Table Of ContentElectromagnetic Boundary Conditions and Di(cid:2)erential Forms
Karl F(cid:2) Warnick(cid:3) Richard H(cid:2) Selfridge and David V(cid:2) Arnold
Abstract(cid:2)We develop a new representation for electromagnetic boundary con(cid:4)
ditions involving a boundary projection operator de(cid:5)ned using the interior and
exterior products of the calculus of di(cid:6)erential forms(cid:2) This operator expresses
boundary conditions for (cid:5)elds represented by di(cid:6)erentialforms of arbitrary degree(cid:2)
Withvectoranalysis(cid:3)the(cid:5)eldintensityboundary conditionsrequirethecrossprod(cid:4)
uct(cid:3) whereas the (cid:7)ux boundary conditions use the inner product(cid:2) With di(cid:6)erential
forms(cid:3) the (cid:5)eld intensityand (cid:7)ux density boundary conditions are expressed using
a single operator(cid:2) This boundary projection operator is readilyapplied in practice(cid:3)
so that our work extends the utility of the calculus of di(cid:6)erential forms in applied
electromagnetics(cid:2)
Warnick(cid:2)et al(cid:2) (cid:3) March (cid:4)(cid:3)(cid:2) (cid:3)(cid:5)(cid:5)(cid:6)
(cid:8)(cid:2) INTRODUCTION
In this paper we derive a new formulation for the boundary conditions at a discontinuity
in the electromagnetic(cid:5)eld using di(cid:6)erential forms(cid:2) The utility of the calculus of di(cid:6)erential
forms in electromagnetic (cid:5)eld (cid:9)EM(cid:10) theory has been demonstrated by Deschamps (cid:11)(cid:8)(cid:12)(cid:3) Bal(cid:4)
domir (cid:11)(cid:13)(cid:12)(cid:3) Schleifer (cid:11)(cid:14)(cid:12)(cid:3) Thirring (cid:11)(cid:15)(cid:12)(cid:3) Burke (cid:11)(cid:16)(cid:3) (cid:17)(cid:12) and others(cid:2) The intent of this paper is to
extend the range of engineering problems for which di(cid:6)erential forms are useful by providing
a practical means of working with boundary conditions(cid:2)
Thirring (cid:11)(cid:15)(cid:12) and Burke (cid:11)(cid:16)(cid:3) (cid:17)(cid:12) treat boundary conditions using the calculus of di(cid:6)erential
forms(cid:2) Thirring(cid:18)s approach is similar to ours(cid:3) but we extend his methods by introducing a
boundary projection operator to express boundary conditions for forms of arbitrary degree
in a space of arbitrary dimension(cid:2) The expressions for junction conditions on (cid:5)eld intensity
and (cid:7)ux density(cid:3) for example(cid:3) are identical in form(cid:2)
For many electromagnetic quantities(cid:3) the vector representation and the representation as
a di(cid:6)erential form are duals(cid:3) so that their components di(cid:6)er only by metrical coe(cid:19)cients(cid:2)
This is not the case for the surface current and charge density twisted forms yielded by the
boundary projection operator(cid:2) It is simpler to compute(cid:3) for example(cid:3) total current through
a path using the surface current twisted (cid:8)(cid:4)form than using the usual surface current vector(cid:2)
In Sec(cid:2) (cid:13) we derive an expression for boundary sources at a (cid:5)eld discontinuity using the
boundary projection operator(cid:2) In Sec(cid:2) (cid:14) we provide a simple computational example to
illustrate the method(cid:2) Due to the unfamilarity of most engineers with di(cid:6)erential forms and
associated notation(cid:3) our treatment is more elementary than is usual(cid:2) Accordingly(cid:3) we also
