## Abstract

The Eulerian approach to continuum mechanics does not make use of a body manifold. Rather, all fields considered are defined on the space, or the space–time, manifolds. Sections of some vector bundle represent generalized velocities which need not be associated with the motion of material points. Using the theories of de Rham currents and generalized sections of vector bundles, we formulate a weak theory of forces and stresses represented by vector-valued currents. Considering generalized velocities represented by differential forms and interpreting such a form as a generalized potential field, we present a weak formulation of pre-metric, *p*-form electrodynamics as a natural example of the foregoing theory. Finally, it is shown that the assumptions leading to *p*-form electrodynamics may be replaced by the condition that the force functional is continuous with respect to the flat topology of forms.

## 1. Introduction

In this paper, we make an attempt to provide a mathematical setting that is general enough to encompass both stress theory in continuum mechanics and classical field theories of physics. In fact, the proposed setting generalizes both. In the continuum mechanics context, we propose a weak formulation of the fundamentals of continuum mechanics that does not use the traditional notion of a material point. In the context of field theories, we present a weak formulation of *p*-form electrodynamics. Here, by ‘weak’ we mean that we use the geometric counterpart of the theory of Schwartz distributions—the theory of de Rham currents and some extensions thereof. In particular, we consider objects such as vector-valued forms on manifolds that may be as irregular as Borel measures and their weak derivatives. The proposed setting does not use either a metric structure or a connection on the manifolds under consideration.

### (a) An example

To demonstrate the simplest conceivable example of the relation between electrodynamics and the theory of stress in continuum mechanics [1], we recall that, for classical continuum mechanics, one assumes that a force on a body is given in terms of a vector field **b** defined in the physical space and a surface force defined on the boundary of . The virtual power of the forces on for a virtual velocity field *w* is given by
1.1
It is further recalled that if the dependence of on the body satisfies Cauchy's postulates, then there is a 3×3 tensor field, the Cauchy stress field *σ*, such that where **n**(*x*) is the outwards pointing normal to the boundary of at . Thus, one has
1.2
where the standard definition of the transpose has been used.

It is also noted that the balance of moment of momentum implies traditionally that the stress tensor *σ* is symmetric. However, we want to examine the case where *σ* is skew symmetric. In this case, using the Levi–Civita symbol, we may define a vector field whose components are given by
1.3
such that . Thus, assuming for simplicity that **b**=0, one has
1.4
Using Gauss's theorem and the identity , we have
1.5
Setting
1.6
so that
1.7
the power may be written in the form
1.8

We finally observe that, in the case where we interpret *w* as the vector potential of magnetostatics, interpret as the magnetic field intensity, interpret as the magnetic field and interpret as the current density, equations (1.6) and (1.7) are simply the restriction of Maxwell's equations to magnetostatics.

The mathematical framework proposed below generalizes the simple analysis of the foregoing example in the following ways: it holds in a space of any dimension; the Euclidean structure used in the definition of the inner and the cross products is not assumed; the tensor field *σ* and the vector fields and may be singular. In particular, one cannot represent all of Maxwell's equations in three dimensions in a simple form as shown above and a four-dimensional formulation is usually adopted. In such a metric-independent formulation of electrodynamics in four-dimensional space–time, the vector potential is replaced by a one-form—a linear functional of tangent vectors. The *n*-dimensional formulation presented makes it possible to consider potentials represented by forms, skew symmetric tensors, of an arbitrary degree *p* in an *n*-dimensional manifold.

### (b) Guiding principles

Continuum mechanics is concerned with the motion of matter in the physical space and with the forces corresponding to such motions. The classical theory is formulated in a three-dimensional Euclidean space representing the physical space. At least since the middle of the twentieth century, a fundamental object of continuum mechanics has been that of a material body which represents the matter whose motion is studied. It has been recognized that, in general, bodies need not have natural reference configurations in space, and, as such, they should be represented mathematically as differentiable manifolds. The notion of a body point makes sense because of the principle of material impenetrability which implies that the configurations of a body in space are assumed to be embeddings. In fact, as early as [2], the configurations of a body in space were described as charts on the body manifold. We will refer to this point of view of continuum mechanics as the Lagrangian approach.

A basic notion of force theory in continuum mechanics is the concept of the stress tensor. The mathematical proof of existence of stress, the assumptions needed for the proof and the mathematical properties of the resulting stress object have been studied since the nineteenth century, notably by Cauchy in the 1820s, by Piola in the 1840s, and are still the subject of research (e.g. [2–10]). In the standard continuum mechanics literature, the existence of stress is proved on the basis of what is referred to as Cauchy's postulates, or mathematically equivalent assumptions.

