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A distributional formulation of electromagnetic fields and sources in the presence of media discontinuities

A self-contained formulation of Maxwell's theory of electromagnetic fields and sources is presented in the language of distributional forms. Properties of a fundamental double-form of bi-degree (p,p) for p ≥ 0 are reviewed in order to establish a computational framework for analysing equations involving the Hodge-de Rham operator for p–forms on space or spacetime and singular sources. A constructive approach to Dirac distributions on (moving) submanifolds embedded in R3 is developed in terms of (Leray) forms generated by the geometry of the embedding and illustrative examples presented from problems in electrostatics, magnetostatics and radiating point charges. The formulation offers a straightforward analysis of the relativistic jump conditions across static and moving interfaces where certain fields become discontinuous and provides a general methodology for electromagnetic modeling where possibly time dependent sources of certain physical attributes, such as electric charge, electric current and polarization or magnetization, are concentrated on localized regions in space or spacetime.


Tribute
It is with great pleasure that I dedicate this article to Tony Sudbery. Since our student days we have maintained a most valuable friendship and I am most grateful to Tony for all the things that he has helped me with in Physics and Mathematics. Many years ago I discovered that he was one of the world's experts in the theory of Quaternions over the complex field and this was something I then knew little about. I was using this algebra in generalized theories of gravitation and thanks to Tony's guidance I was able to make sense of this branch of mathematics in the gravitational context. Later he helped me understand Octonions as we worked together on Octonionic Geometry and Simple Supergravity in 11-Dimensions. Today his breadth of expertise roams many new and exciting fields. I count myself fortunate to have such a valuable colleague and friend.

Introduction
Mathematical distributions [1] play an all pervading role in the physical sciences. They underpin the theoretical formulation of quantum mechanics and many linear systems in classical field theory and are used to extend the solution space for linear partial differential equations to include non-smooth fields with singularities. They also offer a useful tool in electromagnetic modeling where the sources of certain physical attributes, such as electric charge, electric current, polarization or magnetization, are concentrated on localized regions in space [2], [3], [4], [5], [6], [16], [17] or spacetime. These aspects of distributions often occur together in problems in electromagnetic theory and this article explores this symbiosis in the unifying language of exterior double-forms. Such a formulation has direct relevance to many contemporary design problems in accelerator science where one is confronted with a number of difficult issues in electromagnetic computation. The effective resolution of such problems is often contingent on reliable computationally expensive numerical analysis which can greatly benefit from reliable analytic treatments. The formulation below will be illustrated by a number of applications in applied electro-magnetics.

Notation
The formulation exploits the geometric language of exterior differential forms [7] since this is ideally suited to accommodate local changes of coordinates that can be used to simplify the description of boundary and initial-value problems and naturally encapsulates intrinsic global properties of domains. In the following the distinction between smooth (C ∞ ) forms on some regular domain and those with possible singularities or discontinuities will be important. Furthermore smooth forms with compact support will play a pivotal role. These will be referred to as test forms [8], [9], [10] and distinguished below by a superposed hat. It will also be useful sometimes to notationally distinguish p−forms on different manifolds. Thus α M (p) will denote a p−form 1 on some manifold M while γ (p,q) (X, Y ) will denote a double-form of bi-degree (p, q) (see section 4 ) on the product manifold X × Y . Each n−dimensional manifold M will be assumed orientable and endowed with a preferred n-form (M ) 1 induced from a metric tensor g M . One then has [7] the linear Hodge operator (M ) that maps p−forms to (n − p)-forms on M . If g M has signature t g one may write where η ij = diag(±1, ±1, . . . ± 1) and with t g = det(η ij ) and {e i M } is a set of basis 1−forms in ΓT (M ). If α and β are integrable p−forms on U ⊂ R n denote Furthermore if one of the forms has compact support on U (and belongs to the space of Schwartz test functions) an alternative notation is When U = R n the bracket subscript will be omitted.  If (α,φ) is well defined i.e. α is integrable with respect to the bracket (., . ) then one says that α D is a regular Schwartz p−form distribution associated with α. Not all distributions are regular and associated with smooth forms.
Suppose Z is a smooth vector field on R n then Z ≡ g(Z, −) denotes a smooth 1− form 2 . Using the property = t g η n−1 (6) where the involution η on forms satisfies η α (p) then it is straightforward to verify the algebraic identities. α (q) The operator δ ≡ −1 d η is a formal adjoint of the exterior derivative d with respect to the bracket (., .) in the sense that Similarly δ α (p) [ψ with respect to a smooth vector field Z by where ∇ denotes covariant differentiation with respect to the Levi-Civita connection for which In terms of d and δ one defines 3 the Hodge de Rham operator ∆ = d δ + δ d. A number of important identities follow by integrating the Leibnitz relation involving smooth forms From these elementary identities further useful identities involving singular double-forms can be obtained.