providean introductionto theinteriorproduct and twistedformsinan Appendix(cid:2) This work
shows that the calculus of di(cid:6)erential forms is useful for practical EM problems involving
boundary conditions(cid:2)
(cid:13)(cid:2) BOUNDARY CONDITIONS
In this section we derive an expression for sources on a boundary where the electromag(cid:4)
Warnick(cid:2)et al(cid:2) (cid:4) March (cid:4)(cid:3)(cid:2) (cid:3)(cid:5)(cid:5)(cid:6)
netic (cid:5)eld is discontinuous(cid:2) We (cid:5)nd that generalized boundary conditions can be given
using a boundary projection operator(cid:2) We then discuss the resulting boundary conditions
for magnetic (cid:5)eld intensity and electric (cid:7)ux density(cid:2)
(cid:13)(cid:2)(cid:8) Representing Surfaces With (cid:8)(cid:4)forms
In an n(cid:4)dimensional space(cid:3) a (cid:8)(cid:4)form is represented graphically by (cid:9)n(cid:4)(cid:8)(cid:10)(cid:4)dimensional hy(cid:4)
persurfaces(cid:2) In n(cid:4)space(cid:3) a boundary is also an (cid:9)n(cid:4)(cid:8)(cid:10)(cid:4)dimensional hypersurface(cid:2) Thus(cid:3) we
can use (cid:8)(cid:4)forms to represent boundaries(cid:2) If a continuous function f(cid:9)x(cid:0)(cid:2)(cid:3)(cid:3)(cid:3)(cid:2)xn(cid:10) vanishes
(cid:9)or is constant(cid:10) along a boundary(cid:3) then the (cid:8)(cid:4)form df graphically has a surface that lies
on the boundary(cid:2) The surface of a paraboloid re(cid:7)ector antenna(cid:3) for example(cid:3) is given by
(cid:2) (cid:2)
x y (cid:20)az (cid:21) (cid:22)(cid:3) so that the unnormalized boundary (cid:8)(cid:4)form is (cid:13)xdx (cid:13)ydy (cid:20)adz(cid:2) A
(cid:0) (cid:0) (cid:0) (cid:0)
randomly rough surface is written h(cid:9)x(cid:2)y(cid:10)(cid:20)z (cid:21) (cid:22)(cid:3) giving the boundary (cid:8)(cid:4)form dh(cid:20) dz(cid:2)
(cid:0) (cid:0)
For the remainder of this paper we will use the notation
df df
n (cid:21) (cid:9)(cid:8)(cid:10)
(cid:2) df
df df
j j
q
where is the interior product(cid:3) which is discussed in the Appendix(cid:2) The (cid:8)(cid:4)form n is dual to
the usual surface normal vector n(cid:23)(cid:2)
(cid:13)(cid:2)(cid:13) Derivation of the Boundary Projection Operator
Inthreedimensions(cid:3)letE betheelectric(cid:5)eldintensity(cid:8)(cid:4)form(cid:3)H themagnetic(cid:5)eldintensity
(cid:8)(cid:4)form(cid:3) D the electric(cid:7)ux density (cid:13)(cid:4)form(cid:3) B the magnetic(cid:7)ux density (cid:13)(cid:4)form(cid:3) J the electric
current density (cid:13)(cid:4)form and (cid:4) the charge density (cid:14)(cid:4)form(cid:2) Then Maxwell(cid:18)s laws are
(cid:5)
dE (cid:21) B
(cid:0)(cid:5)t
(cid:5)
dH (cid:21) D(cid:20)J (cid:9)(cid:13)(cid:10)
(cid:5)t
dD (cid:21) (cid:4)
dB (cid:21) (cid:22)(cid:3)
Each equation equates the exterior derivative of a di(cid:6)erential form to the sum of a source
and a nonsingular (cid:5)eld term(cid:3) of which one or both may vanish(cid:2) Recognizing that Maxwell(cid:18)s
Warnick(cid:2)et al(cid:2) (cid:7) March (cid:4)(cid:3)(cid:2) (cid:3)(cid:5)(cid:5)(cid:6)
laws have a common form(cid:3) we can derive an expression for the boundary sources that is the
same for both (cid:5)eld intensity (cid:9)(cid:8)(cid:4)forms(cid:10) and (cid:7)ux density (cid:9)(cid:13)(cid:4)forms(cid:10)(cid:2)
