Electromagnetic Green Functions Using Differential Forms

17 Pages·1.252 MB·English

by  Warnick K., Arnold D.

Electromagnetic Green Functions Using Di(cid:2)erential Forms
Karl F(cid:2) Warnick and David V(cid:2) Arnold
Short Title(cid:3) Electromagnetic Green Forms
Department of Electrical and Computer Engineering
(cid:4)(cid:5)(cid:6) Clyde Building
Brigham Young University
Provo(cid:7) UT(cid:7) (cid:8)(cid:4)(cid:9)(cid:10)(cid:11)
Warnick(cid:2) et al(cid:2) (cid:3) September (cid:3)(cid:4)(cid:2) (cid:3)(cid:5)(cid:5)(cid:6)
Abstract(cid:2) In this paper we redevelop the scalar and dyadic Green functions of
electromagnetic theory using di(cid:12)erential forms(cid:2) The Green dyadic becomes a dou(cid:13)
ble form(cid:7) which is adi(cid:12)erential form in one space with coe(cid:14)cients thatare forms in
another space(cid:7) or a di(cid:12)erential form(cid:13)valued form(cid:2) The results presented here corre(cid:13)
spond closely with the usual dyadic treatment(cid:7) but are clearer and more intuitive(cid:2)
Many of the usual expressions using green functions in vector notation require a
surface normal(cid:15) with the Green forms the surface normal is unnecessary(cid:2) We illus(cid:13)
trate the formalism by computing scattering from a randomly rough conducting
surface and deriving the Green form for a dielectric half(cid:13)space(cid:2) We also de(cid:16)ne the
interior derivative(cid:7) which is equivalent to the coderivative but for a constant met(cid:13)
ric has a computational rule dual to that of the exterior derivative and simpli(cid:16)es
calculation in coordinates(cid:2) This work makes available some of the tools that have
not yet been presented in the language of di(cid:12)erential forms but are essential in
applied electromagnetics(cid:2)
Warnick(cid:2) et al(cid:2) (cid:7) September (cid:3)(cid:4)(cid:2) (cid:3)(cid:5)(cid:5)(cid:6)
(cid:17)(cid:2) INTRODUCTION
In this paper we treat Green function methods in electromagnetic (cid:18)EM(cid:19) (cid:16)eld theory using
the calculus of di(cid:12)erential forms(cid:2) The calculus of di(cid:12)erential forms has been applied to EM
theory by Deschamps (cid:20)(cid:17)(cid:21)(cid:7) Baldomir (cid:20)(cid:11)(cid:21)(cid:7) Schleifer (cid:20)(cid:22)(cid:21)(cid:7) Thirring (cid:20)(cid:4)(cid:21)(cid:7) Burke (cid:20)(cid:5)(cid:7) (cid:9)(cid:21)(cid:7) Bamberg (cid:20)(cid:23)(cid:21)(cid:7)
Ingarden and Jamio(cid:24)lkowksi (cid:20)(cid:8)(cid:21)(cid:7) Parrott (cid:20)(cid:6)(cid:21) and others(cid:2)
Several authors have advocated the use of the calculus of forms in engineering EM theory(cid:7)
but some important tools for applied problems have not been developed(cid:2) In (cid:20)(cid:17)(cid:10)(cid:21) the authors
presented a representation of EM boundary conditions using di(cid:12)erential forms(cid:2) In this
work we develop another tool well suited for practical use(cid:7) the Green form in the (cid:18)(cid:22)(cid:25)(cid:17)(cid:19)
representation(cid:2)
As proposed by Thirring (cid:20)(cid:4)(cid:21)(cid:7) the EM Green function becomes a double form(cid:2) Double
forms are de(cid:16)ned by de Rham in (cid:20)(cid:17)(cid:17)(cid:21)(cid:2) Green forms are treated in the mathematics literature
