Research Article Green s Second Identity for Vector Fields

Transcription

1 International Scholarl Research Network ISRN Matheatical Phsics Volue 2012, Article ID , 7 pages doi: /2012/ Research Article Green s Second Identit for Vector Fields M. Fernández-Guasti Laboratorio de Óptica Cuántica,Departaentode Física, Universidad Autonoa Metropolitana-Iztapalapa, Apartado Postal , México, DF, Mexico Correspondence should be addressed to M. Fernández-Guasti, fg@xanu.ua.x Received 2 Ma 2012; Accepted 20 June 2012 Acadeic Editors: U. Kulshreshtha, P. Ro, and D. Singleton Copright q 2012 M. Fernández-Guasti. This is an open access article distributed under the Creative Coons Attribution License, which perits unrestricted use, distribution, and reproduction in an ediu, provided the original work is properl cited. The second derivative of two vector functions is related to the divergence of the vector functions with first order operators. Nael, P 2 Q Q 2 P P Q Q PP Q Q P. 1. Introduction Green s second identit establishes a relationship between second and the divergence of first order derivatives of two scalar functions p 2 q q 2 p p q q p, 1.1 where p and q are two arbitrar scalar fields. This identit is of great iportance in phsics because continuit equations can thus be established for scalar fields such as ass or energ. It has been called forth to obtain a scalar wave energ densit 1. It is also invoked in the classical 2, 3 as well as the quantu 4, 5 tie-dependent haronic oscillator in order to obtain an exact invariant 6. In optics, it is also used to derive the integral theore of Kirchhoff in scalar diffraction theor. Although the second Green s identit is alwas presented in vector analsis, onl a scalar version is found on textbooks. Even in the specialized literature, a vector version is not easil found. In vector diffraction theor, two versions of Green s second identit are introduced. One variant invokes the divergence of a cross product 7 9 and states a relationship in ters of the curl-curl of the field P Q Q P Q P P Q. This equation can be written in ters of the Laplacians using the well-known identit Q Q 2 Q, P 2 Q Q 2 P Q P P Q P Q Q P. 1.2

2 2 ISRN Matheatical Phsics However, the ters Q P P Q could not be readil written in ters of a divergence. The other approach introduces bivectors; this forulation requires a dadic Green function 10, 11. It is the purpose of this counication to establish an equivalent Green s identit for vector fields involving the Laplacians of vector functions written out in ters of the divergence operator. 2. Divergence of Two Vector Fields Consider that the scalar fields in 1.1 are the Cartesian coponents of vector fields, that is, P p ê and Q q ê. Each coponent obes an equation of the for of 1.1. Suing up these equations, we obtain ] [p 2 q q 2 p [ p q q p ]. 2.1 The LHS according to the definition of the dot product a be written in vector for as ] [p 2 q q 2 p P 2 Q Q 2 P. 2.2 The RHS is a bit ore awkward to express in ters of vector operators. Due to the distributivit of the divergence operator over addition, the su of the divergence is equal to the divergence of the su, that is, p q q p p q q p. Recall the vector identit for the gradient of a dot product 12 P Q P Q Q P P Q Q P 2.3 which, written out in vector coponents, is given b P Q p q p q q p. 2.4 This result is siilar to what we wish to evince in vector ters except for the inus sign. Since the differential operators in each ter of 2.3 act either over one vector sa p s or the other q s, the contribution to each ter ust be p q P QP Q, q p Q PQ P. 2.5

3 ISRN Matheatical Phsics 3 These results are rigorousl proven to be correct in Appendix A through evaluation of the vector coponents. Therefore, the RHS of 2.1 can be written in vector for as p q q p P QP Q Q P Q P. 2.6 Putting together these two results, a theore for vector fields analogous to Green s theore for scalar fields is obtained P 2 Q Q 2 P P Q P Q Q P Q P. 2.7 Reassuringl, fro 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 P Q Q P P Q P Q Q P; Green s vector identit can then be rewritten as P 2 Q Q 2 P P Q Q P P Q P Q Q P. 2.8 Since the divergence of a curl is zero, the third ter vanishes and the identit can be written as P 2 Q Q 2 P P Q Q P P Q Q P. 2.9 This result should prove useful when the divergence and curl of the fields can be established in ters of other quantities, as is the case in electroagnetis. There are several particular cases of interest of this expression: if the fields satisf Helholtz equation, the LHS of 2.9 is zero. Thus, a conserved quantit with zero divergence is obtained; if the fields are curl-free so that the can be written in ters of the gradients of scalar functions α and β, expression 2.9 becoes α 2 β [ ] β 2 α α 2 β β 2 α Another identit that a prove useful is obtained fro the divergence of 2.3 P Q P Q Q P P Q Q P, 2.11 invoking the Green s vector identit 2.7 derived above; the Laplacian of the dot product can be expressed in ters of the Laplacians of the factors 2 P Q P 2 Q Q 2 P 2 Q P Q P If the substitution of the vector identit 2.7 is perfored eliinating the ters Q P Q P, the Laplacian of the dot product is 2 P Q P 2 Q Q 2 P 2 P QP Q. 2.13