Let (cid:6) be a p(cid:4)form with p (cid:7) n (cid:9)where n is the dimension of space(cid:10) that represents a (cid:5)eld
with a (cid:9)p(cid:20)(cid:8)(cid:10)(cid:4)form (cid:8) as a source(cid:3) so that
d(cid:6) (cid:21) (cid:9) (cid:20)(cid:8) (cid:9)(cid:14)(cid:10)
(cid:0)
where (cid:9) is nonsingular(cid:2) Let f (cid:21) (cid:22) represent a boundary(cid:3) where f is C and vanishes only
along the boundary(cid:2) Let (cid:6) equal (cid:6)(cid:2) for f (cid:10) (cid:22) and (cid:6)(cid:0) for f (cid:7) (cid:22)(cid:2)
We can write (cid:6) (cid:21) (cid:9)(cid:6)(cid:2) (cid:6)(cid:0)(cid:10)(cid:11)(cid:9)f(cid:10)(cid:20)(cid:6)(cid:0)(cid:3) where (cid:11) is the unit step function(cid:2) Then
(cid:0)
(cid:9) (cid:20)(cid:8) (cid:21) d (cid:9)(cid:6)(cid:2) (cid:6)(cid:0)(cid:10)(cid:11)(cid:9)f(cid:10)(cid:20)(cid:6)(cid:0)
f (cid:0) g
(cid:21) (cid:12)(cid:9)f(cid:10)df (cid:9)(cid:6)(cid:2) (cid:6)(cid:0)(cid:10)(cid:20)(cid:11)(cid:9)f(cid:10)d(cid:9)(cid:6)(cid:2) (cid:6)(cid:0)(cid:10)(cid:20)d(cid:6)(cid:0)(cid:3) (cid:9)(cid:15)(cid:10)
(cid:3) (cid:0) (cid:0)
(cid:21) (cid:12)(cid:24)(cid:9)f(cid:10)n (cid:9)(cid:6)(cid:2) (cid:6)(cid:0)(cid:10)(cid:20)(cid:11)(cid:9)f(cid:10)d(cid:9)(cid:6)(cid:2) (cid:6)(cid:0)(cid:10)(cid:20)d(cid:6)(cid:0)
(cid:3) (cid:0) (cid:0)
where (cid:12) is the Dirac delta function and (cid:12)(cid:24)(cid:9)f(cid:10) is (cid:12)(cid:9)x(cid:0) x(cid:0)(cid:3)(cid:10) (cid:12)(cid:9)xn xn(cid:3)(cid:10) such that the point
(cid:0) (cid:4)(cid:4)(cid:4) (cid:0)
(cid:0) n (cid:2)(cid:4)(cid:5)f(cid:6)
(cid:9)x(cid:3)(cid:2)(cid:3)(cid:3)(cid:3)(cid:2)x(cid:3)(cid:10) lies on the boundary and (cid:12)(cid:9)f(cid:10) (cid:21) (cid:2) The singular parts of both sides of (cid:9)(cid:15)(cid:10)
df df
must be equal(cid:3) so that q
(cid:0)
(cid:8) (cid:21) (cid:12)(cid:24)(cid:9)f(cid:10)n (cid:9)(cid:6)(cid:2) (cid:6)(cid:0)(cid:10) (cid:9)(cid:16)(cid:10)
(cid:3) (cid:0)
(cid:0)
where (cid:8) is the singular part of (cid:8)(cid:3) representing the boundary source along f (cid:21) (cid:22)(cid:2) Since the
(cid:0)
source (cid:8) is con(cid:5)ned to the boundary(cid:3) it can be written (cid:11)(cid:15)(cid:12)
(cid:0)
(cid:8) (cid:21) (cid:12)(cid:24)(cid:9)f(cid:10)n (cid:8)s (cid:9)(cid:17)(cid:10)
(cid:3)
(cid:0)
where (cid:8)s is a p(cid:4)form(cid:3) the restriction of (cid:8) to the boundary(cid:2) Integrating (cid:9)(cid:16)(cid:10) and (cid:9)(cid:17)(cid:10) over a
region containing the boundary shows that the equality
n (cid:8)s (cid:21) n (cid:9)(cid:6)(cid:2) (cid:6)(cid:0)(cid:10)(cid:3) (cid:9)(cid:25)(cid:10)
(cid:3) (cid:3) (cid:0)
must hold on the boundary(cid:2) We then take the interior product of both sides of (cid:9)(cid:25)(cid:10) by n and
apply the identity (cid:9)(cid:14)(cid:22)(cid:10) to obtain
n (cid:9)n (cid:9)(cid:6)(cid:2) (cid:6)(cid:0)(cid:10)(cid:10) (cid:21) n (cid:9)n (cid:8)s(cid:10)
(cid:3) (cid:0) (cid:3)
(cid:21) (cid:9)n n(cid:10) (cid:8)s n n (cid:8)s(cid:3) (cid:9)(cid:26)(cid:10)
(cid:3) (cid:0) (cid:3)
Warnick(cid:2)et al(cid:2) (cid:8) March (cid:4)(cid:3)(cid:2) (cid:3)(cid:5)(cid:5)(cid:6)
Because n is normalized(cid:3) n n (cid:21) (cid:8)(cid:2) Since (cid:8)s is by de(cid:5)nition con(cid:5)ned to the boundary(cid:3)
