Three-vector and scalar field identities and uniqueness theorems in ...

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Theorem U1. The divergence and curl of a twice continuouslydifferentiable (static) three-vector field, whichvanishes sufficiently rapidly at infinity, uniquely determinesthe three-vector field over an unbounded volumeV of R 3 .In other words, we must specify∇ × F(x, y, z) = j(x, y, z),∇ · F(x, y, z) = ρ(x, y, z),(1a)(1b)over the volume V. In an electromagnetic context j isa circulation current density and ρ is a source chargedensity. A proof of a version of this theorem over a finitevolume of R 3 is given in Ref. 11.The second uniqueness theorem, which is typicallycalled a Helmholtz theorem, can be stated as follows: 6Theorem H1. A general continuous three-vector fielddefined everywhere in R 3 that along with its first derivativesvanishes sufficiently rapidly at infinity may beuniquely represented as a sum of an irrotational and asolenoidal part, up to a possible additive constant vector.A proof of theorem H1 is given in Ref. 8. To prove thistheorem it is sufficient to prove only that the three-vectorF can be written as 11F(x, y, z) = −∇Φ(x, y, z) + ∇ × A(x, y, z). (2)Then the three-vector identity∇ × ∇Φ = 0, (3)and the irrotational field assumption ∇ × F I = 0 impliesthat F I = −∇Φ is irrotational. And the three-vectoridentity∇ · ∇ × A = 0, (4)and the solenoidal field assumption ∇ · F S = 0 impliesthat F S = ∇ × A is solenoidal. Thus Eq. (2) is a sum ofan irrotational and a solenoidal part.The most straightforward proof of Eq. (2) (see Refs. 9and 10) and therefore of theorem H1 is to obtain a Helmholtzidentity 8 that is of the same form as Eq. (2). Aproof of this identity is presented here so that our analysisis self contained. This proof is based on the assumptionthat there exists a solution for a three-vectorF of an inhomogeneous vector Poisson equation in Cartesiancoordinates, and that each separate Cartesian componentof F is a solution of a scalar Poisson equation.The scalar Poisson equation can then be solved in termsof a two-point scalar Green’s function G(r, r ′ ) that connectsits unit delta function source located at the sourcepoint r ′ = (x ′ , y ′ , z ′ ) to a measurement at the field pointr = (x, y, z). Because∇ 2 14πr = −δ3 (r − r ′ ), (5)for all r and r ′ with r ≡ |r − r ′ |, the Poisson equationdefining the Green’s function,∇ 2 G(r, r ′ ) = −δ 3 (r − r ′ ), (6)leads to G(r, r ′ ) = 1/4πr. The delta function property ofthe vector identity (5) can be applied to any well-behavedfunction F(x, y, z) as∫F(x, y, z) = F(x ′ , y ′ , z ′ )δ 3 (r − r ′ )dV ′ (7a)V∫′ ( ) −1= F(x ′ , y ′ , z ′ )∇ 2 dV ′ . (7b)V 4πr′Because the Laplacian operator acts only on the fieldpoint coordinates, it can be brought outside of the integrationover the source point coordinates. We can usethe well known three-space identity∇ 2 A = ∇(∇ · A) − ∇ × (∇ × A) (8)to rewrite Eq. (7b) as [typo corrected from journal article]∫F(x, y, z) = − ∇ ∇ · F(x′ , y ′ , z ′ )dV ′V 4πr′∫+ ∇ × ∇ × F(x′ , y ′ , z ′ )dV ′ , (9)V 4πr′where one of the ∇ operators has been brought backinto each of the integrals, which is allowed because theyoperate only on the field coordinates. Although Eq. (9)is in a form like Eq. (2) as required to prove theorem H1,it is necessary to obtain an identity that is suitable toprove the three-space theorem U1 as well. If we use thevector identities 10∇ · (φA) = A · ∇φ + φ∇ · A,∇ × (φA) = −A × ∇φ + φ∇ × A,(10a)(10b)(note that the divergence and curl equal zero with respectto the field coordinates of a vector A which is a functiononly of the source coordinates) and use the identity∇(1/r) = −∇ ′ (1/r), we find∫F(x, y, z) = ∇ F(x ′ , y ′ , z ′ ) · ∇ ′ 1V 4πr dV ′′∫+∇ ×V ′ F(x ′ , y ′ , z ′ ) × ∇ ′ 14πr dV ′ . (11)The vector identities (10) are now applied again, this timeon the integrands of Eq. (11) where the ∇ operators acton the source coordinates. We obtain four terms; the onescontaining ∇ · (φA) and ∇ × (φA), respectively, becomesurface integrals via the following identities, 10 (that is,letting T = φA),∫∮ ∮∇ · T dV = T · dS = T · n dS, (12a)VSS∫∮∮∇ × T dV = n × T dS = − T × n dS, (12b)VSwhere n is the unit surface normal of S bounding V .These integrals vanish as r → ∞ for the field F, whichis assumed to fall off sufficiently rapidly at infinity. TheS2
final two terms are now of the proper form yielding thedesired identity∫∇ ′ · F(x ′ , y ′ , z ′ )F(x, y, z) = − ∇dV ′V 4πr∫′ ∇ ′ × F(x ′ , y ′ , z ′ )+ ∇ ×dV ′ (13)V 4πr′over all Euclidean three-space.A proof of Eq. (13) can be traced back to Stokes. 8 Nevertheless,Eq. (13) is sometimes referred to as a “Helmholtzidentity” (see Ref. 10). For an extended versionof Eq. (13) in a finite volume of R 3 the surface integralterms can be retained. 