A fundamental principle we apply in our formulation of the Lagrangian approach is the representation of a mechanical system by its configuration space—the set of all admissible configurations. In classical mechanics of systems having a finite number of degrees of freedom, the configuration space is a differentiable manifold having the corresponding dimension. A motion of the system may be viewed as a curve in the configuration manifold, and generalized velocities are represented by tangent vectors to the manifold. Forces are viewed as linear functionals acting on generalized velocities to produce virtual power. It is observed that this structure does not use either a metric structure or a connection on the configuration space.

Evidently, for such a global formulation of continuum mechanics, the configuration space of a continuous body in the physical space is infinite dimensional. While every linear functional acting on a finite dimensional vector space is also continuous, in the infinite dimensional situation one has to be specific as to the topology used for the configuration space and for the tangent spaces containing the generalized velocities. We regard this issue as the source of the fundamental difference between the mechanics of systems with finite numbers of degrees of freedom and continuum theories.

Within the Lagrangian settings, one may consider the configuration space even in the general situation where both body and physical space are modelled by differentiable manifolds devoid of a metric or a connection. The collection of embeddings of the body manifold in the space manifold may be given a structure of a Banach manifold—a manifold with charts valued in a Banach space. Thus, the notion of virtual power, the action of a generalized force on generalized velocities, is fundamental and is the only physical notion used in the theory.

As claimed in [6], admitting embeddings as configurations makes it natural to use the *C*^{1}-topology for the configuration space and for its tangent spaces. For this reason, we adopt the *C*^{1}-topology and its analogues for the spaces of test-sections, throughout. The general structure described above together with this choice of topology contain the essence of stress theory in continuum mechanics.

A standard representation theorem for functionals, which are continuous relative to the *C*^{1}-topology, results in the existence of a stress object that may be as irregular as a measure. (See, for example, [11] for a concrete situation where irregularities that require the modelling of stresses by measures may occur.) The representing mathematical object is identified with a stress because the representation procedure implies the following. The evaluation of the stress object on the derivative of the generalized velocity, as represented by the jet of the corresponding vector field, is equal to the power of the force functional—a generalization of the principle of virtual work in continuum mechanics. Just as one would expect, without any constitutive information, a force does not determine a unique representing stress. On the other hand, it is emphasized that the significance of the representation of forces by measures stems from the fact that measures are the most irregular objects that may be naturally restricted to sub-bodies of the body. As such, a stress induces a force system in the body.^{1}

In spite of the advantages of the Lagrangian point of view, the notion of a material point, which is conserved in all configurations of a body, is not general enough (e.g. [15]). It is not clear what would be the meaning of material points for the continuum description of growing bodies or chemically reacting mixtures. Thus, instead of modelling virtual velocities as vector fields associated with the motion of material points, one may consider flux fields of some extensive property in space. (See [16] for the discussion of the kinematics in this case.) Thus, similarly to classical field theories, as in the physics terminology, the body manifold is no longer used. To model generalized velocities, instead of tangent vector fields over the body manifold, one considers the sections of some general vector bundle over the space manifold. We refer to this approach as the Eulerian point of view of continuum mechanics. (See [17,13] for the corresponding analysis of continuous stress fields.) A global formulation of Eulerian continuum mechanics, with the analogue of the *C*^{1}-topology for the generalized velocities, enables a generalized formulation of electrodynamics as a special case. For electrodynamics, differential *p*-forms, generalizing the vector potential field, model generalized velocities.

It is noted also that, in this work, we follow the standard paradigm of continuum mechanics by which the least possible constitutive information is considered. Thus, for example, we do not consider either a Lagrangian function or a constitutive relation between the Faraday form and the Maxwell form of electrodynamics.

### (c) Outline

We start by motivating the use of the *C*^{1}-topology and presenting the immediate consequences in §2, where we review the global point of view of Lagrangian continuum mechanics. In classical continuum mechanics, the stress object plays two roles. It acts on the derivative of the virtual velocity to produce power and it determines the traction fields on sub-bodies. In continuum mechanics on manifolds, for the case of differentiable stress fields, there are two distinct mathematical objects that play these roles. We refer to the object that performs power when acting on the jet of the generalized velocity as a variational stress, and we refer to the object that determines the traction on sub-bodies as the traction stress. The variational stress determines the traction stress, but, in general, the converse is false. The properties of the traction stress and its relation to the variational stress have been studied in the works mentioned above only for differentiable stress distributions. The relation between the two objects is reviewed in §3.