n (cid:8)s (cid:21) (cid:22)(cid:2) Graphically(cid:3) the surfaces of (cid:8)s are perpendicular to the boundary because (cid:8)s can
contain no factor proportional to n(cid:2) Applying n (cid:8)s (cid:21) (cid:22) to (cid:9)(cid:26)(cid:10)(cid:3) we have
(cid:8)s (cid:21) n (cid:9)n (cid:9)(cid:6)(cid:2) (cid:6)(cid:0)(cid:10)(cid:10) (cid:9)(cid:27)(cid:10)
(cid:3) (cid:0)
which is the central result of this paper(cid:2)
Eq(cid:2) (cid:9)(cid:27)(cid:10) applies to both nontwisted and twisted forms(cid:2) As noted below(cid:3) electromagnetic
sources are conveniently represented by twisted forms(cid:2) If (cid:8)s is a twisted form(cid:3) we must
provide an outer orientation (cid:9)see Appendix A and Burke (cid:11)(cid:17)(cid:12)(cid:10) for (cid:8)s(cid:2) The orientation is given
by
(cid:9)(cid:8)s(cid:2)(cid:28)s(cid:10) n (cid:8)s (cid:21) (cid:28) (cid:9)(cid:8)(cid:22)(cid:10)
f g(cid:3) (cid:3)
where (cid:9)(cid:8)s(cid:2)(cid:28)s(cid:10) is a nontwisted (cid:9)n p (cid:8)(cid:10)(cid:4)form specifying the outer orientation of (cid:8)s and
f g (cid:0) (cid:0)
(cid:28) is a volume element (cid:9)n(cid:4)form(cid:10) serving as an orientation for the surrounding n(cid:4)space(cid:2) (cid:28)s
is a volume element in the boundary(cid:3) but need not be found in order to obtain the outer
orientation of (cid:8)s(cid:2) Given any boundary(cid:3) there are two possible choices for n(cid:2) The orientation
speci(cid:5)ed by(cid:9)(cid:8)(cid:22)(cid:10) is easily seen to be independentof that choice(cid:2) In right(cid:4)handed coordinates(cid:3)
an equivalence can be made between inner and outer orientations(cid:3) and the orientation of (cid:8)s
can be taken as that of the (cid:9)p(cid:20)(cid:8)(cid:10)(cid:4)form n (cid:8)s(cid:2)
(cid:3)
In Sections (cid:13)(cid:2)(cid:15) and (cid:13)(cid:2)(cid:16)(cid:3) where (cid:9)(cid:27)(cid:10) is specialized to surface current and surface charge
densities(cid:3) we provide a simpler means for orienting surface sources(cid:2) We (cid:5)nd that the need
for an orientation for (cid:8)s corresponds precisely to conventions used with the vector calculus
when integrating Js and the scalar surface charge density qs(cid:2)
Since (cid:9)(cid:27)(cid:10) applies to any situation for which a law of the form (cid:9)(cid:14)(cid:10) is valid(cid:3) we can use
Maxwell(cid:18)s laws (cid:9)(cid:13)(cid:10) to write at a boundary f (cid:21) (cid:22)(cid:3)
n (cid:9)n (cid:9)E(cid:2) E(cid:0)(cid:10)(cid:10) (cid:21) (cid:22)
(cid:3) (cid:0)
n (cid:9)n (cid:9)H(cid:2) H(cid:0)(cid:10)(cid:10) (cid:21) Js
(cid:3) (cid:0)
n (cid:9)n (cid:9)D(cid:2) D(cid:0)(cid:10)(cid:10) (cid:21) (cid:4)s (cid:9)(cid:8)(cid:8)(cid:10)
(cid:3) (cid:0)
n (cid:9)n (cid:9)B(cid:2) B(cid:0)(cid:10)(cid:10) (cid:21) (cid:22)
(cid:3) (cid:0)
where Js is the surface current twisted (cid:8)(cid:4)form and (cid:4)s is the surface charge twisted (cid:13)(cid:4)form(cid:2)
In four(cid:4)space we have dF (cid:21) (cid:22) and d(cid:13)F (cid:21) j(cid:3) where F (cid:21) B(cid:20)E dt(cid:3) (cid:13)F (cid:21) D H dt and
(cid:3) (cid:0) (cid:3)
Warnick(cid:2)et al(cid:2) (cid:6) March (cid:4)(cid:3)(cid:2) (cid:3)(cid:5)(cid:5)(cid:6)
j (cid:21) (cid:4) J dt(cid:2) We can express all four boundary conditions as
(cid:0) (cid:3)
n (cid:9)n (cid:9)F(cid:2) F(cid:0)(cid:10)(cid:10) (cid:21) (cid:22)
