Dyadic Green s Function

Transcription

1 EECS 730 Winter 2009 c K. Sarabandi Dyadic Green Function A mentioned earlier the application of dyadic analyi facilitate imple manipulation of field ector calculation. The ource of electromagnetic field i the electric current which i a ector quantity. On the other hand mall-ignal electromagnetic field atify the linearity condition and therefore the behaior of the field can be decribed in term of the ytem impule repone. Since both the input (excitation current) and the output (field quantitie) of the ytem are ector quantitie, the impule repone of the ytem mut be a dyadic quantity. In what follow the deriation of dyadic Green function (impule repone for free pace) i preented. Then the Fourier repreentation of the Green function i deried which expree the field of an infiniteimal current ource in term of a continuou pectrum of plane wae. Thi form of the dyadic Green function i ueful for further deelopment of dyadic Green function for more complicated media uch a a dielectric half-pace medium or a tratified (multi-layer) dielectric medium. Conider an arbitrary time-harmonic electric current ditribution J e in an unbounded homogeneou medium with permittiity ǫ and permeability µ. Starting from the Maxwell equation, the ector wae equation for the electric field can be obtained and i gien by: E(r) k 2 E(r) iωµj e (r) () where J e (r) i the impreed olumetric current ditribution. A hown preiouly the electric field i uually calculated indirectly from the electric Hertz potential and i gien by: where E(r) ( k 2 + ) Π e (r) (2) Π e (r) iz k J e (r )g(r, r )d (3) The electric field expreion gien by (2) i alid for all r in thi medium including ource point. Here g(r, r ) e ik r r 4π r r

2 i the calar Green function atifying the calar wae equation ( 2 + k 2) g(r, r ) δ(r r ) (4) Let u now conider an infiniteimal current ource along ˆx direction gien by J e (r) ˆx iωµ δ(r r ) ˆx ikz δ(r r ) (5) According to (3) and (2) the reulting electric field can be obtained from G x (r, r ) ( + k 2 ) g(r, r )ˆx where G x (r, r ) denote the impule repone to an x-directed excitation. In a imilar manner the electric field in repone to infiniteimal y-directed and z-directed current are gien by G y (r, r ) ( + ) g(r, r )ŷ k 2 G z (r, r ) ( + ) g(r, r )ẑ k 2 Uing the compact dyadic notation, the electric field due to an arbitrary oriented ˆp (along ˆp) infiniteimal current δ(r ikz r ) can be obtained from: E p (r, r ) G (r, r ) ˆp where G (r, r ) G x (r, r )ˆx + G y (r, r )ŷ + G z (r, r )ẑ i referred to a the dyadic Green function of free-pace. The explicit expreion for G (r, r ) i gien by G (r, r ) ( + ) g(r, r )(ˆxˆx + ŷŷ + ẑẑ) k ( 2 + ) g(r, r ) k 2 I where I i the unit dyad (idemfactor). Noting that ( ψ I ) ψ I ψ for any differentiable calar function ψ, the expreion for the dyadic Green function i gien by 2

3 ( I G (r, r ) + ) k g(r, r ) (6) 2 Referring to () each ector component of G (r, r ) ( G q (r, r ); q x, y, z ) atifie G q (r, r ) k 2 G q (r, r ) ˆqδ(r r ) (7) By juxtapoing a unit ector ˆx, ŷ, or ẑ at the poterior poition of the three ector equation gien by (7) and umming thee equation, we obtain G (r, r ) k 2 G (r, r ) I δ(r r ) (8) Deriation of Field Quantitie From the Dyadic Green Function Conider a homogeneou medium bounded by a cloed urface S which include an arbitrary electric current ditribution J e (r). Uing the ector wae equation () and (8) in conjunction with the ector-dyadic Green theorem gien by P Q ( P) ] Q d (9) (ˆn P) Q +(ˆn P) Q ] d. an explicit expreion for the electric field due to the impreed electric current can be obtained. By letting P E(r) and Q G (r, r ) it can eaily be hown that E(r ) ikz J e (r) G (r, r )d (0) (ˆn E(r)) G (r, r ) + (ˆn E(r)) G ] (r, r ) d. Noting that E(r) ikzh(r), (0) can be written a E(r ) ikz J e (r) G (r, r )d () ikz(n H(r)) G (r, r ) + (ˆn E(r)) G ] (r, r ) d 3