(cid:18)see (cid:20)(cid:17)(cid:11)(cid:21) and its references(cid:19)(cid:7) and Thirring gives the time(cid:13)dependent Green form for electro(cid:13)
dynamics in Minkowski spacetime(cid:2) Our Green form has the same components as the Green
dyadic in Kong (cid:20)(cid:17)(cid:22)(cid:21) and therefore is easily related to the usual methods in applied electro(cid:13)
magnetics(cid:2) We derive expressions for the Green double form in terms of the scalar Green
function(cid:7) the electric (cid:16)eld due to a surface current density and the Stratton(cid:13)Chu formula(cid:2)
The use of di(cid:12)erential forms makes the results presented here clearer in certain ways
than the usual vector and dyadic treatment(cid:2) In obtaining expressions for observed (cid:16)elds
in terms of the Green forms(cid:7) the product rule for the exterior derivative takes the place of
several vector identities(cid:2) This makes the derivation much cleaner(cid:2) The dyadic expression for
observed (cid:16)elds due to tangential (cid:16)elds along the surface of an observation region using the
Green dyadic includes a surface normal(cid:2) With the corresponding expression using the Green
form(cid:7) the surface normal is unnecessary(cid:2)
In this paper we also de(cid:16)ne the interior derivative(cid:7) which is equivalent to the standard
coderivative (cid:20)(cid:23)(cid:21)(cid:7) but simpli(cid:16)es calculations in coordinates when the metric is constant(cid:2) The
computational rule which we propose is dual to that of the exterior derivative(cid:2)
In Sec(cid:2) (cid:11) we review operations on forms and treat double forms brie(cid:26)y(cid:2) In Sec(cid:2) (cid:22) we solve
Maxwell(cid:27)s laws of electromagnetics in terms of the Green double form and the scalar Green
Warnick(cid:2) et al(cid:2) (cid:8) September (cid:3)(cid:4)(cid:2) (cid:3)(cid:5)(cid:5)(cid:6)
function(cid:2) Finally(cid:7) in Sec(cid:2) (cid:4) we illustrate the method by computing scattered (cid:16)elds from a
rough conducting surface and deriving the Green form for a dielectric half(cid:13)space(cid:2) This work
shows that the calculus of di(cid:12)erential forms can be used in all applications to which Green
functions and dyadics are suited(cid:2)
(cid:11)(cid:2) DEFINITIONS
In this section we give de(cid:16)nitions and notation to be used in Sec(cid:2) (cid:22) to derive the Green
forms(cid:2) We de(cid:16)ne the interior and exterior derivatives(cid:7) the interior and exterior products and
the Laplace(cid:13)de Rham operator(cid:2) Double (cid:17)(cid:13)forms are also introduced in this section(cid:2)
(cid:11)(cid:2)(cid:17) Operators
The exterior derivative d is de(cid:16)ned in (cid:20)(cid:23)(cid:21) and elsewhere(cid:2) It can be represented formally as
(cid:2) i
d (cid:0) dx (cid:2) (cid:18)(cid:17)(cid:19)
i
(cid:2)x
(cid:0) n
where x (cid:3)(cid:3)(cid:3)x are coordinates on an n(cid:13)dimensional space and the summation convention is
i j
used(cid:2) The exterior product (cid:2) is the antisymmetrized tensor product(cid:7) so that dx (cid:2) dx (cid:28)
j i i i