(cid:3) (cid:0)
n (cid:9)n (cid:9)(cid:13)F(cid:2) (cid:13)F(cid:0)(cid:10)(cid:10) (cid:21) js (cid:9)(cid:8)(cid:13)(cid:10)
(cid:3) (cid:0)
where js (cid:21) (cid:4)s Js dt and units are suitably normalized(cid:2)
(cid:0) (cid:3)
The operator n n might be termed the boundary projection operator(cid:2) Graphically(cid:3) this
(cid:3)
operator removes the component of a form with surfaces parallel to the boundary(cid:2) The
boundary projection of a (cid:8)(cid:4)form has surfaces perpendicular to the boundary(cid:2) The boundary
projection of a (cid:13)(cid:4)form has tubes perpendicular to the surface at every point(cid:2)
(cid:13)(cid:2)(cid:14) Decomposition of Forms at a Boundary
Any form (cid:14) of degree less than the dimension of the space we are working in can be decom(cid:4)
(cid:2)(cid:0)
posed using the boundary projection operator and its complementaryoperation (cid:13) n n (cid:13)(cid:3)
(cid:3)
so that
(cid:2)(cid:0)
(cid:14) (cid:21) n (cid:9)n (cid:14)(cid:10)(cid:20)(cid:13) (cid:11)n (cid:9)n (cid:13)(cid:14)(cid:10)(cid:12) (cid:9)(cid:8)(cid:14)(cid:10)
(cid:3) (cid:3)
where the second termon the right(cid:4)hand side is the component of (cid:14) with surfaces parallel to
the boundary(cid:2) This identity can be proved by expressing (cid:14) in terms of orthonormal (cid:8)(cid:4)forms
i i i
dx at a point(cid:3) verifying (cid:9)(cid:8)(cid:14)(cid:10) for n (cid:21) dx and extending to the general case n (cid:21) nidx (cid:3)
(cid:2)
where i(cid:9)ni(cid:10) (cid:21) (cid:8) using the linearity of the exterior and interior products(cid:2) The operator
(cid:13)(cid:2)(cid:0)n nP (cid:13) can be used to obtain the arbitrary part of a (cid:5)eld at a boundary(cid:2)
(cid:3)
We discuss special cases of the boundary projection operator for (cid:8)(cid:4)forms and (cid:13)(cid:4)forms
below(cid:2)
(cid:13)(cid:2)(cid:15) Surface Current
Surface current is represented by a twisted (cid:8)(cid:4)form(cid:2) Fig(cid:2) (cid:8) shows how this (cid:8)(cid:4)form is obtained
fromthemagnetic(cid:5)eldintensityataboundary (cid:9)Fig(cid:2)(cid:8)a(cid:10)(cid:2) Fig(cid:2)(cid:8)bshowsthe(cid:8)(cid:4)form(cid:9)H(cid:2) H(cid:0)(cid:10)(cid:2)
(cid:0)
(cid:9)Note that the (cid:8)(cid:4)forms H(cid:2) and H(cid:0) are both de(cid:5)ned above and below the boundary even
though each represents the (cid:5)eld on one side of the boundary only(cid:2) Thus(cid:3) we can consider
Warnick(cid:2)et al(cid:2) (cid:9) March (cid:4)(cid:3)(cid:2) (cid:3)(cid:5)(cid:5)(cid:6)
H(cid:2) H(cid:0) as being de(cid:5)ned for all space(cid:3) even though only its value on the boundary is of
(cid:0)
interest(cid:2)(cid:10) Fig(cid:2) (cid:8)c shows n (cid:9)n (cid:9)H(cid:2) H(cid:0)(cid:10)(cid:10)(cid:2) Fig(cid:2) (cid:8)d is the restriction of this (cid:8)(cid:4)form to the
(cid:3) (cid:0)
boundary(cid:3) along with the corresponding vector Js(cid:2)
Graphically(cid:3) the boundary projection operator removes any component of H(cid:2) H(cid:0) with
(cid:0)
surfacesparalleltotheboundary(cid:2) Physically(cid:3)thisexpressesboththerequirementthatsurface
current can only (cid:7)ow along the boundary and the arbitrariness of the normal component of
H(cid:2) H(cid:0) at the boundary(cid:2)
(cid:0)
Most forms in EM theory(cid:3) such as E(cid:3) H(cid:3) D and B(cid:3) are dual to the corresponding vector