4 To find an expreion for the magnetic field, we tart with the ector wae equation for the magnetic field gien by H(r) k 2 H(r) J e (r) (2) Again by letting P H(r) and Q G (r, r ) in (9), we obtain H(r ) Je (r) ] G (r, r )d (3) (ˆn H(r)) G (r, r ) + (ˆn H(r)) G ] (r, r ) d Applying the dyadic identity (a b) a b a b, The olume integral in (3) can be written a Je (r) ] G (r, r )d { J e (r) G ] (r, r ) + J e (r) G } (r, r ) d Uing the diergence theorem J e (r) G ] (r, r ) d ˆn J e (r) G ] (r, r ) d ˆn J e (r) ] G (r, r ) d and Maxwell equation H(r) iky E(r) + J e (r) in (3), the expreion for the magnetic field reduce to H(r ) + J e (r) G (r, r )d (4) { iky n E(r) ] G (r, r ) (ˆn H(r)) G } (r, r ) d 4

5 2 Field Quantitie Generated from Magnetic and Electric Current Equation () and (4) proide the electric and magnetic field quantitie in a bounded region originated from an electric current ditribution and a certain urface field quantitie at the urface of thi bounded region. In thi ection thee reult are extended to allow for the exitence of both electric and magnetic current. Thi can eaily be done by firt obtaining the field expreion uing a magnetic current ditribution a the excitation. The duality relation can be employed to find the field quantitie for a magnetic current excitation from thoe gien by () and (4). We firt point out that the magnetic dyadic Green function for an unbounded homogeneou medium i the ame a the electric one. Apply the duality relation to () and (4) the following expreion are obtained H m (r ) iky J m (r) G (r, r )d (5) iky (ˆn E m (r)) G (r, r ) + (ˆn H m (r)) ] G (r, r ) d E m (r ) J m (r) G (r, r ) d (6) ikz(ˆn H m (r)) G (r, r ) + (ˆn E m (r)) G ] (r, r ) d Superpoition of () and (6) and (4) and (5) proide the total field within S and are gien by E(r ) H(r ) + ikzj e (r) G (r, r ) J m (r) G ] (r, r ) d (7) ikz(ˆn H(r )) G (r, r ) + (ˆn E(r )) G ] (r, r ) d J e (r) G (r, r ) + iky J m (r) G ] (r, r ) d (8) iky (ˆn E(r)) G (r, r ) (ˆn H(r)) G ] (r, r ) d 3 Radiation Condition For Dyadic Green Function The contribution from the urface integral of (3) and (4) hould anih a the urface approache infinity according to the radiation condition firt potulated by Sommerfeld. 5

6 The electric field far away from the ource and oberation point atifie lim r { E(r) ikˆr E(r) } 0 (9) r The magnetic field alo atifie an identical equation. Uing (9) for G x (r, r ), G y (r, r ), and G z (r, r ) and by juxtapoing unit ector ˆx, ŷ, and ẑ at the poterior poition of each equation repectiely and then adding the three reulting equation we get { lim r G (r, r ) ikˆr } G (r, r ) 0 (20) r which i known a the radiation condition for the free-pace dyadic Green function. 4 Explicit Form of The Dyadic Green Function The compact form of the dyadic Green function which i gien by I G (r, r ) + ] e ik r r k 2 4π r r can be expreed in any deired coordinate ytem. For example, in Carteian coordinate ytem, where x ˆx + yŷ + zẑ, the dyadic Green function can be repreented, in matrix form, in the following manner G (r, r ) k x 2 x y x z k y x y 2 y z 2 z x 2 z y k z 2 e ik r r 4πk 2 r r (2) It i quite obiou from (2) that G (r, r ) i a ymmetric dyad, i.e. Therefore, for any ector V, we hae: G ] T G (r, r ) (r, r ) (22) V G (r, r ) G (r, r ) V 6

7 Alo noting that 0, G (r, r ) can eaily be ealuated a follow G (r, r ) g(r, r ) I ( I + ) ] I k g(r, r ) g(r, r )] 2 which in Carteian coordinate ytem take the following form G (r, r ) y 0 z 0 z y x 0 x eik r r 4π r r (23) which i obiouly anti-ymmetric (any dyad of the form C I i anti-ymmetric). Another expanded form of G (r, r ) can be obtained by noting that g(r) d dr g(r) R (ik R )g(r) R ( ik ) g(r) R ˆR, where R r r and Hence, ˆR r r r r. ( g(r) ik ) ] ( g(r) ˆR + ik ) g(r) R R ˆR, (24) ˆR can be calculated eaily noting that, But (R) I and therefore ( ˆR) ( ) R (R) ( ) R R + R R ( ˆR) ( I ) ˆR ˆR R After ome algebraic manipulation it can be hown that 7