6,9,10 Equation (13) is of the formof Eq. (2) and can therefore be considered as completingthe proof of theorem H1, provided that the integrals arewell defined. For the integrals to be well defined we mustassume that the field F vanishes sufficiently rapidly atinfinity. As noted in Ref. 6, passing the remaining vectorderivatives over the field point coordinates for a twicecontinuously differentiable vector field F into each of therespective integrals of Eq. (13) can improve the convergenceproperties of the integrands.To prove theorem U1 using the Helmholtz identity (13)for the (static) three-vector field F(r) we postulate theexistence of a second three-vector field G(r), which alsosatisfies Eqs. (1) and (13). That is, we replace F(r) onthe left-hand side of Eqs. (1a) and (1b) by G(r), leavingthe right-hand side of Eqs. (1a) and (1b) unchanged. Thethree-vector field F(r) is unique if we can show thatW(r) ≡ F(r) − G(r) = 0. (14)If we take the divergence of W and use Eq. (1b), weobtain∇ · W = ∇ · F − ∇ · G = ρ − ρ = 0 (15)for all r in R 3 . We next take the curl of W and useEq. (1a) to obtain∇ × W = ∇ × F − ∇ × G = j − j = 0 (16)for all r in R 3 . The substitution of the results (15) and(16) for the three-vector field W into the Helmholtz identity(13) yields the result W(r) = 0 which implies [viathe definition of W(r) in Eq. (14)] that F(r) = G(r)everywhere in R 3 . This proof implies that the (static)three-vector field F is uniquely determined by Eqs. (1a)and (1b), thus proving theorem U1 on the uniqueness of(static) three-vector fields in terms of their curl and divergence,that is, in terms of a vector and a scalar (source)field, respectively. [For a finite volume of R 3 the normalcomponents W·n and tangential components W×n alsovanish in a similar fashion via a surface charge density σand surface current density K, as can be shown for examplein an explicit calculation for the case of a massive(static) three-vector field. 6 ]We now state the obvious. Equations (1a) and (1b) associatedwith theorem U1 are not equivalent to Maxwell’sequations, which uniquely specify the electromagneticfields. They do not contain any partial time derivativeterms and do not contain any coupling between differentthree-vector fields. Therefore, Eqs. (1a) and (1b)and theorem U1 do not hold for time-varying electricand magnetic fields, which are examples of hyperbolicallypropagating 4 time-varying three-vector fields. Therefore,theorem U1 does not hold for all time-varying threevectorfields because there is at least one counterexample,that is, electromagnetism.In contrast, the Helmholtz theorem H1 is quite differentin scope. It is essentially a projection theorem whichuses three-vector analysis to project out the longitudinal(irrotational) parts (which have zero curl) and transverse(solenoidal) parts (which have zero divergence) ofan arbitrary three-vector field, and states that any threevectorfield can be represented as a sum of these twoparts. To show that this theorem holds even for timedependentthree-vectors, it is sufficient to verify the stepsemployed in deriving the Helmholtz identity (13), becauseit is of the proper form of Eq. (2), which is sufficientto prove theorem H1. Following this line of reasoning,first observe that the delta function integral propertyin Eq. (7b) holds even for well-behaved time-varyingthree-vector functions F(x, y, z, t) because the space andtime variables are independent, and consequently spaceand time integrations are performed separately. As forEq. (8), it is a three-vector identity over the Euclideanvector space R 3 and over the associated scalars in thereal number field R which applies to any (even timedependent)three-vector because it is based on multiplicationof vectors by scalars (a defining property of a vectorspace) and on the two principal ways that vectors canbe multiplied by each other in R 3 , that is, dot and crossproducts. [More formally, for A as a time-dependentvariable, Eq. (8) forms a one parameter family of vectorfields parametrized by t. For each time t, Eq. (8) holds.The same can be said for Eq. (7b).] Equation (8) is oftenused in deriving the spatial parts of wave equations andis the principal vector identity that leads to the decomposition(2). For the same reasons the vector identities(10) and integral identities (12) are valid for time-varyingfields. Therefore, the Helmholtz identity (13) holds evenfor time-varying three-vector fields and so Helmholtz’stheorem H1 follows for them as well via Eq. (2).Helmholtz’s theorem H1 has been used frequently withtime-varying fields in both classical and quantum mechanicalcontexts. Rohrlich 3 has demonstrated that thisHelmholtz decomposition holds even for electromagneticsources, provided that the sources are bounded in space.This latter requirement is essentially equivalent to thecase of the general fields discussed here which are assumedto vanish sufficiently rapidly at infinity in unboundedR 3 . The relevant requirement in a bounded volumeV of R 3 is that the vector (or scalar) field must besufficiently smooth, 6 that is, a twice continuously differentiablefunction on the union of V and its boundingsurface S, in order that a Helmholtz identity in a finite3
Page 4 and 5: volume and its associated finite vo
Page 6 and 7: set the divergence of Eq. (29) equa
Page 8 and 9: V. UNIQUENESS THEOREMS FOR THREE-VE
Page 10: a) Electronic mail: dalew@ics.mq.ed

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