quantity(cid:3)so that components di(cid:6)er only by metricalcoe(cid:19)cients(cid:2) The twisted surface current
(cid:8)(cid:4)form Js(cid:3) however(cid:3) is not the dual of the vector Js(cid:2) The nontwisted (cid:8)(cid:4)form with the same
components as Js does not satisfy the simple de(cid:5)nition given below in (cid:9)(cid:8)(cid:15)(cid:10)(cid:2)
The surface current (cid:8)(cid:4)form Js can be de(cid:5)ned in terms of the (cid:7)ow vector v of a surface
charge distribution (cid:4)s(cid:2) The current density(cid:13)(cid:4)formJ is J (cid:21) v (cid:4)(cid:2) The surface currentdensity
is Js (cid:21) v (cid:4)s(cid:2) If (cid:4)s (cid:21) qdxdy and the (cid:7)ow (cid:5)eld v (cid:21) vy(cid:23)(cid:3) then Js (cid:21) vy(cid:23) qdxdy (cid:21) qvdx(cid:2)
(cid:0) (cid:0)
The (cid:8)(cid:4)form dual to Js would be qvdy(cid:2)
The integral of the surface current density over a path should yield the total current
through thepath(cid:2) The(cid:8)(cid:4)formJs as obtainedusing theboundary projectionoperator satis(cid:5)es
this de(cid:5)nition(cid:29)
I (cid:21) Js (cid:9)(cid:8)(cid:15)(cid:10)
P
Z
where P is a path(cid:2) The sense of I is with respect to the direction of the (cid:13)(cid:4)form n s(cid:3)
(cid:3)
where s is the (cid:8)(cid:4)form dual to the tangent vector s of P (cid:9)so that s is a (cid:8)(cid:4)form with surfaces
perpendicular to the path P and oriented in the direction of integration(cid:10)(cid:2)
The simple integral in (cid:9)(cid:8)(cid:15)(cid:10) replaces a much more cumbersome vector expression(cid:2) The
total current through P using the usual surface current vector is
I (cid:21) Js (cid:9)n(cid:23) ds(cid:10) (cid:9)(cid:8)(cid:16)(cid:10)
P (cid:4) (cid:5)
Z
where the sense of I is relative to the direction of n(cid:23) ds(cid:2) Note that this is the same as the
(cid:5)
reference direction n s of (cid:9)(cid:8)(cid:15)(cid:10)(cid:2)
(cid:3)
The integral in (cid:9)(cid:8)(cid:15)(cid:10) is evaluated in practice by the method of pullback(cid:2) If a surface S is
parameterized by x (cid:21) (cid:6)(cid:0)(cid:9)u(cid:2)v(cid:10)(cid:3) y (cid:21) (cid:6)(cid:2)(cid:9)u(cid:2)v(cid:10)(cid:3) z (cid:21) (cid:6)(cid:7)(cid:9)u(cid:2)v(cid:10) where u and v range over some
Warnick(cid:2)et al(cid:2) (cid:10) March (cid:4)(cid:3)(cid:2) (cid:3)(cid:5)(cid:5)(cid:6)
subsetT oftheu(cid:30)v plane(cid:3)thentheintegralof(cid:14) (cid:21) (cid:14)(cid:0)(cid:9)x(cid:2)y(cid:2)z(cid:10)dx(cid:20)(cid:14)(cid:2)(cid:9)x(cid:2)y(cid:2)z(cid:10)dy(cid:20)(cid:14)(cid:7)(cid:9)x(cid:2)y(cid:2)z(cid:10)dz
over S is
(cid:3)
(cid:14) (cid:21) (cid:6) (cid:14)
S T
Z Z
(cid:21) (cid:14)(cid:0)(cid:9)(cid:6)(cid:0)(cid:2)(cid:6)(cid:2)(cid:2)(cid:6)(cid:7)(cid:10)d(cid:6)(cid:0) (cid:20)(cid:14)(cid:2)(cid:9)(cid:6)(cid:0)(cid:2)(cid:6)(cid:2)(cid:2)(cid:6)(cid:7)(cid:10)d(cid:6)(cid:2) (cid:20)(cid:14)(cid:7)(cid:9)(cid:6)(cid:0)(cid:2)(cid:6)(cid:2)(cid:2)(cid:6)(cid:7)(cid:10)d(cid:6)(cid:7) (cid:9)(cid:8)(cid:17)(cid:10)
T
Z
where the superscript denotes the pullback operation(cid:2) After pullback(cid:3) the integrand be(cid:4)
(cid:6)
comes a (cid:8)(cid:4)form in du and dv(cid:2) Partial derivatives of the coordinate transformation enter