4 4 ISRN Matheatical Phsics 3. Conclusions Green s second identit relating the Laplacians with the divergence has been derived for vector fields. No use of bivectors or dadics has been ade as in soe previous approaches. In diffraction theor, the vector identit was stated before in ters of the curl. However, this earlier forulation had the drawback that the Laplacian could not be invoked without involving extra ters. As a corollar, the awkward ters in 1.2 can now be written in ters of a divergence b coparison with 2.9 P Q Q P P Q Q P. 3.1 This result can be verified b expanding the divergence of a vector ties a scalar for the two addends on the RHS. The condition iposed b Helholtz equation 2 P k 2 P can be readil incorporated in the present forulation of Green s second identit. This result is particularl useful if the vector fields satisf the wave equation. Appendices A. Derivation b Coponents In order to evaluate P Q P Q, A.1 consider the first ter in three-diensional Cartesian coponents P Q p p qx z ê x q ê q z ê z A.2 that a be written as P Q p q p p z q p z ê x q ê p p z ê z. A.3 The curl in the second ter is R qz Q Q q qx ê x q z q ê q x ê z. A.4

5 ISRN Matheatical Phsics 5 The cross product is P R Q p R q z pz R q ê x pz R q x px R q z ê px R q p R q x ê z. A.5 The second ter is then q P Q p q x qz p z q qx p z q z ê x q q x ê qx q z qz p q ê z A.6 that expands to P Q p q p p z p z p z p z q p q x p p ê x ê q ê z. A.7 Evaluate P Q P Q in the x direction P Q P Q x p p z p q p p z p z ê x, A.8 canceling out ters P Q P Q x p q p z ê x. A.9

6 6 ISRN Matheatical Phsics Analogous results are obtained in the other directions so that P Q P Q p p p q p z q p z q p z ê x ê ê z A.10 that a be written out in vector for as P Q P Q P Q P Q P Q. A.11 However, the ters can be rearranged as P Q P Q êx ê êz q q q p êx p ê p êz p z êx p z ê p z êz, A.12 and thus P Q P Q q x p q p z q z. A.13 An equivalent procedure for Q P Q P gives Q P Q P q x q p q z p z. A.14 B. Scalar Case If we take one coponent vectors, for exaple, P ê x, Q q x ê x, the vector relationship 2.7 becoes [ ] 2 q x q x 2 êx q x êx P Q Q P. B.1

7 ISRN Matheatical Phsics 7 Since Q /ê / ê z, P Q ê ê z B.2 and Q P q x / ê q x /ê z. Therefore, 2 q x q x 2 [p x q x q x ], B.3 and we recover Green s second identit for the functions,q x. Acknowledgent I a grateful to A. Caacho Quintana and the referees for useful suggestions for iproving this paper. References 1 M. Fernández Guasti, Copleentar fields conservation equation derived fro the scalar wave equation, Phsics A, vol. 37, no. 13, pp , R. K. Colegrave and M. A. Mannan, Invariants for the tie-dependent haronic oscillator, Journal of Matheatical Phsics, vol. 29, no. 7, pp , M. Fernández Guasti and A. Gil-Villegas, Orthogonal functions invariant for the tie-dependent haronic oscillator, Phsics Letters A, vol. 292, no. 4-5, pp , M. Fernández Guasti and H. Moa-Cessa, Aplitude and phase representation of quantu invariants for the tie dependent haronic oscillator, Phsical Review A, vol. 67, Article ID , pp. 1 5, I. A. Pedrosa and I. Guedes, Quantu states of a generalized tie-dependent inverted haronic oscillator, International Modern Phsics B, vol. 18, no. 9, pp , H. R. Lewis, Classical and quantu sstes with tie-dependent haronic-oscillator-tpe hailtonians, Phsical Review Letters, vol. 18, no. 13, pp , A. E. H. Love, The integration of the equations of propagation of electric waves, Philosophical Transactions of the Roal Societ of London A, vol. 197, pp. 1 45, J. A. Stratton and L. J. Chu, Diffraction theor of electroagnetic waves, Phsical Review, vol. 56, no. 1, pp , N. C. Bruce, Double scatter vector-wave Kirchhoff scattering fro perfectl conducting surfaces with infinite slopes, Optics, vol. 12, no. 8, Article ID , W. Franz, On the theor of diffraction, Proceedings of the Phsical Societ A, vol. 63, no. 9, p. 925, C.-T. Tai, Kirchhoff theor: scalar, vector, or dadic? IEEE Transactions on Antennas and Propagation, vol. 20, no. 1, pp , G. Arfken, Matheatical Methods for Phsicists, Acadeic Press, New York, NY, USA, 1966.

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