8 {( 3 G (r, r ) k 2 R 3i ) ( 2 kr ˆR ˆR + + i kr ) I } g(r) (25) k 2 R 2 5 Far Field Expreion of Dyadic Green Function In the ealuation of field away from the ource where r r i much larger than typical dimenion of the ource, imple expreion for the field quantitie are uually obtained. Keeping only the term of the order of, the far-field expreion for the free-pace R r dyadic Green function can be obtained from (25) and i gien by G (r, r ) I e ik r r ˆrˆr] 4πr I e ikr ˆrˆr] 4πr e ikˆr r (26) Equation (26) indicate that the field quantitie do not poe a radial component. 6 Fourier Repreentation of The Free-Space Dyadic Green Function Another ueful repreentation of the dyadic Green function i it Fourier Tranform where the field repone to an impule excitation i expreed in term of a continuou pectrum (angular) of plane wae. Thi expanion in term of plane wae i ueful ince the cattering olution of many problem to plane wae excitation i known. Uing the plane wae olution together with the uperpoition principle, the olution to any arbitrary ource can be obtained. The tarting point i equation (4). Let u aume, without lo of generality, that the ource point i at the origin. Then ( ) 2 x y z + 2 k2 0 g(r) δ(r) (27) The Fourier tranform of g(r), repreented by g(k), i gien by + g(k) g(r)e i(kxx+kyy+kzz) dxdy dz 8

9 Conerely g(r) in term of it Fourier tranform i obtained from g(r) + (2π) 3 g(k)e i(kxx+kyy+kzz) dk x dk y dk z (28) Subtituting (28) in (27) and noting that δ(r) + (2π) 3 e i(kxx+kyy+kzz) dk x dk y dk z g(k) can be ealuated and i gien by g(k) k 2 x + k2 y + k2 z k2 o (29) Although the 3-dimenional Fourier tranform can be ued to expre the field quantitie in term of plane wae ( e ik r), it i not uually ued becaue all three component of the propagation ector are independent, that i, the frequencie of thee plane wae are not necearily the ame. To contrain the propagation ector k the integration with repect to one of the ariable mut be carried out. We conider integration of (28) oer k z, that i I(z) + 2π k 2 z h2eikzz dk z ; h 2 k 2 o k 2 x k 2 y (30) Conidering the behaior of g(r), we expect that I(z) approache zero at z ±. Thi i jutifiable if we let h to be complex with Imh] > 0. Such aumption i common and correpond to a lightly loy media. After ealuation of the integral, the lole condition i retored by allowing Imh] 0. With thi aumption the location of the pole of the integrand (30) are hown in Figure. The contour of integration i aumed to be along the real axi. For z 0 the contour can be cloed in the upper half-plane with a emi-circle of a large radiu (R ) noting that Imk z ] > 0 (a radiation condition requirement). In thi cae the integrand along the emi-circle contour i zero. For z 0, the contour can be cloed in the lower half-plane. Uing Cauchy reidue theorem to the contour integral, I(z) can eaily be ealuated and i gien by I(z) i { e ihz z 0 2h e ihz z 0 i 2h eih z where h k 2 o k2 x k2 y. Replacing h with k z and keeping in mind that k z i no longer an independent parameter, (28) take the following form 9

10 Im k z ] Path of integration for z > 0 k z h k z h Re k z ] Figure : k z -plane and the location of the pole of integrand (26). where g(r) k r i (2π) 2 + k xˆx + k y ŷ xˆx + yŷ k z k 2 k 2 ρ, Imk z] > 0 e +ik r +ik z z dk 2k z (3) k 2 ρ k 2 x + k 2 y To find G (r), deriatie of g(r) mut be ealuated, howeer, it hould be noted that the deriatie of g(r) with repect to z i dicontinuou, that i, z g(r) (2π) 2 where f(z) i a tep function + 2 eik r +ik z z dk f(z) f(z) { z > 0 z < 0 Further differentiation with repect to z will gie a dirac delta function 2 z 2g(r) (2π) 2 + e ik r dk δ(z) 0