Double-Forms on R n
The notion of a double-form was introduced in order to analyse problems in potential theory in R 3 [11]. Double-forms have since been used by Duff [12,13] and others as a powerful tool to study exterior equations on more general manifolds. A double-form of bi-degree (p, q) over the product manifold X × Y may be regarded for each point in X as a p-form valued q−form on Y or for each point in Y as a q-form valued p−form on X . To illustrate these notions suppose X = Y = R n . In the natural coordinate system (x i ) for X and (y i ) for Y one has 4 : The symmetric product is defined so that: and this implies γ for all γ, γ . The Euclidean metric tensors in natural coordinates and their associated Hodge maps will be designated: and (X) then act naturally on forms in the appropriate space.
The fundamental double-form of bi-degree (p, p) is defined as 4 If f : R n × R n → R we denote its image by f (X, Y ) with a similar notation for double-forms The singularity at x = y in γ (p,p) (X, Y ) permeates many of the integrands of integrals that feature below. Such integrals, when they exist, are then defined implicitly as limits. In general integrals U α of n-forms α, over a domain U ⊂ R n , that contain such a singularity are understood as Lim where B is an n-ball of radius centred on the location of the singularity. For any smooth p−form α on Y the double-form γ (p,p) (X, Y ) may be used to define the typepreserving integral operator Γ: α X → (Γα) Y mapping p−forms on X to p−forms on Y : where From (13) and (14) it follows that for all test formsφ X , the differential operators d X and δ X commute with the integral operator Γ: If one writes 5 Vn is the volume of the unit n-sphere in R n . and notes that Γ is a symmetric operator: (Γα, β) = (α, Γβ) = (Γβ, α) then it follows from Green's identity applied to forms with compact support that: and Γ is seen to be an inverse of ∆.
For boundary value problems it is useful to introduce singular double-forms on U that differ from the fundamental double-forms γ = 0. Such double-forms inherit the singularity structure of γ (p,p) (X, Y ) but can be tailored to satisfy certain conditions on the boundary ∂U.
As an example suppose one requires p−form solutions to the generalized Poisson equation that vanish on ∂U. Denote by and Suppose more generally now that G p XY is a double-form satisfying chosen conditions on ∂U. For any smooth p−form on R n define the integral operator G by If y ∈ U the integral is singular at x = y and One may use (16) together with the limits (derived by explicit computation in spherical polar coordinates with origin at the center of B ): where S is the n − 1 dimensional surface of the ball B , to verify In this expression, ν = 0 if y is outside U, ν = 1 2 if y ∈ ∂U and ν = 1 if y ∈ U If the first term on the right of this equation is written in terms of the source S It should be stressed however that this representation is not an explicit solution in general since δφ, dφ, φ and φ cannot in general be assigned values arbitrarily on ∂U. Nevertheless this representation is the cornerstone of many developments in potential theory [2]. The above identities and their subsequent uses are often attributed to Green.

Electromagnetic Fields in Spacetime
Maxwell's equations for an electromagnetic field in an arbitrary medium can be written where F ∈ SΛ 2 M is the Maxwell 2-form, G ∈ SΛ 2 M is the excitation 2-form and j ∈ SΛ 3 M is the 3-form electric current source 6 on a spacetime M endowed with a metric (signature (−1, +1, +1, +1)) and associated Hodge map . The electric 4-current j describes both (mobile) electric charge and effective (Ohmic) currents in a conducting medium. To close this system, "electromagnetic constitutive relations" relating G and j to F are necessary. A unit future-pointing timelike 4-velocity vector field U ∈ ΓT M may be used to describe an observer frame on spacetime and its integral curves model idealized observers. The electric field e ∈ SΛ 1 M and magnetic induction field b ∈ SΛ 1 M associated with F and U are defined by Thus i U e = 0 and i U b = 0 and since g(U, U ) = −1 Likewise the displacement field d ∈ SΛ 1 M and the magnetic field h ∈ SΛ 1 M associated with G are defined with respect to U by and i U d = 0 and i U h = 0. It follows from these definitions that where E = (e ∧ U ) and D = (d ∧ U ).