In §4, we motivate the transition from the Lagrangian point of view of continuum mechanics to the Eulerian counterpart. In order to adapt the global Lagrangian formulation to the case where the base manifold is the space manifold, which is not compact, while still keeping the level of irregularity of measures, notions from geometric measure theory should be used. Thus, in §5, we introduce the basics of the de Rham theory of currents [18] and generalized sections of vector bundles as in [19]. We also combine the two theories into what we refer to as vector-valued currents, which we find useful later on.

Using the language of §5, we analyse in §6 weak stress theory within the Eulerian setting.

As an example, we present in §7 a weak formulation of pre-metric (e.g. [20–22,]), *p*-form electrodynamics (e.g. [23,24]), as a natural particular case of the theory. It is observed that current formulations of *p*-form electrodynamics use additional geometric structure and consider smooth fields. In the transition to electrodynamics, we interpret the generalized velocity fields as variations of a *p*-form potential. (See [25], an earlier version of which was communicated to me in 2014, for the use of such a variation in magnetostatics.) The basic assumption for the application to *p*-form electrodynamics, a generalization of the assumption made in the foregoing example that the stress is skewsymmetric, is that the stress object performs power only on the antisymmetric part of the jet of a *p*-form—its exterior derivative. Finally, we show that this assumption also follows from a choice of topology—the flat topology [26,27]—on the space of *p*-test forms—variations of potentials. This completes the bridge between Lagrangian continuum mechanics and electrodynamics.

## 2. Global invariant stress theory: the Lagrangian approach

The Lagrangian formulation of continuum mechanics considers configurations of a material body *B* in space . In traditional analyses, is considered as a three-dimensional Euclidean space while the body is usually identified with the image of a reference configuration, mostly an assumed stress-free configuration.

In this work, we assume that both body and space are modelled by general differentiable manifolds devoid of either a metric structure or a connection. The body manifold is assumed to be a compact manifold with corners.

This section introduces the notation and reviews the global metric-invariant Lagrangian point of view of first-order continuum mechanics as a particular case of the theory presented in [6].

### (a) Kinematics and forces

In accordance with a standard assumption of continuum mechanics, that of material impenetrability, a configuration of a body *B* in space is modelled by a *C*^{1}-embedding, . Thus, the global Lagrangian approach to continuum mechanics considers, as a central object, the configuration space
2.1
Here, is the set of *C*^{1}-mappings , and denotes the set of *C*^{1}-embeddings.

For a compact manifold with corners *B*, the collection, , of *C*^{1}-mappings, , may be given the structure of a Banach manifold. For , let
2.2
be the pullback of the tangent bundle by *κ*. The infinite dimensional vector space of *C*^{1}-sections of (2.2) has a Banachable topology. For example, let be a finite vector bundle atlas on *κ***TS* so that {*U*_{ζ}} is a finite open cover of *B*, are the coordinates of *x*_{ζ}∈*U*_{ζ}, and represents , *n*=dim *B*, . Then, the atlas induces a *C*^{1}-norm on by
2.3
where the supremum is taken over all *i*=1,…,*n*, *β*,*γ*=1,…,*m*, *x*_{ζ}∈*U*_{ζ}, *ζ*=1,…,*Z*. A different choice of an atlas will give a different, but equivalent, norm. Another way to construct a norm is by prescribing a Riemannian metric with the corresponding Levi–Civita connection on .

A Banach manifold structure may be defined on (see [28,29] for example), where a neighbourhood of is represented by an open subset of . For example, a tubular neighbourhood of may be used. The tangent space may be identified naturally with . Equivalently, may be identified with the collection of vector fields along *κ*, that is, mappings such that .

It may be shown (e.g. [28,30,31]) that the subset of embeddings is open in . As a result, for an embedding *κ*,
2.4
Elements of a tangent space are interpreted mechanically as virtual velocity fields. Indeed, their representation by vector fields as in (2.4) is in accordance with the traditional notions of fields of virtual velocity or virtual displacements.

It is observed that the above may be extended to the category of Lipschitz manifolds, Lipschitz mappings and embeddings. (See [32] and its application to continuum mechanics in [4].) We also note that the function spaces we use—either the *C*^{1}-function space or the space of Lipschitz mappings, the Sobolev space —are not suitable for the formulation of existence theorems for the resulting partial differential equations. For such purposes, the Sobolev spaces *W*^{1,p}, , are commonly used.