11 i (2π) 2 + e ik r +ik k z z z 2 dk f2 (z) But f 2 (z) and (2π) 2 + e ik r dk δ(x)δ(y) and therefore 2 z2g(r) δ(r) i (2π) 2 + k z 2 ei(k r +k z z ) dk x dk y (32) Subtituting (3) in (6) and uing (32) a imple expreion for G (r) can be obtained. Interchanging the order of differentiation and integration and noting that, and x y can be replaced with ik z z, ik y and ±ik z ( + ign for z > 0 and ign for z < 0 ) repectiely it can eaily be hown that G (r) ẑẑ δ(r) + k 2 i + 8π 2 i + 8π 2 k z I kk k 2 ] e ik r d k z > 0 k z I KK k 2 ]e ik r d k z < 0 (33) where k k xˆx + k y ŷ + k z ẑ K k xˆx + k y ŷ k z ẑ Equation (33) i the expanion of the free-pace dyadic Green function in term of a continuou pectrum of plane wae (monochromatic) propagating along the ector k and K which are in general complex quantitie. Propagation ector K i the mirror image of k in x-y plane ( K k 2(k ẑ)ẑ ) and repreent plane wae propagating along the ẑ direction. Another ueful repreentation appropriate for planar boundarie can be obtained by decompoing the ector in G (r) into TE and TM component. Recognizing that k/k i a unit ector (ˆk) the horizontal (TE) and ertical (TM) unit ector are, repectiely, defined by ê(k z ) ˆk ẑ ˆk ẑ k y ˆx k x ŷ k 2 x + k 2 y k ρ (k yˆx k x ŷ) ĥ(k z ) ê ˆk k z kk ρ (ˆxk x + ŷk y ) + k ρ k ẑ The triplet ( ĥ, ê, ˆk ) form an orthonormal ytem, and therefore

12 I ˆkˆk êê + ĥĥ (34) A imilar orthonormal ytem can be formed with ˆK K k intead of ˆk. In thi ytem the horizontal and ertical unit ector are gien by: and a before ê( k z ) ĥ( k z ) ê ˆK ˆK ẑ ˆK ẑ ê(k z) I ˆK ˆK ê(k z )ê(k z ) + ĥ( k z)ĥ( k z) (35) Inerting (34) and (35) into (33) and tranlating the ource from the origin to r, the free-pace dyadic Green function take the following form: G (r, r ) ẑẑ k δ(r 2 r )+ i + 8π 2 i + 8π 2 k z êê + ĥ(k z )ĥ(k z) ] e ik (r r ) d k z > z êê k z + ĥ( k z )ĥ( k z) ] (36) e ik (r r ) d k z < z Thi form of the dyadic Green function i uually not appropriate for numerical ealuation, epecially when z z << λ. In thi cae the conergence rate of the integral i ery poor. 7 Dyadic Green Function for Two-Dimenional Problem In ome electromagnetic cattering problem where the geometry of the problem i independent of one coordinate ariable the formulation of the problem can be made omewhat impler. Without lo of generality let u aume that the catterer geometry i independent of z. In thi cae the catterer i a cylinder of arbitrary cro-ection whoe axi i along ẑ. Since there i no ariation with repect to z, all field quantitie take the z dependence of the excitation. Suppoe J(r) J(ρ)e ik ziz 2

13 where ρ xˆx + yŷ. The differential operator can alo be made explicit with repect to z which i replaced by ik zi, that i t + ik zi ẑ where t ˆx + x yŷ. Since the dependence of J(r) with repect to z i explicit, in field calculation the integral with repect to z can be carried out. Therefore the 2-D dyadic Green function i gien by + G (ρ, ρ ) G (r, r )e ik ziz dz ( I + ) + 4π k 2 e ik r r ziz r r eik dz Uing the identity + e ik ρ ρ 2 +(z z ) 2 ziz dz iπh () ρ ρ 2 + (z z 0 (k ρ ρ ρ )e ik ziz ) 2eik (37) where k ρ k 2 k 2 zi, the 2-D dyadic Green function take the following form: G (ρ, ρ ) i I + ( t 4 k 2 t + ik zi t ẑ + ik ziẑ t kziẑẑ )] 2 H () 0 (k ρ ρ ρ )e ikziz. (38) In matrix form (38) become G (ρ, ρ ) + k 2 2 x 2 k 2 2 x y k 2 y x k 2 y 2 ik zi k 2 x ik zi k 2 y ik zi k 2 ik zi k 2 kρ 2 k 2 0 x y i 4 H() 0 (k ρ ρ ρ )e ik ziz (39) 8 Fourier Repreentation of 2-D Dyadic Green Function A procedure imilar to what wa hown for 3-dimenional dyadic Green function can be followed to obtain the Fourier repreentation of 2-D dyadic Green function. The Fourier repreentation can alo be obtained in a impler way uing the following identity H () 0 (k ρ (x x ) 2 + (y y) 2 ) + e ikx(x x )+ik y y y dk x (40) π k y 3