Time dependent Maxwell Systems in Space
Although the formulation above has a natural setting in a curved spacetime in this article we restrict to a Minkowski spacetime which can be globally foliated by 3-dimensional spacelike hyperplanes. The Minkowski metric induces a metric (and associated Hodge map #) with Euclidean signature on each hyperplane. Furthermore each hyperplane contains events that are deemed simultaneous with respect to a clock attached to any integral curve of a future-pointing unit time-like vector field U = 1 c 0 ∂ ∂t on spacetime and the Hodge map induces a the Hodge map on each hyperplane by the relation The spacetime Maxwell system can now be reduced to a family of exterior systems on R 3 . Each member is an exterior system involving forms on R 3 depending parametrically on time t. Let the g−orthogonal 3 + 1 split of the 4-current 3-form with respect to the foliation be with i ∂ ∂t J (2) = 0. Then, from (48) Here and below an over-dot denotes (Lie) differentiation with respect to the parameter t anḋ α ≡ L ∂ ∂t α. It is convenient to introduce on each hyperplane the (Hodge) dual forms: , D , B , H , j := # J (2) so that the 3 + 1 split of the spacetime covariant Maxwell equations (48) with respect to d B (2) = 0, All p-forms (p ≥ 0) in these equations are independent of dt but have components that may depend parametrically on t.
For a simple isotropic medium at rest in the frame U , b = µ h (1) and d (1) = e (1) where = r 0 , µ = µ r µ 0 . Thus in terms of e (1) , h , E , H : d e (1) = −µḢ (2) , It is straightforward to show from this exterior system that the fields e (1) and h Thus for homogeneous media with zero conductivity these equations each reduce to driven wave equations. These equations simplify considerably for non-dispersive simple homogeneous isotropic media where , µ are constant scalars. Locally on spacetime one has an equivalence class of 1−forms whose elements A differ by the addition of any exact 1−form. Then since F is closed one may write locally F = d A.
Decomposing A (with respect to a particular U ) into 0−form φ and 1-form A potentials yields The Maxwell system then reduces to In a gauge with δA − µφ = 0 the Maxwell system above requires that the potentials must satisfy for sources satisfying (57). Physical solutions correspond to those satisfying physically motivated boundary conditions (in both space and time). These conditions are discussed in later sections. Alternatively in a gauge with δA = 0 (73) In these equations on R 3 with the Euclidean metric, the Hodge map on all forms satisfies ## = 1 so δ = #d # η and ∆ = #d # ηd + d #d #η. When attention is devoted to exterior systems on R 3 it is unnecessary to maintain the notational distinction between d and d, δ and δ, ∆ and ∆.

Electrostatics in R 3 with Sources in Domains with Boundaries
Integral operators analogous to Γ may be used to solve electrostatic boundary value problems in R 3 involving the Hodge-de Rham operator ∆ above and electric fields independent of t. As a basic example consider the Dirichlet problem of finding the electrostatic 0−form potential φ on U ⊂ R 3 that must satisfy The solution may be represented in terms of the integral operator G defined by and is given by where with γ and If the data is smooth it defines regular distributionsρ D ,φ D with supports on U and ∂U respectively byρ Thus for singular distributional sourcesρ D one then has distributional Dirichlet solutions: If the eigenvalues λ M and Dirichlet eigenfunctions Φ N of ∆ on U can be found by solving then a traditional way to satisfy the conditions required of G XY is to express it as a Fourier expansion in the eigenmodes Φ M (Y ) for each point of Y ∈ U.
If the real modes are labeled by a discrete index set M the expansion takes the form If such modes (with support on U) are ortho-normalised so that Since the solutions Φ N are zero on ∂U it follows that This series must be regarded as weakly convergent and the summations become integrations when the eigenvalues are continuous. For example consider U to be the space between two perfectly conducting parallel plates at z = 0 and z = L, separated by a distance L in vacuo. Taking cartesian coordinates {x, y, z} for This gives If the sourceρ is not smoothly distributed in U one needs further technology to evaluate the solution and this will be developed below.