One can adopt now a general setting for classical mechanics whereby a generalized force at a configuration is an element (e.g. [33], for the finite dimensional case). The action *F*(*w*), , is interpreted as the virtual power that the force performs for the given virtual velocity. Thus, with the consequences of adapting it to the infinite dimensional case, we apply this framework to continuum mechanics.

In various theories of materials with micro-structure, the state of a material point is not determined only by its position in space. Additional parameters, internal degrees of freedom or order parameters, are used to indicate the physical state of the material (e.g. [34]). Such parameters are valued in an assumed manifold which is generally devoid of a metric structure. Thus, a configuration of the body is generalized to a mapping . In order to include such theories in our analysis, we generalize the vector bundle to an arbitrary vector bundle and we view the Banachable space of sections *C*^{1}(*W*) as the space of generalized velocities. We will retain the notation as in (2.3). Accordingly, a force will be modelled by an element of the dual *C*^{1}(*W*)*.

We contend that the foregoing setting together with the analysis of the representation of force functionals constitute the fundamental structure for the general stress theory of continuum mechanics. It is observed that no structure has been used in addition to the differential topological structure of the manifolds.

### (b) Stress representation of forces

The following presents a standard representation procedure for elements of *C*^{1}(*W*)* in terms of measures on *B* valued in the dual of the jet bundle .

We first consider the vector space of continuous sections of the jet bundle
2.5
and endow it with a Banachable space structure. An element *A*∈*J*^{1}*W* is represented under a vector bundle chart in the form , *i*,*j*=1,…,*n*, *β*,*γ*=1,…,*m*. If *A*=(*j*^{1}*w*)(*x*)∈(*π*^{1})^{−1}(*x*), then, *w*^{β}=(*w*(*x*))^{β} and . In analogy with the construction of (2.3), with the same vector bundle atlas on *W*, we assign to a continuous section , the norm
2.6
Again, a different atlas will induce a different, but equivalent, norm.

As a result, the jet extension mapping *j*^{1}:*C*^{1}(*W*)→*C*^{0}(*J*^{1}*W*), represented locally by , is a linear isometry for the norms specified. Evidently, *j*^{1} is not surjective as sections of the jet bundle need not be holonomic, i.e. need not be the jet of a vector field in *C*^{1}(*W*).

Let *F* be a generalized force, that is, an element of *C*^{1}(*W*)*. Then, is a linear and continuous functional on Image *j*^{1}. By the Hahn–Banach theorem, *F*∁(*j*^{1})^{−1} may be extended to an element *S*∈*C*^{0}(*J*^{1}*W*)*. By the Riesz representation theorem, such an element *S* is a measure which is valued in the dual of the jet bundle. If a different atlas had been chosen, the fact that equivalent topologies would be induced on the spaces of sections implies that *F*∁(*j*^{1})^{−1} is still continuous. The construction is therefore independent of the choice of an atlas. It follows that
2.7
We refer to *S* as a *variational stress* (measure). We also refer to equation (2.7), exhibiting the condition that the stress *S* represents the force *F*, as the equilibrium equation. The equilibrium equation may be written explicitly in the form
2.8
which is the analogue of the principle of virtual work in continuum mechanics. (We will be more specific and will make changes to the notation pertaining to the representation by measures in §5.)

As the jet mapping is not surjective, the representation of a force by a stress is not unique. A stress representing a force contains more information than the force. While there is no natural restriction of forces to subsets of the body, a stress measure may be restricted naturally to Borel sets. For any Borel subset *P*, in particular an *n*-dimensional submanifold of *B*, one has a corresponding force *F*_{P} given by
2.9

## 3. Stresses represented by smooth densities

It is assumed now that the variational stress measure *S*, valued in the dual of the jet bundle, is given in terms of a smooth section, which we still denote as *S*, of the vector bundle . Here, is the vector bundle whose fibre at *x* is the space of linear mappings . If {**e**_{β}} is a local basis of *W*, *S* is represented locally in the form
3.1
Here, *S*_{β1⋯n} and represent the components of two different arrays. The first corresponds to the values of the section *w* and the second corresponds to the derivatives of the local representatives of the section. In order to simplify the notation, we make a distinction between the two objects only by the indices affixed to them. We will use the notation *S*⋅*j*^{1}*w*, for the *n*-form defined by (*S*⋅*j*^{1}*w*)(*x*)=*S*(*x*)(*j*^{1}*w*(*x*)), which is given locally by
3.2
The integral of this *n*-form over the body manifold gives the virtual power that the given force performs on *w*.