naturally via the exterior derivatives of (cid:6)(cid:0)(cid:3) (cid:6)(cid:2) and (cid:6)(cid:7)(cid:2) Eq(cid:2) (cid:9)(cid:8)(cid:17)(cid:10) is easily generalized to
integrals of (cid:13)(cid:4)forms(cid:2)
(cid:13)(cid:2)(cid:16) Surface Charge
Surface charge densitydue to discontinuouselectric(cid:7)ux densityat a boundary is represented
by the twisted (cid:13)(cid:4)form (cid:4)s (cid:21) n (cid:9)n (cid:9)D(cid:2) D(cid:0)(cid:10)(cid:10)(cid:2) The boundary projection operator removes
(cid:3) (cid:0)
any component of D(cid:2) D(cid:0) with tubes parallel to the boundary(cid:3) as shown in Fig(cid:2) (cid:13)(cid:2)
(cid:0)
The (cid:13)(cid:4)form (cid:4)s obtained using the boundary projection operator di(cid:6)ers from the usual
value qs (cid:21) n(cid:23) (cid:9)D(cid:2) D(cid:3)(cid:10) because (cid:4)s is a (cid:13)(cid:4)form(cid:3) whereas qs is a scalar(cid:2) The total charge on
(cid:4) (cid:0)
an area A of a boundary with surface charge (cid:4)s is
Q (cid:21) (cid:4)s(cid:3) (cid:9)(cid:8)(cid:25)(cid:10)
A
Z
Q is positive for positive surface charge if the (cid:13)(cid:4)form (cid:14) that satis(cid:5)es n (cid:14) (cid:21) (cid:28) also satis(cid:5)es
(cid:3)
A(cid:14) (cid:10) (cid:22)(cid:3) where (cid:28) is the standard volumeelement(cid:3) dxdydz in rectangular coordinates(cid:2) This
R
corresponds exactly to the convention of choosing dS in Q (cid:21) AqsdS (cid:9)where qs is the usual
R
surface charge density scalar(cid:10) such that AdS is positive(cid:2) The sign of the charge represented
nR(cid:4)(cid:3)s
by (cid:4)s can also be found by computing (cid:2)
(cid:8)
(cid:13)(cid:2)(cid:17) Comparison to Burke(cid:18)s Pullback Method
Burke (cid:11)(cid:17)(cid:12) derives an expression for boundary sources using the method of pullback(cid:2) In his
notation(cid:3) boundary conditions have the form (cid:11)H(cid:12) (cid:21) Js and (cid:11)D(cid:12) (cid:21) (cid:4)s(cid:2) The square brackets
Warnick(cid:2)et al(cid:2) (cid:11) March (cid:4)(cid:3)(cid:2) (cid:3)(cid:5)(cid:5)(cid:6)
denote the sum of the pullback of H above the boundary to the boundary and the pullback
of H below the boundary to the boundary(cid:3) so that
(cid:3) (cid:3)
Js (cid:21) (cid:11)H(cid:12) (cid:6)(cid:2)H(cid:2) (cid:20)(cid:6)(cid:0)H(cid:0) (cid:9)(cid:8)(cid:26)(cid:10)
(cid:2)
where (cid:6)(cid:2) and (cid:6)(cid:0) are functions of the space above and below the boundary into theboundary(cid:2)
The pullback method has a concise and elegant proof in (cid:11)(cid:17)(cid:12)(cid:2) By transforming to a coordi(cid:4)
(cid:0) n (cid:0)
nate systemx (cid:2)(cid:3)(cid:3)(cid:3)(cid:2)x such that a boundary is given byx (cid:21) (cid:22)(cid:3) it can be shown that Burke(cid:18)s
(cid:0)
formulation is equivalent to n n for n (cid:21) dx (cid:2) Although the pullback boundary conditions
(cid:3)
are mathematically very natural(cid:3) the boundary projection operator has the advantage that
(cid:9)as with the usual vector formulation(cid:10) the boundary sources are always expressed in the
same coordinates as the (cid:5)elds(cid:2)
(cid:14)(cid:2) EXAMPLE
Aconductingboundary liesalongthesurfacez (cid:21) cosy(cid:2) Abovetheboundary themagnetic
(cid:5)eld is H(cid:2) (cid:21) H dx(cid:2) Below the (cid:5)eld is zero(cid:2) This is shown in Fig(cid:2) (cid:14)a(cid:2)