Magnetostatics in R 3 with Smooth Sources
Since magnetic charge is absent in Nature a typical problem in magnetostatics is the determination of a static magnetic field from a stationary electric current. The magnetostatic equations are a subset of Maxwell's equations and may be written: in terms of forms on R 3 . If the current is localized in space (vanishing at ∞) the boundary conditions can be accommodated by using γ  Thus in free space with a smooth sourceJ = 0 one has B = dA (1) in regular domains and in vacuo #B = µ0h where µ0 is the permeability of free space.
Now for a smooth (regular) sourceJ IfJ is a smooth 1−form in R 3 the integralJ D [Γψ] furnishes a solution for A. However if the current has support on a curve in R 3 it must be regarded as a distributional 1− form source not associated with a smooth 1−form. Similarly the charge density in the electrostatic problem may be restricted to a curve or surface in R 3 in which case a distributional sourceρ D not associated with a smooth 0−form must be specified.

Dirac Distributions on Submanifolds
To define singular distributions with support on (possibly disjoint) submanifolds of a manifold M it is convenient to describe each submanifold parametrically as an embedding in M . Suppose that a distribution has support on a collection of submanifolds {S . It is sufficient to establish the distributional framework for a single component of a general chain so henceforth the label (r) will be omitted.
If M is endowed with a metric g (of arbitrary signature and associated Hodge map ), the forms {df j } give rise to a class of forms Ω f (n−k) on M defined with respect to 1 by This class restricts to a natural class of measures on Σ n−k f . Thus each representative induces the measure ω f . Members of the class on M are equivalent if they differ by any combination of the df j . This gauge freedom is of no significance on Σ n−k f since the df j vanish there under pull-back. Thus it is sufficient to represent the class by an element satisfying In those situations where the leaves form orthogonal families, i.e.
where |df j | 2 = df j (df j ) = 0 and the sign of |df j | is chosen so that the direction of the vector field df j ≡ g −1 (df j , −) is in the direction of increasing values of f j . Since df j = 0 one has on Σ n−k f two fields of "unit normals": For non-orthogonal leaves an Ω f can be chosen directly from (103). This condition is equivalent to the equation which yields freedom to choose ω f with non-singular coordinate components when restricted and defined by Furthermore if Y is a smooth vector field in the vicinity of Σ n−k f one may define the directional It follows that where for orthogonal leaves: Since If one regards any point p 0 ∈ M as a 0−dimensional space then the original singular Dirac distribution δ D p 0 (0) with support on p 0 is defined by: It follows that The above singular 0−form Dirac distributions can be used to construct singular p−form Dirac distributions with support on submanifolds in various ways. Thus in a spherical polar chart for R 3 a 1−form 0-layer Dirac distribution on the unit 2−sphere centred at the origin may be represented as where µ θ and µ φ are smooth functions on the sphere. A 1−form 1−layer Dirac distribution on the sphere (associated with some vector field Y on R 3 ) may take the form and a 2−form 1−layer Dirac distribution takes the form for smooth q and s respectively. More generally in coordinates adapted to a p−dimensional submanifold in R n , one has for j = 0, 1, . . . p, the j−form, "r-layer" singular Dirac distribution with support on Σ n−k

Line and Surface Currents in R 3
The source of a smooth magnetostatic field is the smooth current 2-form J in R 3 and the total electric current passing through a surface S ⊂ R 3 is I[S] = S J, measured in amps in MKS units. In some circumstances this current density may be concentrated in the vicinity of 1dimensional submanifolds Σ 1 ⊂ R 3 or 2-dimensional submanifolds Σ 2 ⊂ R 3 . Electric current filaments on material curves in space (wires) and current sheets in material surfaces in space are idealizations of such localized sources and may be modeled by singular distribution-valued 1forms with support on curves and surfaces respectively. In these situations new physical current densities are introduced so that the total current in a segment of wire or a region of a surface is finite and measurable. These singular distributional sources can be mathematically modeled in terms of the distributions δ Σ r f for r = 1, 2 and a vector field W in R 3 with support on Σ r f . For a wire described by the curve with image Σ 1 f introduce the distributional 1-form where I 0 is a smooth function on the wire. Hence and with J = #J: is zero on Σ 1 f . It does not make sense to ascribe a physical dimension to I D belonging to ΓΛ 0 Σ 1 f is the total current (amps) flowing along the wire at any point s ∈ Σ 1 f In a similar way, for the 1-form distribution on R 3 : ) on Σ 2 f . Herȇ ) ∈ ΓΛ 1 Σ 2 f is a smooth 1-form (measured in amps) on the surface Σ 2 f . If C 1 ∈ R 3 is any space-curve lying on this surface then C 1κ (1) is the total current in amps crossing this curve in the direction W , on Σ 2 f . This may be compared with the definition above of the total current (in amps), S J, crossing the surface S ∈ R 3 in the direction #J. If C 1 ∂ ∂s is a unit tangent vector 8 to C 1 then i ∂ ∂sκ (1) is the surface current density in amp m −1 .
It is worth noting that I 0 ∈ ΓΛ 0 R 3 , κ 0 ∈ ΓΛ 0 R 3 , W ∈ ΓT R 3 and Σ 1 f , Σ 2 f are the primary objects used to define the distributions I D