### (a) The vertical sub-bundle and the vertical projection

The jet bundle has a natural vector sub-bundle, the vertical sub-bundle, defined as follows. Let be the vector bundle morphism over the identity such that, for a section *w* of *W*, . Then, the vertical sub-bundle is and we use *ι*_{V}:*V* *J*^{1}*W*→*J*^{1}*W* to denote the natural inclusion. The vertical sub-bundle is isomorphic to *L*(*TB*,*W*) and its elements are of the form , which we will also write as in view of this isomorphism.

The inclusion *ι*_{V} induces a projection
3.3
whereby *S*↦*S*°*ι*_{V}.

### Remark 3.1

We recall that the mapping
3.4
given by *e*(*γ*)(*ω*)=*γ*∧*ω* is an isomorphism. For the case where *p*=1, its inverse is given by the contraction , . Note that is related to the usual left contraction,
3.5
by .

We will use the same notation for the contraction 3.6 defined by .

### (b) Traction stresses, Cauchy's formula

Consider the composition
3.7
We will refer to *σ*=*p*_{σ}(*S*) as the *traction stress* corresponding to the variational stress *S*. Locally, *σ* is represented by so that represents the (*n*−1)-form *σ*⋅*w*. In the expressions above, a superimposed ‘hat’ represents an omitted term. Locally, *p*_{σ} is given by where, .

The traction stress may be restricted to submanifolds of *B* of dimension *n*−1. For such a submanifold, ,
3.8
is naturally defined by restricting the forms *σ*⋅*w* to vectors tangent to . In particular, for each sub-body *P*⊂*B*, *S* induces a unique surface force
3.9
The last equation is the generalization of the Cauchy formula.

### (c) The divergence of the variational stress and the field equations

For the differentiable variational stress field *S*, the divergence div*S* is a section of defined by the condition div*S*⋅*w*=d(*p*_{σ}(*S*)⋅*w*)−*S*⋅*j*^{1}*w*. It is observed that div*S*⋅*w* is indeed independent of the derivatives of *w* and its local representatives are given by . Finally, the body force field **b**, a section of , is defined by **b**=−div *S*, which has the same form as the traditional equilibrium equation. Hence, for every sub-body *P*⊂*B*,
3.10
which is, again, a generalization of the principle of virtual power.

## 4. The Eulerian point of view of continuum mechanics

A somewhat generalized point of view of continuum mechanics is offered by the Eulerian formulation. The Eulerian formulation dispenses with the assumptions regarding the existence of body points and a body manifold. Instead, one considers a particular extensive property in the *n*-dimensional space manifold , such as the electric charge or the mass of a particular constituent of a mixture. A flux field *ω* of that extensive property, an (*n*−1)-form, assumes the role of a generalized velocity field. It is observed that, if the density of the extensive property is given by an *n*-form *θ* which does not vanish where *ω* does not vanish, there is a unique vector field *w* satisfying the condition . Evidently, the vector field *w* may be interpreted as a velocity field and its integral lines may be interpreted as body points. In fact, even if no specific volume element *θ* is given, the flux form *ω* determines a family of one-dimensional submanifolds which one may interpret as body points [16].

Rather than restricting ourselves to sections of , as suggested by the discussion above, we consider for now a general vector bundle . In accordance with the Eulerian point of view, we model a force functional *F* as a continuous, linear functional on the space of differentiable sections of *W* having compact supports. Using notions from the theory of de Rham currents [18,26] and section distributions [19], [35], pp. 236–239, [36], pp. 339–341, we find it useful to introduce vector-valued currents as a means for setting up a theory of global Eulerian continuum mechanics that allows for stresses that are as irregular as measures.

## 5. Generalized sections and vector-valued currents

We start by reviewing the notation pertaining to de Rham currents [18,26] and generalized sections of vector bundles as in [19].

### (a) de Rham currents

A *C*^{r}-test *p*-form on the manifold is a differential form on of class *C*^{r} having a compact support, where . The vector space of such test forms is denoted here by . A locally convex topology is induced on by considering local representatives under an atlas and using an analogous procedure to that used for test functions in the theory of distributions. That is, a sequence of forms (*ψ*^{a}), *a*=1,2,…, converges to zero if, for a given atlas on , all elements of the sequence are supported in a compact subset *K* of a coordinate neighbourhood and
5.1
where *I*=(*i*_{1},…,*i*_{n}) is a multi-index with |*I*|=*i*_{1}+⋯+*i*_{n}, and are the components of the representative of *ψ*^{a} with an increasing multi-index λ [18]. A *p*-current of order *r* on the manifold is a continuous and linear functional on and the space of *p*-currents of order *r* is denoted here by . Using partitions of unity, one shows that a current is determined by its restrictions to forms supported in coordinate neighbourhoods.