14 where k y k 2 ρ k 2 x. Subtituting (40) in (39) and after ome algebraic manipulation it can be hown that G (ρ, ρ ) ŷŷ k δ(ρ 2 ρ )e ikziz + ie ik zi z 4π ie ik zi z 4π + I k y kk e ikx(x x )+k y(y y ) dk x y > y I ] (4) k y KK e ikx(x x ) k y(y y )] dk k 2 x y < y k 2 ] where k k xˆx + k y ŷ + k zi ẑ K k xˆx k y ŷ + k zi ẑ. A before k can be conidered a the propagation ector of a plane wae going along poitie y direction and K i that of a wae going along y direction. k zi i a fixed known quantity. 9 Symmetrical Property of Dyadic Green Function Symmetrical property of dyadic Green function allow for imple ealuation of the dyadic Green function when the location of ource and oberation point are interchanged. To demontrate thi property the dyadic-dyadic Green econd identity gien by Q] T P Q ] T P d (42) { Q ] T ( ˆn P ) + Q] T ( ˆn P) } d will be employed. Let u conider two ituation where in each cae the ource location i at r a and r b repectiely. The dyadic Green function for each cae mut atify G (r, r a ) k 2 G (r, r a ) δ(r r a ) I (43) G (r, r b ) k 2 G (r, r b ) δ(r r b ) I (44) Subtituting G (r, r a ) for Q and G (r, r b ) for P in (42) and uing the radiation condition at infinity and equation (43) and (44) it can readily be hown that 4

15 G ] T G (r a, r b ) (r b, r a ) (45) That i, the dyadic Green function when the ource i at r b and the oberation point i at r a i the tranpoe of the Green function when the ource point i at r a and the oberation point i at r b. Although the proof i gien for the free pace Green function, (40) i a general reult. Equation (2) and (25) how that the free pace Green function i ymmetric, i.e., G T G (r, r ) (r, r )] In iew of (45) for free pace Green function we hae G (r, r ) G (r, r) (46) Howeer, it hould be noted that (46) i not a general reult. 0 Dyadic Green Function for Piece-Wie Homogenou Media In the preiou ection we conidered the propertie of dyadic Green function for homogeneou media. In practice howeer, the medium of interet i often complex which may be compoed of many homogeneou media uch a the one hown in Figure 2. For thee problem it i uually deired to derie the expreion for a dyadic Green function which atifie the neceary field boundary condition at the interface. The field quantitie in repone to a olumetric electric current ditribution J e (r) in the unbounded inhomogeneou medium are imply gien by E(r ) ik n Z n H(r ) J e (r) G (r, r ) d (47) J e (r) G (r, r )]d (48) The imple form of (47) i obtained by impoing certain boundary condition on G (r, r ). To derie thee boundary condition conider a imple medium compoed of two homogeneou media. Suppoe the ource exit only in medium where we hae 5

16 µ 2 2 µ P.E.C. µ 3 3 Figure 2: A complex medium compoed of a number of homogeneou media and perfect electric conductor. and in the econd medium E (r) k 2 E (r) iωµ J (r) (49) E 2 (r) k 2 2 E 2(r) 0 (50) Let u denote the expreion for the dyadic Green function in medium by G () (r, r ) which atifie G () (r, r ) k 2 G () (r, r ) I δ(r r ) (5) and the dyadic Green function in medium 2 by G (2) (r, r ) which atifie G (2) (r, r ) k 2 2 G (2) (r, r ) 0. (52) The application of the ector-dyadic Green econd identity to (49) and (5) gie: + E (r ) iωµ J (r) G () (r, r )d (53) iωµ (ˆn H (r )) G () (r, r ) + (ˆn E (r)) G () ] (r, r ) d where i the boundary between the two media. 6