Magnetostatic Fields from a Singular Distributional Stationary Current Source
Suppose a uniform electric current confined to a wire in R 3 is modeled in terms of a 1−form 0−layer Dirac distribution δ Σ 1 f with support on the wire. A distributional sourceJ D will generate a distributional 1−form A D . Thus define A D by SupposeJ D has a support on the locus described by f = (f 1 , f 2 ) with f 1 = r − a, and f 2 = z in cylindrical coordinates (r, φ, z). This will be used to model a uniform stationary current source in a circular loop of radius a ifJ and where In cylindrical polar coordinates, x = (r cos φ, r sin φ, z), y = (r cos φ , r sin φ , z ) and the Euclidean metric tensor is Hence dφ = 1 or with Ψ = φ − φ and noting that the integrand is an even periodic function of Ψ: with magnitude independent of φ. For a constant currentȊ in the circular loop the constant λ = 4π µ 0Ȋ a. In MKS units the physical unit for J (2) and hence h (1) is the ampere and that forJ is µ 0 ampere/m. The methodology here illustrated for a circular planar coil is directly applicable to any open or closed current carrying conductor of arbitrary shape in space and possibly composed of piecewise smooth connected segments.
To verify that when A D [ψ] = X A ∧ #ψ the distribution A D lies in the gauge satisfying δA D = 0 one must compute = a d φ and Σ 1 f (Y ) is a closed curve. This is physically equivalent to the statement that the current in the coil is "conserved". 9 The term in the integrand proportional to sin(φ − φ ) integrates to zero.

Magnetostatic Fields from a Steady Helical Line Current
The derivation above of the potential for the magnetic field due to a circular line current was somewhat labored. However no further steps are required to find the potentials due to more complex geometries once one parametrises the source in terms of the geometry of its support in space. Suppose a helical wire with pitch p > 0 and radius a is the space curve (r = a, φ = √ 1−p 2 a σ, z = p σ) in cylindrical polar coordinates. The parameter σ is the arc-length parameter so 0 < σ < L for a helix of length L. It is convenient to introduce P > 0 with a 2 P 2 + p 2 = 1. Then the unit tangent to the helix is the direction W = p ∂ ∂z + a P ∂ ∂φ . If one chooses to describe the helix in terms of the foliating set f = {f 1 = r − a, f 2 = z − p P φ} this yields a Leray form with support on the helical wire yields the magnetostatic potential 1−form For a constant currentȊ 0 in the helix the constant λ = 4π µ 0Ȋ0 a. The potential A yields a global description of the field in terms of elliptic integrals that depend on the geometry of the helix specified by the parameters L, a and p or P . Such parameters offer natural scales that are useful for defining dimensionless variables that in turn can be used to generate multipole or asymptotic expansions of the above integrals. The practical generation of high intensity uniform magnetic fields by carefully designing coils with complex helical windings is of paramount importance in the construction of undulators and free electron lasers.