We recall the following operations defined for currents. The boundary ∂*R* of a *p*-current *R* of order *r* is the (*p*−1)-current, of order *r*+1, defined by ∂*R*(*ω*)=*R*(d*ω*). It is noted that the choice of topology on the space of test forms makes the boundary operator continuous.

For a *p*-current *R* and a *k*-form *ψ*, with *k*≤*p*, the (*p*−*k*)-current defined by for any (*p*−*k*)-test form *ω*.

The restriction of a *p*-current of order *r* to a coordinate neighbourhood may be represented in the form
5.2
where for each increasing (*n*−*p*)-multi-index, , *R*_{λ} is an *n*-current of order *r*. Thus, . In particular, *n*-currents are simply distributions and 0-currents are generalized *n*-forms or generalized densities.

### (b) Generalized sections

Following the notation in Guillemin & Sternberg [19], pp. 305–308, for a vector bundle , denotes the space of *C*^{r}-sections of *W* with compact supports. A locally convex topology is induced by a vector bundle atlas on *W* as for currents. One defines as the space of generalized sections of *W* of order *r*. For example, a smooth section *u* of *W* defines an element of *C*^{−r}(*W*) by
5.3
Here, *w** is a section of *W**and *ρ* is an *n*-form.

We will refer to elements of as *section distributions* of *W*, or *W*-*valued distributions*, of order *r*. Thus, we note that section distributions belong to , as the space of *n*-multi-covectors is one-dimensional. Using a partition of unity, we may write an element by its restrictions to vector bundle charts, the local representatives of which are of the form , where {**e**^{β}} is a local basis for the vectors in *W* and *F*_{1⋯n} are distributions, or *n*-currents, of order *r*.

### (c) Vector-valued currents

While the definition of generalized sections seems to apply to sections of only, one may substitute for *W* in the definition to obtain . Thus, in view of the isomorphism *e* of remark 3.1, we set
5.4
and we refer to the elements of this space as *vector-valued p-currents* of order

*r*or as

*generalized*(

*n*−

*p*)-

*covector-valued forms*. In particular, it is noted that, for the case , one retrieves de Rham currents. For the cases

*p*=0 and

*p*=

*n*, one retrieves the expressions for generalized sections and section distributions, respectively.

For a vector-valued *p*-current *R* of order *r* and a *C*^{r}-section *w* of *W*, one obtains a *p*-current *R*⋅*w* by setting (*R*⋅*w*)(*ψ*)=*R*(*w*⊗*ψ*) for any test *p*-form *ψ*. The restriction of a vector-valued current to a vector bundle chart is of the form , where λ is an increasing (*n*−*p*)-multi-index and *R*_{βλ} are *n*-currents. Thus, for a section *w*, Given a section *χ* of , which is supported in the same vector bundle chart, having the form , where *μ* is an increasing *p*-multi-index, one has .

Consider a smooth vector bundle morphism *A*:*W*→*V* over the identity on . For a section , the pushforward is given by setting *C*^{r}(*A*)(*w*⊗*ψ*)(*x*)=*A*|_{x}(*w*(*x*))⊗*ψ*(*x*). Evidently, the mapping
5.5
is continuous and linear for all 0≤*p*≤*n*. Thus, we have a pullback, the dual mapping
5.6
It may be natural to introduce the notation *C*^{−r}(*A**):=*C*^{r}(*A*)*.

## 6. Vector-valued currents and stress theory

As indicated above, the Eulerian point of view of continuum mechanics considers a vector bundle as a fundamental object and the space of generalized velocities is of *C*^{1}-sections of this vector bundle. In a more general study, one could consider the space , for *r*>1. However, such theories lead to high-order continuum mechanics and hyper-stresses which we do not consider in this work.

A force, a continuous linear functional on the space of generalized velocities, is therefore an element . The same procedure as in §2b is used for the representation of forces. One uses the jet extension to represent each force *F* by some non-unique element —a variational stress measure—in the form
6.1
Here, the dual mapping may be written as
6.2

Thus, for a stress *S* representing the force *F*, one has
6.3
where we use the same notation for the functional *S* and the measure representing it.