(cid:3)dx (cid:2) dx and dx (cid:2) dx (cid:28) (cid:10)(cid:2) (cid:18)Often the wedge between di(cid:12)erentials is dropped(cid:15) there is an
impliedwedgebetweenthedi(cid:12)erentialsintheintegrandofanymultipleintegral(cid:2)(cid:19) Thepartial
(cid:2) (cid:2) (cid:2)f (cid:2)f
derivatives (cid:2)xi actonthecoe(cid:14)cientsofaform(cid:7)sothatinR (cid:7)d(cid:18)f dx(cid:19) (cid:28) (cid:2)y dy(cid:2)dx(cid:25)(cid:2)z dz(cid:2)dx
since dx(cid:2) dx (cid:28) (cid:10)(cid:2) We de(cid:16)ne the interior derivative in the euclidean metric similarly(cid:7)
(cid:2) i
d (cid:0) dx (cid:18)(cid:11)(cid:19)
i
(cid:2)x
(cid:2) (cid:2)f
where is the interior product de(cid:16)ned in (cid:20)(cid:17)(cid:10)(cid:21)(cid:2) In R we have d (cid:18)f dx(cid:19) (cid:28) (cid:2)x since dx dx (cid:28) (cid:17)
and dy dx (cid:28) dz dx (cid:28) (cid:10)(cid:2)
The interior product is de(cid:16)ned to be the contraction of a vector with a k(cid:13)form (cid:18)which is
(cid:3)
a totally antisymmetric (cid:18)k(cid:19) tensor(cid:19)(cid:2) In this paper we use the euclidean metric(cid:7) so we can
extend this de(cid:16)nition to the interior product of a (cid:17)(cid:13)form and k(cid:13)form easily since vectors and
ij i(cid:0) ik
(cid:17)(cid:13)forms have the same components(cid:2) The interior product of dx and dx (cid:2)(cid:4)(cid:4)(cid:4)(cid:2) dx is zero
Warnick(cid:2) et al(cid:2) (cid:4) September (cid:3)(cid:4)(cid:2) (cid:3)(cid:5)(cid:5)(cid:6)
j(cid:0)(cid:0) i(cid:0) ij(cid:0)(cid:0) ij(cid:2)(cid:0) ik
for ij not equal to any of i(cid:0)(cid:3)(cid:3)(cid:3)ik(cid:7) otherwise it is (cid:18)(cid:3)(cid:17)(cid:19) dx (cid:2)(cid:4)(cid:4)(cid:4)(cid:2)dx (cid:2)dx (cid:2)(cid:4)(cid:4)(cid:4)(cid:2) dx
for (cid:17) (cid:5) j (cid:5) k(cid:2) Thus(cid:7) by (cid:18)(cid:11)(cid:19) the interior derivative of a form is computed by moving each
di(cid:12)erential in turn to the leftmost position by alternating the sign of the form each time two
di(cid:12)erentials are swapped(cid:7) removing that di(cid:12)erential and taking the corresponding partial
derivative(cid:2)
(cid:0)
The interior derivative is equivalent up to a sign to the coderivative de(cid:16)ned in (cid:20)(cid:4)(cid:7) (cid:23)(cid:21) and
elsewhere(cid:7)
k(cid:4)(cid:0) (cid:0)(cid:0)
d (cid:4) (cid:28) (cid:18)(cid:3)(cid:17)(cid:19) (cid:5) d(cid:5)(cid:4) (cid:18)(cid:22)(cid:19)
(cid:2)
where k is the degree of (cid:4) and (cid:5) is the Hodge star operator(cid:2) In R with the euclidean metric(cid:7)
(cid:0)(cid:0)
(cid:5)(cid:17) (cid:28) dxdydz(cid:7) (cid:5)dx (cid:28) dy (cid:2) dz(cid:7) (cid:5)dy (cid:28) dz (cid:2) dx(cid:7) (cid:5)dz (cid:28) dx(cid:2) dy and (cid:5) (cid:28) (cid:5)(cid:2) Note that
k(cid:4)(cid:0)
the interior derivative contains the sign (cid:18)(cid:3)(cid:17)(cid:19) naturally(cid:2) For a nonconstant metric(cid:7) such
as would arise in curvilinear coordinates(cid:7) (cid:18)(cid:22)(cid:19) replaces (cid:18)(cid:11)(cid:19) as the de(cid:16)nition of the interior
derivative(cid:2)
The interior derivative is easier to compute with than the coderivative(cid:7) as illustrated by
the following example(cid:2) We (cid:16)rst use the coderivative to (cid:16)nd