Time Dependent Electromagnetic fields and Smooth Sources
When the smooth electromagnetic sources ρ and J depend on t they may generate time dependent electromagnetic fields that must satisfy (62), (63), (64), (65). These equations offer a well-posed initial-boundary value problem that is traditionally approached by finding timedependent potentials φ and A that satisfy the source-driven wave-equations (71), (72) (subject to initial and boundary conditions) in the Lorentz gauge. When the sources are distributional one seeks distributional potentials subject to similar conditions. The distributional formulation given above for real static field configurations generalizes without difficulty to the time dependent situation. The essential modification is to define functionals on the space of t dependent complex test forms on R 3 and exploit the Fourier transform [14] of such forms. Thus ifψ is a t-dependent test p-form on X = R 3 its Fourier transform is the ω-dependent test p-form Fψ on X = R 3 defined by For fields in the vacuum the fundamental t-dependent (p, p) type double-form is the real part of the Fourier transform of 10 Motivated by (155) the distribution C D associated with the smooth p−form C(X, t) is defined by Hence Thus finally whereJ D is the distribution associated with the smooth time-dependent p−formJ . If one now contemplates a distributional sourceJ D that is not associated with a smooth time-dependent p−form on R 3 then (161) offers a particular distributional solution C D to (150) based on a retarded extension of the fundamental solution γ The above solution is immediately applicable to the problem of finding the retarded electromagnetic potentials A and φ in free space, where c 0 2 µ = 1, for any smooth time dependent source. Thus from section (6) one sees that with a smooth sourceJ = − ρ in equation (150) the smooth 0−form solution φ associated with C D describes a free-space scalar potential solution to (72) and that with a smooth sourceJ = µ#J in equation (150) the smooth 1−form solution A associated with C D describes a free-space vector potential solution to (71).
In the absence of losses due to conduction the convective current spacetime 3−form is given as j = ρ 0 V where the unit future-pointing time-like vector field V convects proper-charge density ρ 0 . In a local spacetime co-basis {c 0 d t, e 1 , e 2 , e 3 } adapted to U the source velocity The analysis above also offers distributional solutions to (71) and (72) in terms of scalar and 1−form distributional sources. The electromagnetic field associated with an arbitrarily moving charge point source is space can now be modeled in terms of a moving Dirac distribution in space.

Time Dependent Electromagnetic fields and Distributional Sources: Moving Dirac Distributions in R 3
The above section formulates a distributional solution to (150) associated with a t− dependent p-form distributional sourceJ D on R 3 . 0−layer singular distributions associated with electromagnetic sources constrained to moving curves or surfaces in space can be constructed in terms of a family of distributions δ Σ 3−k f ×R (0) defined for k = 1, 2 by is a moving point p 0 (t) in X = R 3 . Then the worldline history of the moving point is encoded into the support of a distribution δ p 0 ×R If the history of p 0 ∈ X is parameterized by t it is the spacetime curve Then and One may now easily calculate is the 0−form associated with the distributional source δ p 0 ×R (0) [Γ Rβ ] at the field point Y at the instant t. It is straightforward to calculate the Jacobian in terms of x 0 (t) and its derivativė x 0 (t): The instantaneous Newtonian 3−velocity of the point support is v(t) ≡ẋ 0 (t) and if one introduces the Euclidean unit vector connecting the field point y at time t to the source point x 0 (t )at the earlier time t =t (Y, t), then the Jacobian inverse takes the form where t ≡t (Y, t) solves the equation t = t − |x 0 (t )−y| c 0 and g denotes the Euclidean metric tensor on R 3 By contrast to the smooth electromagnetic sourcesJ discussed in section (13) one models a moving point source (with electric charge q and 3−velocity v(t)) for the scalar potential φ D bȳ and for the potential 1−form A D byJ It then follows immediately from (161) that the distribution φ D may be associated with while the distribution A D may be associated with where t =t (Y, t) as above. These are the classic Lienard-Weichert potentials for a moving point charge [2]. It is of interest to verify from #J D = − vρ D that the distributions A D and φ D satisfy the gauge condition δ A D + 0 µ 0φ D =0. It may also be noted that by using the appropriate fundamental double-forms in place of γ (0,0) and γ (1,1) these potentials maintain their structure for solutions in D ⊂ Y that satisfy appropriate boundary conditions on ∂D.