Next, we consider the weak analogues of the objects and mappings considered in §3. A force *F* is represented locally in the form , where *F*_{β1⋯n} are *n*-currents, or distributions, of order 1. The variational stress measure *S* is represented locally in the form
6.4
where the elements *S*_{β1⋯n} and are measures, i.e. *n*-currents of order 0. For a differentiable section *w* of *W*, *S*⋅*j*^{1}*w* is the generalized *n*-form—a zero-current—represented locally by
6.5

The inclusion of the vertical sub-bundle in the jet bundle induces . Locally, *C*^{0}(*ι*_{V}) is represented by so that, for every continuous section *χ* of the vertical bundle supported in the domain of a chart, .

Next, we consider the right contraction
6.6
given by , . In the definition of the traction stress below, we use the contraction rather than so that our notation conforms better with the standard operations on currents. In particular, the contraction is the inverse of the isomorphism *e*, which is repeatedly used below due to the representation (5.2).

The dual of the contraction is , and, using (2.2) for the vector bundle and *p*=*n*, one has .

In analogy with *p*_{σ} introduced in §3, and using the same notation for the global object, we set , by .

The action of the contraction may also be viewed as follows. Let (for example, *S*^{+}=*C*^{0}(*ι*_{V})*(*S*)). However, the transposition isomorphism induces the isomorphism
6.7
which is identical to .

To obtain the local expression for *σ*=*p*_{σ}(*S*), given a section *w*, we view *σ*⋅*w* as a one-current so that its action on a one-form *φ* is
6.8
where *S*^{+}=*C*^{0}(*ι*_{V})*(*S*). Hence, locally,
6.9
so that
6.10

The motivation for the scheme of notation is demonstrated by equation (6.10). The various relations retain their form in the case where the vector-valued currents are represented by differential forms.

For , we will refer to as the *traction stress*associated with the variational stress *S*.

### Remark 6.1

Consider an oriented (*n*−1)-dimensional submanifold and let be the natural inclusion. It is noted that we do not have analogues of equations (3.8) and (3.9). Differential forms are pulled back under mappings between manifolds, such as the inclusion in this context. However, currents, such as *σ*⋅*w*, are pushed forward.

Nevertheless, the zero-current ∂(*σ*⋅*w*) contains information analogous to the Cauchy restriction as in (3.8) and (3.9). Consider the case where, for a differentiable section *σ*_{0} of , the traction stress is given by
6.11
where *B* is an *n*-dimensional submanifold with boundary of . Then, for every function *u*,
6.12
Thus, by choosing a function *u* that vanishes in most of the interior of *B* and equals 1 on ∂*B*, the boundary integral is approximated.

We define the generalized divergence of a variational stress *S* so that the generalized density (∂iv *S*)⋅*w* is given by
6.13

The action of the zero-current ∂(*σ*⋅*w*) on a function *u* supported in a vector bundle chart may be evaluated as follows:
6.14
where
6.15
is viewed as a one-current for each *β*, is viewed as a zero-current and (6.5) was used. It follows that, locally,
6.16

In order to write the power in a form resembling the traditional equilibrium equations of continuum mechanics, one may set **b**=−∂iv *S*.

We thus have for the generalized densities *S*⋅*j*^{1}*w*=−**b**⋅*w*−∂(*σ*⋅*w*) and
6.17
which differs from the traditional expression by the extra minus sign on the right-hand side.

## 7. Form-conjugate forces and *p*-form electrodynamics

Among the various vector bundles over one may consider the bundle , which offers some richer structure. For example, it may be shown [17,13] that, in the case *r*=*n*−1, a differentiable traction stress density may be represented by a section of , a genuine tensor field.

### (a) Form-conjugate forces

In this section, we consider the case , 0≤*p*≤*n*. We will refer to forces in this case as *form-conjugate forces*. Since the generalized velocity fields are represented by differential forms now, we will change the notation *w* to for a generic test field.

In the case where , a form-conjugate force
7.1
Hence, a form-conjugate force is a *p*-current of order 1, which may be represented by an (*n*−*p*)-generalized form of order 1.

A variational stress *S* will belong to and a traction stress . (Note that, evidently, cannot be identified with a space of forms.) For each test form *α*, *σ*⋅*α* is a generalized (*n*−1)-form—a one-current.

