]