Distributional Analysis of Fixed Junction Conditions in R 3
The formulation in this section begins by assuming that all electromagnetic sources are contained within a static region U ⊂ R 3 . This region is partitioned into compact static regions some of which are empty, U = I U I where each U I is the image of some 3− chain. Some regions will serve to support distinct physical media characterized by electromagnetic constitutive properties such as permittivity, permeability and conductivity. Others may contain the vacuum or smooth volume distributions of electric charge or current. Some components of some electromagnetic fields in the vicinity of adjacent domains containing such sources will not be continuous. All regions may also include singular sources distributed on points, curves or surfaces, where electromagnetic fields may be singular. In this manner one can apply Stoke's theorem to regular forms on regular domains and account for discontinuities and singularities in terms of different types of singular distributions. Let α(U ) denote a smooth p−form with bounded components on some regular domain U ⊂ R 3 in the partition of U. Denote typical test p−forms with arbitrary support on R 3 byΛ and those with support containing W ⊂ R 3 byΛ(W). It will be assumed that all p−chains [0, 1] p → R 3 inherit a fixed orientation from the orienting 3-form #1 on In general ∂U I is a sum of disjoint 2−chains that may include parts of ∂U.
With these preliminaries one may analyze the content of the distributional Maxwell equation where the total distributional Maxwell displacement 2−form D D on R 3 is defined for all test 2−formsβ by and the distributional electric charge density 3−form ρ D on R 3 is defined for all test 3−formŝ γ by The first term on the right hand side above represents a sum over all regular volume distributions of charge density 11 , the second term represents a sum over all point multipoles in U (including 11 ρ(UI ) may be zero on some UI . point charges), the third a sum over all line-distributions of charge on curves C I (given by the locif I 1 = 0,f I 2 = 0) in U and the last term a sum over all surface distributions of charge on surfaces S I (given by the locus f I = 0) in U. Each Q P I is a constant describing the electric charge of point mono-poles or the moments of point multi-poles, λ q C I is a 1−form on C I containing a line-charge density and each σ q S I is a 2−form on S I containing a surface charge density. The distributional equation implies that for all test formsγ on R 3 . By restricting the support ofγ the quantities D, D D P I , D D C I , D D S I can be related to sources with support U I , P I , C I , S I respectively. Now and from (178) one has where {U } is the set of regular domains that contain all or part of U in ∂U , {U } is the set of regular domains that contain all or part of U in ∂U and {U , U } is the set of regular domains that have common interfaces I j (U , U ). If one chooses the support ofγ to lie wholly or partly in a subset of U I that excludes all singular sources one collects the set of regular differential equations d D(U I ) = ρ(U I ) on each such subset U I . If one chooses the support ofγ to contain the interface I j (U , U ) which is part of the support f = 0 of a surface charge with density σ j (and no other singular source) one has: The last term pulls back to zero on f = 0 so: and one infers the pull-back relations: on I j (U , U ). Note the correlation between the pull-back discontinuity (I j (U , U )) * (D(U ) − D(U )) and the chosen orientation of the chain I j (U , U ). If the interface is part of the surface f = 0 in R 3 this pull-back relation can also be expressed in terms of d f and forms injected into R 3 : where I j is the interface discontinuity relative to D, where each h(U I ) is a smooth 1−form on U I . The distributional Maxwell equation involves the time derivative of the total Maxwell displacement distribution,Ḋ D , defined bẏ D D = L ∂t D D and, for all test 2−formsβ, the distributional electric current 2−form is taken to be: The distributions H D C I , H D S I are associated with current line-sources with densityλ C I and current surface sources with density κ S I . Eachλ C I is a 1−form on C I and each κ S I is a 1-form on S I . It follows from on each subset of U I free of singular sources and on each I j (U , U ) containing a surface current 1− form κ j . By injecting the surface current 1−form κ j into R 3 one may write the above junction condition as where I j is the interface discontinuity relative to h and Finally if e D denotes the total distributional Maxwell electric field 1−form defined for all test 1−formsα on R 3 by where each e(U I ) is a smooth 1−form on U I then the distributional Maxwell equation where I j is the interface discontinuity relative to e and [e] = e(U ) − e(U ) for f increasing from U to U . In all these boundary conditions, if U I denotes an idealized perfect conductor then e(U I ), b(U I ), D(U I ), H(U I ) are all zero. However if U I denotes an uncharged idealized perfect insulator then σ 0 = 0 on its interface with another uncharged idealized perfect insulator. If Σ is a charged conducting surface its distribution of surface charge cannot be arbitrarily prescribed unlike the distribution of volume charge or surface charge that can exist in a non-conducting medium.