### (b) Electrodynamics of *p*-form potentials

We consider now the special case where **b**=−∂iv *S*=0, a natural assumption if one interprets as the space–time manifold. It follows immediately from equation (6.17) that the power flux is given by
7.2

The fundamental assumption we make now is that there is a generalized (*n*−*p*−1)-form, equivalently a (*p*+1)-current, of order 0,
7.3
so that, for each test form *α*, the one-current *σ*⋅*α* is given by
7.4
Specifically,
7.5
and
7.6

It follows that the zero-current ∂(*σ*⋅*α*) satisfies, for any test function *u*,
7.7
where we used d(*α*∧*u*)=d*α*∧*u*+(−1)^{p}*α*∧d*u* in the third line. One concludes that
7.8
The power generalized density may be written now as
7.9

One may introduce the notation 7.10 so that 7.11 The power density becomes 7.12

For the case where the generalized form *g* is actually a differentiable (*n*−*p*−1)-form,
7.13
Thus,
7.14
The second line is identical to (7.9), as expected, and the first line motivates the definition of the (*n*−*p*)-form
7.15

One may make the following interpretations of the various variables. The *p*-form *α* is viewed as a potential or a variation thereof, *f* is interpreted the Faraday (*p*+1)-form, *g* is viewed as the Maxwell (*n*−*p*−1)-measure-valued generalized form—a (*p*+1)-current—and is the de Rham current representing the electric current density. It is noted that , a *p*-current, is no longer a measure (a current represented by integration in the terminology of Federer [26]). Thus, it may represent objects such as the derivative of the Dirac measure. We conclude that the equations above represent a generalization of Maxwell's equations for a pre-metric, weak formulation of *p*-form electrodynamics.

### (c) Flat forces

The foregoing conclusions may be obtained using a somewhat different approach. A variational stress will be said to be *flat* if it is of the form
7.16
where , and we view each of the terms as a zero-current.

Let be given by *j*^{1}*α*(*x*)↦(*α*(*x*),d*α*(*x*)), which can be represented by the antisymmetrization operator. Then,
7.17
and equation (7.16) is equivalent to the condition that *S* is represented in the form .

Using the isomorphism (3.4), the variational stress *S* is flat if and only if there is a *p*-current and a (*p*+1)-current , such that
7.18
as zero-currents.

We will say that a *force* is *flat* if it may be represented by a flat variational stress. It follows that a force *F* is flat if and only if
7.19
with *S*_{0}, *S*_{1}, *H* and *G* as above. Thus, one has the representation of the force in the form
7.20
As an immediate consequence,
7.21

The condition (7.19) may be viewed also as a requirement that the current *F* is continuous relative to the flat topology on the space of test forms. Specifically, one may consider the space containing *p*-forms with compact supports which are continuous and the exterior derivatives of which are continuous. Just like for the case of currents, one may define the topology on in analogy with the space of test forms but replacing the condition (5.1) for a sequence to converge to zero, with the condition
7.22
Hence, the flat topology is sensitive to the external derivative only and not the complete jet of forms. The space of flat forces is the dual space as we show below. Consider the mapping
7.23
(See Federer [26], pp. 367–368 for an analogous construction for flat chains.) The mapping is linear and injective and it is an isomorphism of the topological vector space into the Cartesian product (as in the proof of the representation of distributions having a finite order (e.g. [37], pp. 259–260)). Thus, is a continuous linear functional which may be extended to an element
7.24
One concludes that there exist two currents of order 0, and , such that (7.20) holds.

We note that the introduction of flat forces above follows closely the definition of flat chains in [26] (see also [27]), hence the terminology we use. The important difference is that flat chains are more regular than flat forces as they are limits, in the flat topology, of normal currents—currents that are represented by measures and the boundaries of which are measures. It is shown in [26] that flat chains may actually be represented by Lebesgue integrable fields.

Returning to the representation of flat forces, with equations (7.8), (7.20) may be written in the form
7.25
In analogy with the assumption that **b**=0 in §7b, it is assumed now that *H*+(−1)^{p}∂*G*=0, and one arrives at
7.26
which is equivalent to (7.6).

One concludes that the weak formulation of *p*-form pre-metric electrodynamics may be thought of as a representation procedure of flat forces.

## Competing interests

The author declares that he has no competing interests.

## Funding

I received no funding for this study.

## Acknowledgements

This work was partially supported by the Pearlstone Center for Aeronautical Engineering Studies and the H. Greenhill Chair for Theoretical and Applied Mechanics at Ben-Gurion University.

## Footnotes

↵† Dedication: In loving memory of my father Shlomo Segev, Jerusalem, 1930–Beer-Sheva, 2016.

One contribution of 11 to a theme issue ‘Trends and challenges in the mechanics of complex materials’.

↵1 It is noted that an approach analogous to the traditional Cauchy theory for the existence of continuous stress fields may still be formulated on general differentiable manifolds [12–14].

- Accepted October 22, 2015.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.