Academic Editor: P. Roy, D. Singleton, U. Kulshreshtha
Received02 May 2012
Accepted20 Jun 2012
Published07 Aug 2012
Abstract
The second derivative of two vector functions is related to the divergence of the vector functions with first order operators. Namely, .
1. Introduction
Greenβs 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. It has been called forth to obtain a scalar wave energy density [1]. It is also invoked in the classical [2, 3] as well as the quantum [4, 5] time-dependent harmonic oscillator in order to obtain an exact invariant [6]. In optics, it is also used to derive the integral theorem of Kirchhoff in scalar diffraction theory.
Although the second Greenβs 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 Greenβs second identity are introduced. One variant invokes the divergence of a cross product [7β9] and states a relationship in terms of the curl-curl of the field . This equation can be written in terms of the Laplacians using the well-known identity ,
However, the terms could not be readily written in terms of a divergence. The other approach introduces bivectors; this formulation requires a dyadic Green function [10, 11]. It is the purpose of this communication to establish an equivalent Greenβs identity for vector fields involving the Laplacians of vector functions written out in terms of the divergence operator.
2. Divergence of Two Vector Fields
Consider that the scalar fields in (1.1) are the Cartesian components of vector fields, that is, and . Each component obeys an equation of the form of (1.1). Summing up these equations, 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, that is, . Recall the vector identity for the gradient of a dot product [12]
which, written out in vector components, is given by
This result is similar to what we wish to evince in vector terms βexceptβ for the minus sign. Since the differential operators in each term of (2.3) act either over one vector (say βs) or the other (βs), the contribution to each term must be
These results are rigorously proven to be correct in Appendix A through evaluation of the vector components. Therefore, the RHS of (2.1) can be written in vector form as
Putting together these two results, a theorem for vector fields analogous to Greenβs theorem for scalar fields is obtained
Reassuringly, from the vector relationship (2.7), we can go back to the scalar case as shown in Appendix B. The curl of a cross product can be written as ; Greenβs vector identity can then be rewritten as
Since the divergence of a curl is zero, the third term vanishes and the identity can be written as
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 (2.9) 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 and , expression (2.9) becomes
Another identity that may prove useful is obtained from the divergence of (2.3)
invoking the Greenβs vector identity (2.7) derived above; the Laplacian of the dot product can be expressed in terms of the Laplacians of the factors
If the substitution of the vector identity (2.7) is performed eliminating the terms , the Laplacian of the dot product is
3. Conclusions
Greenβs second identity relating the Laplacians with the divergence has been derived for vector fields. No use of bivectors 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 (1.2) can now be written in terms of a divergence by comparison with (2.9)
This result can be verified by expanding the divergence of a vector times a scalar for the two addends on the RHS.
The condition imposed by Helmholtz equation can be readily incorporated in the present formulation of Greenβs second identity. This result is particularly useful if the vector fields satisfy the wave equation.
Appendices
A. 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 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
B. Scalar Case
If we take one component vectors, for example, , the vector relationship (2.7) becomes
Since ,
and . Therefore,
and we recover Greenβs second identity for the functions .
Acknowledgment
I am grateful to A. Camacho Quintana and the referees for useful suggestions for improving this paper.
References
M. Fernández Guasti, βComplementary fields conservation equation derived from the scalar wave equation,β Journal of Physics A, vol. 37, no. 13, pp. 4107β4121, 2004.
R. K. Colegrave and M. A. Mannan, βInvariants for the time-dependent harmonic oscillator,β Journal of Mathematical Physics, vol. 29, no. 7, pp. 1580β1587, 1988.
M. Fernández Guasti and A. Gil-Villegas, βOrthogonal functions invariant for the time-dependent harmonic oscillator,β Physics Letters A, vol. 292, no. 4-5, pp. 243β245, 2002.
M. Fernández Guasti and H. Moya-Cessa, βAmplitude and phase representation of quantum invariants for the time dependent harmonic oscillator,β Physical Review A, vol. 67, Article ID 063803, pp. 1β5, 2003.
I. A. Pedrosa and I. Guedes, βQuantum states of a generalized time-dependent inverted harmonic oscillator,β International Journal of Modern Physics B, vol. 18, no. 9, pp. 1379β1385, 2004.
H. R. Lewis, βClassical and quantum systems with time-dependent harmonic-oscillator-type hamiltonians,β Physical Review Letters, vol. 18, no. 13, pp. 510β512, 1967.
A. E. H. Love, βThe integration of the equations of propagation of electric waves,β Philosophical Transactions of the Royal Society of London A, vol. 197, pp. 1β45, 1901.
N. C. Bruce, βDouble scatter vector-wave Kirchhoff scattering from perfectly conducting surfaces with infinite slopes,β Journal of Optics, vol. 12, no. 8, Article ID 085701, 2010.