17 The application of the ector-dyadic Green econd identity to (50) and (52) proide Noting that iωµ 2 (ˆn H 2 (r)) G (2) (r, r ) + (ˆn E 2 (r)) G (2) ] (r, r ) d 0 (54) ˆn H (r) ˆn H 2 (r) ˆn E (r) ˆn E 2 (r) and uing the following identitie ˆn H (r) G (2) (r, r ) H (r) (ˆn G (2) (r, r )) ˆn E (r) G (2) (r, r ) E (r) ˆn ( G (2) (r, r )) the contribution from the urface integral of (53) can be hown to anih if and µ ˆn G () (r, r ) µ 2ˆn G (2) (r, r ) (55) ˆn G () (r, r ) ˆn G (2) (r, r ) (56) Equation (55) and (56) are the neceary boundary condition for the dyadic Green function. On the urface of perfect electric conductor which mandate ˆn(r) G (r, r) 0. ˆn(r) E(r) 0 The ymmetry property of the dyadic Green function can be hown eaily by following the ame procedure outlined for the free pace dyadic Green function. Let u conider two experiment where in one experiment the ource i placed at an arbitrary point r a and the oberation point i at r in the nth region. In the econd experiment we place the ource point at an arbitrary point r b while keeping the oberation point at the ame location r a hown in Figure 3. In both experiment the dyadic Green function atify (5) (without the upercript). Applying the dyadic-dyadic Green econd identity to 7

18 r b r r a r n th region n th region Experiment # Experiment #2 Figure 3: Location of ource and oberation point in a complex dielectric medium for demontrating the ymmetry property of dyadic Green function. thee dyadic Green function which atify the radiation condition, it can readily be hown that G ] T G (r a, r b ) (r b, r a ). (57) It hould be noted that it i not ery eay to find a dyadic Green function for a general piece-wie homogeneou media with arbitrary boundarie. In uch cae it i more conenient to ue the free pace dyadic Green function with urface integral gien by () and (4). In iew of (57), equation (47) can be written a where we ued the fact that E(r ) i k n Z n G (r, r) J e (r) d J e (r) G G ] T (r, r ) (r, r ) Je (r) and G ] T (r, r ) G (r, r). Now interchanging r with r and ice era we get E(r) ik n Z n G (r, r ) J e (r )d (58) 8

19 Dyadic Green Function For Inhomogeneou Media Conider an inhomogeneou iotropic medium whoe permittiity and permeability are function of poition and are, repectiely, denoted by ǫ(r) and µ(r). A before we are eeking imple expreion for electric and magnetic field for an arbitrary ource uing the impule repone of the medium. It hould be emphaized that the ealuation of the dyadic Green function for thi type of problem, in general, i ery complex; howeer, here a formal analyi i proided. Taking the curl of the modified Amper law, it can be hown that the ector wae equation take the following form ] µ(r) E(r) ω 2 ǫ(r) E(r) iωj e (r) (59) Comparing () and (59)for thi problem one can define a Green function uch that it would atify Noting that ] µ(r) G (r, r ) ω 2 ǫ(r) G (r) µ(r) I δ(r r ) (60) ( ) µ(r) P Q ( ) µ(r) Q P P µ(r) Q The Green econd ector identity can be written a { ] ]} P µ Q Q µ P d (6) { } Q P P Q ˆn d µ from which the ector-dyadic Green econd identity can be obtained and i gien by W.C. Chew,. { ] ] } P µ(r) Q µ(r) P Q d (62) { ( P ˆn ) Q (ˆn P ) } Q d µ(r) 9

20 Subtituting E for P and G for Q in (62), applying (59) and (60), and uing the radiation condition it can eaily be hown that E(r) iωµ(r) J e (r ) G (r, r)d (63) The magnetic field can be obtained from the application of Faraday law (H(r) iωµ(r) E(r)). H(r) µ(r) J(r) Uing the dyadic-dyadic Green econd identity µ(r) G ] (r, r) d (64) ( )] T µ(r) Q P Q] T { Q ] T ( ˆn P ) + µ(r) ( ) µ(r) P d Q] T ( ˆn P ) } d it can eaily be hown that µ(r ) G (r, r ) µ(r) G (r, r)] T. (65) 20