We can represent the surface by f(cid:9)x(cid:2)y(cid:2)z(cid:10) (cid:21) z cosy (cid:21) (cid:22)(cid:2) Computing the normalized
(cid:0)
exterior derivative of this function(cid:3)
d(cid:9) cosy(cid:20)z(cid:10)
n (cid:21) (cid:0)
d(cid:9) cosy(cid:20)z(cid:10)
j (cid:0) j
sinydy (cid:20) dz
(cid:21) (cid:3) (cid:9)(cid:8)(cid:27)(cid:10)
(cid:2)
(cid:8)(cid:20)sin y
q
The boundary projection of H(cid:2) is
Js (cid:21) n n H(cid:2)
(cid:3)
H
(cid:21) (cid:2) (cid:9)sinydy (cid:20) dz(cid:10) (cid:9)sinydydx(cid:20) dzdx(cid:10)
(cid:8)(cid:20)sin y
H (cid:2)
(cid:21) (cid:2) (cid:9)dx(cid:20)sin ydx(cid:10)
(cid:8)(cid:20)sin y
(cid:21) H dx(cid:3) (cid:9)(cid:13)(cid:22)(cid:10)
Fig(cid:2) (cid:14)c shows the (cid:8)(cid:4)form Js (cid:21) Hdx restricted to the boundary(cid:2) The direction of the arrow
along the lines of H dx is the orientation of the (cid:13)(cid:4)form n Js(cid:2)
(cid:3)
Warnick(cid:2)et al(cid:2) (cid:5) March (cid:4)(cid:3)(cid:2) (cid:3)(cid:5)(cid:5)(cid:6)
Compare the (cid:5)nal expression for Js in (cid:9)(cid:13)(cid:22)(cid:10) to the vector surface current obtained for
H
the same (cid:5)eld and boundary(cid:3) Js (cid:21) (cid:0) (cid:9)y(cid:23) sinyz(cid:23)(cid:10)(cid:2) The di(cid:6)erential form Js (cid:21) H dx
p(cid:0)(cid:9)sin y (cid:0)
indicates clearly that the total current crossing a path in the boundary is simply the extent
of the path in the x direction scaled by the factor H(cid:2) This is not obvious from the vector
expression(cid:2)
(cid:15)(cid:2) CONCLUSION
The boundary projection operator allows one to express electromagnetic boundary con(cid:4)
ditions using di(cid:6)erential forms(cid:3) using the same operator for both (cid:5)eld intensity and (cid:7)ux
density(cid:2) The di(cid:6)erent appearances of the (cid:5)eld intensity and (cid:7)ux density boundary condi(cid:4)
tions expressed using vector analysis is merely an artifact of the mathematical language(cid:2)
The di(cid:6)erential forms for boundary sources obtained via the boundary projection operator
di(cid:6)er from the vectors obtained by standard methods(cid:2) The surface current (cid:8)(cid:4)form Js(cid:3) for
example(cid:3)has a more natural de(cid:5)nition than the usual surface current vector Js(cid:2) This (cid:8)(cid:4)form
is readilyintegrated to yieldtotal current over a path(cid:3) whereas a vector perpendicular to the
path and tangent to the boundary is required to evaluate the integral for total current using
the surface current vector(cid:2) The surface current vector also obscures intuitive properties of
the boundary source that are clearly evident when the source is represented by the (cid:8)(cid:4)form
Js(cid:2) Graphically(cid:3) a source form is the intersection of the (cid:5)eld forms with the boundary(cid:3) for
both the (cid:5)eld intensity and (cid:7)ux density cases(cid:2)
This methodfor representingboundary conditions can be easilyapplied in practicalprob(cid:4)
lems(cid:3) and so helps to open the way for the use of di(cid:6)erential forms on a regular basis in
engineering electromagnetics(cid:2)
Acknowlegements(cid:3) This material is based in part upon work supported under a National
Science Foundation Graduate Fellowship to KFW(cid:2)
Warnick(cid:2)et al(cid:2) (cid:3)(cid:12) March (cid:4)(cid:3)(cid:2) (cid:3)(cid:5)(cid:5)(cid:6)