(cid:3)(cid:5)d(cid:5)(cid:18)D(cid:0)dydz (cid:25)D(cid:5)dzdx(cid:25)D(cid:2)dxdy(cid:19)
(cid:2) (cid:2) (cid:2)
(cid:28) (cid:3)(cid:5)(cid:18) dx(cid:25) dy(cid:25) dz(cid:19)(cid:2)(cid:18)D(cid:0)dx(cid:25)D(cid:5)dy(cid:25)D(cid:2)dz(cid:19)
(cid:2)x (cid:2)y (cid:2)z
(cid:28) (cid:3)(cid:5)(cid:18)D(cid:0)xdx(cid:2) dx(cid:25)D(cid:0)ydy(cid:2) dx(cid:25)D(cid:0)zdz (cid:2) dx
(cid:25) D(cid:5)xdx(cid:2) dy(cid:25)D(cid:5)ydy(cid:2) dy(cid:25)D(cid:5)zdz (cid:2) dy
(cid:25) D(cid:2)xdx(cid:2) dz (cid:25)D(cid:2)ydy (cid:2) dz (cid:25)D(cid:2)zdz (cid:2) dz(cid:19)
(cid:28) (cid:18)D(cid:5)z (cid:3)D(cid:2)y(cid:19)dx(cid:25)(cid:18)D(cid:2)x (cid:3)D(cid:0)z(cid:19)dy(cid:25)(cid:18)D(cid:0)y (cid:3)D(cid:5)x(cid:19)dz(cid:3)
Using the de(cid:16)nition of the interior derivative we compute the same result immediately(cid:7)
d (cid:18)D(cid:0)dydz (cid:25)D(cid:5)dzdx(cid:25)D(cid:2)dxdy(cid:19) (cid:28) D(cid:0)ydz (cid:3)D(cid:0)zdy (cid:25)D(cid:5)zdx(cid:3)D(cid:5)xdz (cid:25)D(cid:2)xdy(cid:3)D(cid:2)ydx
where we have removed each di(cid:12)erential in turn(cid:7) after it moving to the left if necessary using
the antisymmetry of the exterior product(cid:7) and taken the corresponding partial derivative(cid:2)
The Laplace(cid:13)de Rham operator (cid:29) is
(cid:29) (cid:28) d d(cid:25)dd (cid:18)(cid:4)(cid:19)
Warnick(cid:2) et al(cid:2) (cid:6) September (cid:3)(cid:4)(cid:2) (cid:3)(cid:5)(cid:5)(cid:6)
(cid:5)
which is a generalization of the vector operator r (cid:2) With the euclidean metric(cid:7) (cid:29) becomes
(cid:2)(cid:3)
(cid:18)(cid:29)(cid:4)(cid:19)i (cid:28) j (cid:2)x(cid:3)j(cid:4)i where the subscript i indexes components of (cid:4)(cid:2) On (cid:17)(cid:13)forms(cid:7) (cid:18)(cid:4)(cid:19) is equiv(cid:13)
P
(cid:5)
alent to the euclidean vector identity r (cid:28) (cid:3)r(cid:6)r(cid:6)(cid:25)rr(cid:2)
The generalized Stokes theorem is
d(cid:4) (cid:28) (cid:4) (cid:18)(cid:5)(cid:19)
ZV Z(cid:2)V
where (cid:4) is a p(cid:13)form and V is a p(cid:25)(cid:17) dimensional region with (cid:2)V as its boundary(cid:2) Also(cid:7) the
interior product of two arbitrary forms a and b satis(cid:16)es
a b (cid:28) (cid:5)(cid:18)(cid:5)b(cid:2)a(cid:19) (cid:18)(cid:9)(cid:19)
where (cid:5) is the Hodge star operator(cid:2)
(cid:11)(cid:2)(cid:11) Double Forms
A double form (cid:20)(cid:17)(cid:17)(cid:21) is a di(cid:12)erential form in one space with coe(cid:14)cients that are forms in
(cid:2) (cid:2)(cid:2)
another space(cid:2) The double forms that we will use in this paper are associated with R (cid:6)R
(cid:2) (cid:2)(cid:2)
where R is the observation space and R is the source space(cid:2) We will use (cid:17)(cid:13)form valued
(cid:17)(cid:13)forms(cid:7) or double (cid:17)(cid:13)forms(cid:7) which can be written in general
(cid:2) (cid:2) (cid:2)
G (cid:28) G(cid:0)(cid:0)dxdx (cid:25)G(cid:0)(cid:5)dxdy (cid:25)G(cid:0)(cid:2)dxdz
(cid:2) (cid:2) (cid:2)
(cid:25) G(cid:5)(cid:0)dydx (cid:25)G(cid:5)(cid:5)dydy (cid:25)G(cid:5)(cid:2)dydz
(cid:2) (cid:2) (cid:2)
(cid:25) G(cid:2)(cid:0)dzdx (cid:25)G(cid:2)(cid:5)dzdy (cid:25)G(cid:2)(cid:2)dzdz (cid:3)
Between the primed and unprimed di(cid:12)erentials there is an implied tensor product rather
than an exterior product(cid:2) The coe(cid:14)cients are functions Gij(cid:18)r(cid:6)r(cid:2)(cid:19) of both the observation
and source coordinates(cid:2)
A double form can be used as a transformation kernel (cid:18)if its coe(cid:14)cients vanish su(cid:14)ciently
(cid:2)(cid:2) (cid:2)
quickly at in(cid:16)nity(cid:19)(cid:2) If we (cid:16)x a double (cid:17)(cid:13)form G(cid:7) we have the transformation from R to R
given by the volume integral
(cid:2)
(cid:4) (cid:28) G(cid:2)(cid:5)(cid:4) (cid:18)(cid:23)(cid:19)
Z
(cid:2) (cid:2)
where (cid:4) is a (cid:17)(cid:13)form and (cid:4) is the transform of (cid:4) due to the kernel G(cid:2) The exterior product
Warnick(cid:2) et al(cid:2) (cid:9) September (cid:3)(cid:4)(cid:2) (cid:3)(cid:5)(cid:5)(cid:6)
(cid:2) (cid:2) (cid:2) (cid:2)(cid:2)
yields a (cid:22)(cid:13)form in dx dy dz which is integrated over R (cid:2) The unprimed di(cid:12)erentials remain(cid:7)
resulting in the (cid:17)(cid:13)form (cid:4) in the observation space(cid:2)
i j
Componentsofadyadic areoftheformx(cid:30) x(cid:30) withnoprime(cid:7) whichdoesnotshow explicitly
the relationship the dyadic can provide to the source and observation spaces(cid:2) The action of
a double form G as a kernel from the source to the observation space is clearly re(cid:26)ected in
i j(cid:2)
the product dx dx of primed and unprimed di(cid:12)erentials in each component(cid:2)
We introduce the identity kernel (cid:7)I where I is the double form
(cid:2) (cid:2) (cid:2)
dxdx (cid:25) dydy (cid:25) dzdz (cid:18)(cid:8)(cid:19)
(cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2)
or dr dr (cid:25)rd(cid:8) r d(cid:8) (cid:25)rsin(cid:8)d(cid:9) r sin(cid:8) d(cid:9) inspherical coordinates(cid:2) (cid:7) isthe three(cid:13)dimensional
(cid:2) (cid:2) (cid:2) (cid:2) (cid:2)
Dirac delta function (cid:7)(cid:18)x(cid:3)x(cid:19)(cid:7)(cid:18)y(cid:3)y (cid:19)(cid:7)(cid:18)z(cid:3)z (cid:19)(cid:2) Using this kernel(cid:7) (cid:7)I(cid:2)(cid:5)(cid:4) (cid:28) (cid:4) j(cid:6)x(cid:3)y(cid:3)z(cid:7) (cid:28) (cid:4)(cid:7)
R
(cid:2)
so that the transformation takes (cid:4) from source to observation space without otherwise
changing its components(cid:2)
(cid:22)(cid:2) THE EM GREEN FORMS
In this section we derive expressions for the electric (cid:16)eld (cid:17)(cid:13)form at an observation point
(cid:0)i(cid:4)t
due to applied sources and (cid:16)elds using the Green forms(cid:2) We consider time(cid:13)harmonic (cid:18)e (cid:19)
(cid:16)elds in an isotropic medium of permittivity (cid:10) and permeability (cid:11)(cid:2)
We write Faraday(cid:27)s and Ampere(cid:27)s laws as
dE (cid:28) i(cid:4)B (cid:18)(cid:6)(cid:19)
d B (cid:28) i(cid:4)(cid:10)(cid:11)E (cid:3)(cid:11)(cid:5)J (cid:18)(cid:17)(cid:10)(cid:19)
where E is the electric (cid:16)eld intensity (cid:17)(cid:13)form(cid:7) B is the magnetic (cid:26)ux density (cid:11)(cid:13)form and J is
the electric current density (cid:11)(cid:13)form(cid:2) The constitutive relations are D (cid:28) (cid:10)(cid:5)E and B (cid:28) (cid:11)(cid:5)H(cid:7)
where D is the electric (cid:26)ux density (cid:11)(cid:13)form and H is the magnetic (cid:16)eld intensity (cid:17)(cid:13)form(cid:2)
By taking the interior derivative of (cid:18)(cid:6)(cid:19) and substituting (cid:18)(cid:17)(cid:10)(cid:19)(cid:7) we obtain
(cid:5)
(cid:18)d d(cid:25)k (cid:19)E (cid:28) (cid:3)i(cid:4)(cid:11)(cid:5)J (cid:18)(cid:17)(cid:17)(cid:19)
Warnick(cid:2) et al(cid:2) (cid:10) September (cid:3)(cid:4)(cid:2) (cid:3)(cid:5)(cid:5)(cid:6)
(cid:5) (cid:5)
where k (cid:28) (cid:4) (cid:11)(cid:10)(cid:2) The Green double (cid:17)(cid:13)form G for Eq(cid:2) (cid:18)(cid:17)(cid:17)(cid:19) then satis(cid:16)es
(cid:5)
(cid:18)d d(cid:25)k (cid:19)G (cid:28) (cid:3)(cid:7)I(cid:3) (cid:18)(cid:17)(cid:11)(cid:19)
Here and below(cid:7) all derivatives will act on primed coordinates unless otherwise noted(cid:7) but
to avoid clutter(cid:7) the derivatives will remain unprimed(cid:2)
(cid:2) (cid:2) (cid:2)
Let V be a volume containing source current density given by the (cid:11)(cid:13)form J (cid:2) Outside V (cid:7)
the electric (cid:16)eld due to the sources is E(cid:7) a (cid:17)(cid:13)form in the observation space(cid:2) Using the unit
kernel (cid:7)I we can write
(cid:2)
E (cid:28) (cid:7)I (cid:2)(cid:5)E (cid:18)(cid:17)(cid:22)(cid:19)
ZV(cid:2)
Substituting Eq(cid:2) (cid:18)(cid:17)(cid:11)(cid:19) into (cid:18)(cid:17)(cid:22)(cid:19)(cid:7) we obtain
(cid:2) (cid:5) (cid:2)
E (cid:28) (cid:20)d(cid:5)dG(cid:2)E (cid:3)k G(cid:2)(cid:5)E (cid:21) (cid:18)(cid:17)(cid:4)(cid:19)
ZV(cid:2)
(cid:0)
where we have used d dG (cid:28) (cid:18)(cid:3)(cid:17)(cid:19) (cid:5)d(cid:5)dG and moved a (cid:5) across the exterior product (cid:18)if
(cid:12) and (cid:13) are both p(cid:13)forms(cid:7) then (cid:5)(cid:12) (cid:2) (cid:13) (cid:28) (cid:12)(cid:2) (cid:5)(cid:13)(cid:7) as can be veri(cid:16)ed easily in coordinates(cid:19)(cid:2)
deg(cid:5)
Using the product rule for the exterior derivative(cid:7) d(cid:18)(cid:14)(cid:2)(cid:15)(cid:19) (cid:28) d(cid:14)(cid:2)(cid:15) (cid:25)(cid:18)(cid:3)(cid:17)(cid:19) (cid:14)(cid:2)d(cid:15)(cid:7) Eq(cid:2)
(cid:18)(cid:17)(cid:4)(cid:19) becomes
(cid:2) (cid:5) (cid:2) (cid:2) (cid:2)
E (cid:28) (cid:20)G(cid:2)(cid:18)d(cid:5)dE (cid:3)k (cid:5)E (cid:19)(cid:25)d(cid:18)(cid:5)dG(cid:2)E (cid:25)G(cid:2)(cid:5)dE (cid:19)(cid:21)(cid:3) (cid:18)(cid:17)(cid:5)(cid:19)
ZV(cid:2)
AfterapplyingthestaroperatortoEq(cid:2) (cid:18)(cid:17)(cid:17)(cid:19)andusingthede(cid:16)nitionoftheinteriorderivative(cid:7)
(cid:2)
we can insert J into (cid:18)(cid:17)(cid:5)(cid:19)(cid:7)
(cid:2) (cid:2) (cid:2)
E (cid:28) (cid:20)i(cid:4)(cid:11)G(cid:2)J (cid:25)d(cid:18)(cid:5)dG(cid:2)E (cid:25)G(cid:2)(cid:5)dE (cid:19)(cid:21)(cid:3) (cid:18)(cid:17)(cid:9)(cid:19)
ZV(cid:2)
Applying the generalized Stokes theorem and using Faraday(cid:27)s law(cid:7) we (cid:16)nd that
(cid:2) (cid:2) (cid:2)
E (cid:28) i(cid:4)(cid:11) G(cid:2)J (cid:25) (cid:18)i(cid:4)(cid:11)G(cid:2)H (cid:25)(cid:5)dG(cid:2)E (cid:19) (cid:18)(cid:17)(cid:23)(cid:19)
ZV(cid:2) Z(cid:2)V(cid:2)
(cid:2)
where the second term takes into account (cid:16)elds on the surface (cid:2)V due to sources outside of
(cid:2)
V (cid:2)
The integrals in (cid:18)(cid:17)(cid:23)(cid:19)(cid:7) like all integrals of di(cid:12)erential forms(cid:7) can be integrated by the
method of pullback (cid:20)(cid:23)(cid:21)(cid:2) This method is completely general and allows forms to be integrated
conveniently over parameterized regions(cid:2)
Warnick(cid:2) et al(cid:2) (cid:11) September (cid:3)(cid:4)(cid:2) (cid:3)(cid:5)(cid:5)(cid:6)
The expression corresponding to (cid:18)(cid:17)(cid:23)(cid:19) using the dyadic Green function G is
E (cid:28) i(cid:4)(cid:11) G(cid:4)J(cid:2)dv(cid:2) (cid:3) (cid:18)i(cid:4)(cid:11)G(cid:4)(cid:20)n(cid:30) (cid:6)H(cid:2)(cid:21)(cid:25)r(cid:6)G(cid:4)(cid:20)n(cid:30) (cid:6)E(cid:2)(cid:21)(cid:19)ds(cid:2)
ZV(cid:2) Z(cid:2)V(cid:2)
(cid:2) (cid:2)
where n(cid:30) is an outward surface normal(cid:2) Because E and H are (cid:17)(cid:13)forms rather than (cid:11)(cid:13)
(cid:2)
forms(cid:7) their exterior product with G behaves di(cid:12)erently than the exterior product G(cid:2)J (cid:2)
(cid:2) (cid:2) (cid:2)
Components of E and H tangent to (cid:2)V naturally do not contribute to the surface integral
in (cid:18)(cid:17)(cid:23)(cid:19)(cid:2) Thus(cid:7) the surface normal is eliminated and a simpler expression results(cid:2) Also(cid:7)
Green(cid:27)s theorem is used in deriving the dyadic result(cid:2) Green(cid:27)s theorem on forms is an
immediate consequence of the product rule for the exterior derivative and the generalized
Stokes theorem(cid:2)
(cid:22)(cid:2)(cid:17) The Scalar Green Function
The scalar Green function g satis(cid:16)es the wave equation for an elementary source (cid:3)(cid:7)(cid:7)
(cid:5)
(cid:18)(cid:29)(cid:25)k (cid:19)g (cid:28) (cid:3)(cid:7) (cid:18)(cid:17)(cid:8)(cid:19)
It can easily be shown that gI satis(cid:16)es Eq(cid:2) (cid:18)(cid:17)(cid:8)(cid:19) for the source (cid:3)(cid:7)I(cid:2) Substituting gI for g
and (cid:3)(cid:7)I for (cid:3)(cid:7) in (cid:18)(cid:17)(cid:8)(cid:19)(cid:7) expanding the Laplace(cid:13)de Rham operator and rearranging gives
(cid:5) (cid:17)
d d(cid:18)gI(cid:19)(cid:25)k (cid:18)gI (cid:25) dd gI(cid:19) (cid:28) (cid:3)(cid:7)I(cid:3) (cid:18)(cid:17)(cid:6)(cid:19)
(cid:5)
k
Since dd (cid:28) (cid:10)(cid:7) this can be rewritten as
(cid:17) (cid:5) (cid:17)
d d(cid:18)gI (cid:25) dd gI(cid:19)(cid:25)k (cid:18)gI (cid:25) dd gI(cid:19) (cid:28) (cid:3)(cid:7)I(cid:3) (cid:18)(cid:11)(cid:10)(cid:19)
(cid:5) (cid:5)
k k
By comparison with Eq(cid:2) (cid:18)(cid:17)(cid:11)(cid:19)(cid:7) we see that
(cid:17)
G (cid:28) (cid:18)(cid:17)(cid:25) dd (cid:19)gI (cid:18)(cid:11)(cid:17)(cid:19)
(cid:5)
k
(cid:5)
up to a solution of (cid:18)d d(cid:25)k (cid:19)H (cid:28) (cid:10) where H is a double (cid:17)(cid:13)form(cid:2) This freedom is used to
satisfy boundary conditions(cid:2) The double (cid:17)(cid:13)form I could be included in the de(cid:16)nition of g(cid:7)
but we prefer to leave it out so that g is the usual scalar Green function(cid:2)
By pullback we can transform Eq(cid:2) (cid:18)(cid:17)(cid:8)(cid:19) to the spherical coordinate system(cid:2) Noting that
g(cid:3) in free space is spherically symmetric(cid:7) we (cid:16)nd that
(cid:5)
(cid:17) d (cid:5)
rg(cid:3) (cid:25)k g(cid:3) (cid:28) (cid:3)(cid:7)(cid:18)r(cid:19)(cid:3) (cid:18)(cid:11)(cid:11)(cid:19)
(cid:5)
rdr
Warnick(cid:2) et al(cid:2) (cid:5) September (cid:3)(cid:4)(cid:2) (cid:3)(cid:5)(cid:5)(cid:6)
The solution of this di(cid:12)erential equation is the usual result
ikr
e
g(cid:3) (cid:28) (cid:3) (cid:18)(cid:11)(cid:22)(cid:19)
(cid:4)(cid:16)r
From this we can compute G(cid:3)(cid:7)
(cid:5)
(cid:17) (cid:2) g(cid:3) (cid:2)
G(cid:3) (cid:28) g(cid:3)I (cid:25) (cid:5) (cid:5) drdr (cid:18)(cid:11)(cid:4)(cid:19)
k (cid:2)r
(cid:2) (cid:2) (cid:2) (cid:2) (cid:2)
which becomes g(cid:3)(cid:18)rd(cid:8) r d(cid:8) (cid:25)rsin(cid:8)d(cid:9) r sin(cid:8) d(cid:9)(cid:19) in the far (cid:16)eld(cid:2)
It is easily veri(cid:16)ed that if g is symmetric in r and r(cid:2) (cid:18)as is the case for reciprocal media(cid:19)(cid:7)
the derivatives in the (cid:16)rst term of Eq(cid:2) (cid:18)(cid:11)(cid:17)(cid:19) can be taken to act on umprimed rather than
(cid:2)
primed coordinates(cid:2) Thus(cid:7) we can write using (cid:18)(cid:17)(cid:23)(cid:19) after neglecting sources outside of V (cid:7)
(cid:17)
E (cid:28) i(cid:4)(cid:11)(cid:18)(cid:17)(cid:25) dd (cid:19) gI (cid:2)J (cid:18)(cid:11)(cid:5)(cid:19)
k(cid:5) ZV(cid:2)
where the derivatives act on unprimed coordinates(cid:2) The Lorentz gauge is d A (cid:28) i(cid:4)(cid:10)(cid:11)(cid:9)(cid:7)
where (cid:9) is the scalar electric potential and A is the magnetic vector potential (cid:17)(cid:13)form(cid:2) In the
Lorentz gauge(cid:7) E (cid:28) i(cid:4)A(cid:3)d(cid:9) together with (cid:18)(cid:11)(cid:5)(cid:19) imply that A (cid:28) (cid:11) V(cid:2)gI (cid:2)J(cid:2)
R
(cid:2)
For a region V containing no sources(cid:7) Eq(cid:2) (cid:18)(cid:17)(cid:23)(cid:19) becomes
(cid:2) (cid:2)
E (cid:28) (cid:18)i(cid:4)(cid:11)G(cid:2)H (cid:25)(cid:5)dG(cid:2)E (cid:19)(cid:3) (cid:18)(cid:11)(cid:9)(cid:19)
Z(cid:2)V(cid:2)
Substituting (cid:18)(cid:11)(cid:17)(cid:19) and using dd (cid:28) (cid:10) gives
(cid:2) i(cid:4)(cid:11) (cid:2) (cid:2)
E (cid:28) (cid:18)i(cid:4)(cid:11)gI (cid:2)H (cid:25) dd gI (cid:2)H (cid:25)(cid:5)dgI (cid:2)E (cid:19)(cid:3) (cid:18)(cid:11)(cid:23)(cid:19)
Z(cid:2)V(cid:2) k(cid:5)
Using the product rule for the exterior derivative(cid:7) (cid:18)(cid:11)(cid:23)(cid:19) can be rewritten as
(cid:2) i(cid:4)(cid:11) (cid:2) (cid:2) (cid:2)
E (cid:28) (cid:18)i(cid:4)(cid:11)gI (cid:2)H (cid:25) (cid:20)d(cid:18)d gI (cid:2)H (cid:19)(cid:3)d gI (cid:2)dH (cid:21)(cid:25)(cid:5)dgI (cid:2)E (cid:19)(cid:3) (cid:18)(cid:11)(cid:8)(cid:19)
Z(cid:2)V(cid:2) k(cid:5)
The second term vanishes by the generalized Stokes theorem(cid:2) Using Ampere(cid:27)s law(cid:7) we obtain
the Stratton(cid:13)Chu formula(cid:7)
(cid:2) (cid:2) (cid:2)
E (cid:28) Z(cid:2)V(cid:2)(cid:20)i(cid:4)(cid:11)gI (cid:2)H (cid:25)(cid:18)(cid:5)dgI(cid:19)(cid:2)E (cid:3)(cid:18)d gI(cid:19)(cid:2)(cid:5)E (cid:21) (cid:18)(cid:11)(cid:6)(cid:19)
which again eliminates the dot and cross products with a surface normal found in the usual
vector expression (cid:20)(cid:17)(cid:22)(cid:21)(cid:2)
Warnick(cid:2) et al(cid:2) (cid:3)(cid:12) September (cid:3)(cid:4)(cid:2) (cid:3)(cid:5)(cid:5)(cid:6)