Distributional Analysis of Moving Junction Conditions
In the previous section the distributional Maxwell system has been written as an exterior system on R 3 in terms of differential forms dependent on time regarded as an evolution parameter. The partition of U was fixed relative to this time parameter. Although this system is fully compatible with a relativistic description of electromagnetism it was not expressed in a manifestly Lorentz covariant form. For many relativistic problems and situations where dynamic interface phenomena are relevant it is more natural to reformulate the distributional descriptions in terms of the spacetime structures introduced in section 4. In this section an outline of this approach will be given. The partitioning of spacetime by a collection of immersions into spacetime is best approached in the language of algebraic topology which is not attempted here and attention is restricted to a simple partition of spacetime by the history of a single compact region of space with source on its spatial boundary. Let Φ be a leaf in a local foliation defining a hypersurface in flat spacetime. Assume further that dΦ = 0 and g(dΦ, dΦ) ≡ dΦ · dΦ) > 0. The unit time-like vector field U on spactime defines a (local) frame associated with a laboratory observer while a unit time-like vector field W on spacetime will be associated below with the observed motion of an interface specified by Φ. At each event p on the hypersurface one also has a unit space-like normal N = dΦ √ dΦ·dΦ . In the following it proves convenient to make notationally explicit the dependence of various fields on U . Hence, any superscripted bold face form ξ U on spacetime is orthogonal to U in the sense i U ξ U = 0.
In this context the distributional Maxwell system in spacetime can be written in a self-evident notation as By arguments in complete analogy to those in the previous section, for a single hypersurface each equation yields an equation of the type for some smooth 2−forms Λ and K in the neighborhood of the hypersurface and p on the hypersurface. Let the orthogonal decomposition of these forms with respect to U be: Thus in the neighborhood of the hypersurface one has One now relates α U and β U to a particular space-like unit vector N U and the instantaneous normal component v N U of the 3−velocity of an interface with respect to N U in the frame U . The space-like direction dΦ induces a time-like vector field W on the hypersurface dΦ = 0 satisfying i W d Φ = 0. The vector field W is unique up to a scaling so with d Φ = 0 one normalises W so that W · W = −1 and writes . The Newtonian 3-velocity field v U of W relative to the frame U is given in terms of U and W by: Let Σ d Φ 3 denote the hypersurface given by Φ = 0 and Σ U 3 a local hypersurface defined by U . In a local chart with U = ∂ t /c 0 , Σ U 3 is the locus t = t 0 for some positive constant t 0 . Within Σ d Φ 3 one may identify a codimension 2 submanifold Σ 2 given by Φ = 0, t − t 0 = 0. The instantaneous space-like unit normal to Σ 2 in Σ d Φ 3 Σ U 3 is denoted N U . In terms of Φ and U one has and then where v N U = v U · N U and W is the solution of W · d Φ = 0, W · W = −1. Thus the generic interface conditions become in terms of normal interface properties in the frame U for any event p ∈ Σ d Φ 3 Σ U 3 . If S U is written in terms of the 1-form s U using the induced Hodge map # on Σ c 0 ∧ N U and with κ U = κ 0 N U + κ U one has the moving interface condition 13 The component of this condition in the direction N U reproduces the condition (208) above on the normal component of [d U ]. Thus for an arbitrarily moving interface relative to an arbitrary observer field U the local interface conditions on the fields e U , b U , d U , h U depend on the local normal velocity component of the interface, the direction of the interface normal and the values of the surface charge and current at that instant.

Conclusions
Using properties of the fundamental double-form of bi-degree (p, p) associated with the Hodgede Rham operator ∆ on differential forms, a distributional framework for analyzing the Maxwell system with singular sources has been developed. A set of r−layer Dirac singular distributions with supports on (moving) embeddings in R 3 has been constructed and finds application in electromagnetic source modeling. The framework has been illustrated by explicitly calculating the fields associated with particular singular sources including those due to an arbitrarily moving point charge in free space. With the aid of r−layer Dirac singular distributions many fields with more complex distributional sources can be readily reduced to quadratures once one parametrises such sources in terms of the geometry of their support in space. Further applications can be found in [16]. The local jump conditions on Maxwell fields across arbitrarily moving interfaces have been deduced directly from the covariant Maxwell equations for media on spacetime 12 In Gibb's 3−vector notation: v N U [b U ] = (N U × [e U ]) 13 In Gibb's 3−vector notation: