Mathematical descriptions of the electromagnetic field

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For the manifestly covariant formulations, see Classical electromagnetism and special relativity and Covariant formulation of classical electromagnetism.

There are various mathematical descriptions of the electromagnetic field that are used in the study of electromagnetism, one of the four fundamental forces of nature. In this article three approaches are discussed, generally the equations are in terms of electric and magnetic fields, potentials, and charges with currents.

Contents

1 Vector field approach
- 1.1 Maxwell's equations in the vector field approach
2 Potential field approach
3 Geometric algebra (GA) formulation
- 3.1 Relation to the vector calculus formalism
4 Differential forms approach
- 4.1 Current 3-form, dual 1-form
- 4.2 Current 1-form, dual 3-form
5 Curved spacetime
- 5.1 Traditional formulation
- 5.2 Formulation in terms of differential forms
6 Classical electrodynamics as the curvature of a line bundle
7 References and notes
8 See also

Vector field approach

Main article: Classical electromagnetism

The most common description of the electromagnetic field to use two three-dimensional vector fields called the electric field and the magnetic field. These vector fields each have a value defined at every point of space and time and are thus often regarded as functions of the space and time coordinates. As such, they are often written as E(x, y, z, t) (electric field) and B(x, y, z, t) (magnetic field).

If only the electric field (E) is non-zero, and is constant in time, the field is said to be an electrostatic field. Similarly, if only the magnetic field (B) is non-zero and is constant in time, the field is said to be a magnetostatic field. However, if either the electric or magnetic field has a time-dependence, then both fields must be considered together as a coupled electromagnetic field using Maxwell's equations.

Maxwell's equations in the vector field approach

Main article: Maxwell's equations

The behaviour of electric and magnetic fields, whether in cases of electrostatics, magnetostatics, or electrodynamics (electromagnetic fields), is governed in a vacuum by Maxwell's equations:

Maxwell's equations (vector fields)
$\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}$	Gauss' law
$\nabla \cdot \mathbf{B} = 0$	Gauss's law for magnetism
$\nabla \times \mathbf{E} = -\frac {\partial \mathbf{B}}{\partial t}$	Faraday's law
$\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0\varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}$	Ampère–Maxwell law

where ρ is the charge density, which can (and often does) depend on time and position, ε₀ is the electric constant, μ₀ is the magnetic constant, and J is the current per unit area, also a function of time and position. The units used above are the standard SI units.

Inside a linear material, Maxwell's equations change by switching the permeability and permittivity of free space with the permeability and permittivity of the linear material in question. Inside other materials which possess more complex responses to electromagnetic fields, these properties can be represented by tensors, with time-dependence related to the material's ability to respond to fast field changes (dispersion (optics), Green–Kubo relations), and possibly also field dependencies representing nonlinear and/or nonlocal material responses to large amplitude fields (nonlinear optics).

Potential field approach

Many times in the use and calculation of electric and magnetic fields, the approach used first computes an associated potential: the electric potential, V, for the electric field, and the magnetic potential, A, for the magnetic field. The electric potential is a scalar field, while the magnetic potential is a vector field. This is why sometimes the electric potential is called the scalar potential and the magnetic potential is called the vector potential. These potentials can be used to find their associated fields as follows:

$\mathbf E = - \mathbf \nabla V - \frac{\partial \mathbf A}{\partial t}$

$\mathbf B = \mathbf \nabla \times \mathbf A$

Maxwell's equations in potential formulation

These relations can be substituted into Maxwell's equations to express the latter in terms of the potentials. Faraday's law and Gauss's law for magnetism reduce to identities (e.g. in the case of Gauss's Law for magnetism, 0 = 0). The other two of Maxwell's equations turn out less simply.

Maxwell's equations (Potential formulation)

$\nabla^2 \varphi + \frac{\partial}{\partial t} \left ( \mathbf \nabla \cdot \mathbf A \right ) = - \frac{\rho}{\varepsilon_0}$

$\left ( \nabla^2 \mathbf A - \frac{1}{c^2} \frac{\partial^2 \mathbf A}{\partial t^2} \right ) - \mathbf \nabla \left ( \mathbf \nabla \cdot \mathbf A + \frac{1}{c^2} \frac{\partial \varphi}{\partial t} \right ) = - \mu_0 \mathbf J$

These equations taken together are as powerful and complete as Maxwell's equations. Moreover, the problem has been reduced somewhat, as between the electric and magnetic fields, each had three components which needed to be solved for, meaning it was necessary to solve for six quantities. In the potential formulation, there are only four quantities, the electric potential and the three components of the vector potential. However, this improvement is contrasted with the equations being much messier than Maxwell's equations using just the electric and magnetic fields.

Gauge freedom

Fortunately, there is a way to simplify these equations that takes advantage of the fact that the potential fields are not what is observed, the electric and magnetic fields are. Thus there is a freedom to impose conditions on the potentials so long as whatever condition is chosen to impose does not affect the resultant electric and magnetic fields. This freedom is called gauge freedom. Specifically for these equations, for any choice of a scalar function of position and time λ, the potentials can be changed as follows:

$\mathbf A' = \mathbf A + \mathbf \nabla \lambda$

$\varphi' = \varphi - \frac{\partial \lambda}{\partial t}$

This freedom can be used to greatly simplify the potential formulation. Generally, two such scalar functions are chosen.

Coulomb gauge

The Coulomb gauge is chosen in such a way that $\mathbf \nabla \cdot \mathbf A' = 0$ , which corresponds to the case of magnetostatics. In terms of λ, this means that it must satisfy the equation

$\nabla^2 \lambda = - \mathbf \nabla \cdot \mathbf A$ .

This choice of function results in the following formulation of Maxwell's equations:

$\nabla^2 \varphi' = -\frac{\rho}{\varepsilon_0}$

$\nabla^2 \mathbf A' - \mu_0 \varepsilon_0 \frac{\partial^2 \mathbf A'}{\partial t^2} = - \mu_0 \mathbf J + \mu_0 \varepsilon_0 \nabla \left ( \frac{\partial \varphi'}{\partial t} \right )$

Several features about Maxwell's equations in the Coulomb gauge are as follows. Firstly, solving for the electric potential is very easy, as the equation is a version of Poisson's equation. Secondly, solving for the magnetic vector potential is particularly hard to calculate. This is the big disadvantage of this gauge. The third thing to note, and something which is not immediately obvious, is that the electric potential changes instantly everywhere in response to a change in conditions in one locality.

For instance, if a charge is moved in New York at 1 pm local time, then a hypothetical observer in Australia who could measure the electric potential directly would measure a change in the potential at 1 pm New York time. This seemingly goes against the prohibition in special relativity of sending information, signals, or anything faster than the speed of light. The solution to this apparent problem lies in the fact that, as previously stated, no observers measure the potentials, they measure the electric and magnetic fields. So, the combination of ∇φ and ∂A/∂t used in determining the electric field restores the speed limit imposed by special relativity for the electric field, making all observable quantities consistent with relativity.

Lorenz gauge

Another gauge (used very often) is the Lorenz gauge. This scalar function λ is chosen such that

$\mathbf \nabla \cdot \mathbf A' = - \mu_0 \varepsilon_0 \frac{\partial \varphi'}{\partial t}$ .

meaning λ must satisfy the equation

$\nabla^2 \lambda - \mu_0 \varepsilon_0 \frac{\partial^2 \lambda }{\partial t^2}= - \mathbf \nabla \cdot \mathbf A - \mu_0 \varepsilon_0 \frac{\partial \varphi}{\partial t}$ .

The Lorenz gauge results in the following form of Maxwell's equations:

$\nabla^2 \mathbf A' - \mu_0 \varepsilon_0 \frac{\partial^2 \mathbf A'}{\partial t^2} = \Box^2 \mathbf A' = - \mu_0 \mathbf J$

$\nabla^2 \varphi' - \mu_0 \varepsilon_0 \frac{\partial^2 \varphi'}{\partial t^2} = \Box^2 \varphi' = - \frac{\rho}{\varepsilon_0}$

The operator $\Box^2$ is called the d'Alembertian (some authors denote by just one square $\Box$ ). These equations are inhomogenous versions of the wave equation, with the terms on the right side of the equation serving as the source functions for the wave. These equations lead to two solutions: advanced potentials (which depend on the configuration of the sources at future points in time), and retarded potentials (which depend on the past configurations of the sources); the former are usually (and sensibly) dismissed as 'non-physical' in favor of the latter, which preserve causality.

As pointed out above, the Lorenz gauge is no more valid than any other gauge, as the potentials themselves are unobservable (with only a few loopholes^{[clarification needed]}, such as the Aharonov–Bohm effect, that still leave gauge invariance intact); any causality exhibited by the potentials will vanish for the observable fields, which are the physically meaningful quantities.

Extension to quantum electrodynamics

Canonical quantization of the electromagnetic fields proceeds by elevating the scalar and vector potentials; φ(x), A(x), from fields to field operators. Substituting 1/c² = ε₀μ₀ into the previous Lorenz gauge equations gives:

$\nabla^2 \mathbf A - \frac 1 {c^2} \frac{\partial^2 \mathbf A}{\partial t^2} = - \mu_0 \mathbf J$

$\nabla^2 \varphi - \frac 1 {c^2} \frac{\partial^2 \varphi}{\partial t^2} = - \frac{\rho}{\varepsilon_0}$

Here, J and ρ are the current and charge density of the matter field. If the matter field is taken so as to describe the interaction of electromagnetic fields with the Dirac electron given by the four-component Dirac spinor field ψ, the current and charge densities have form:^[1]

$\mathbf{J}=-e\psi^{\dagger}\boldsymbol{\alpha}\psi\,\quad \rho=-e\psi^{\dagger}\psi \,,$

where α are the first three Dirac matrices. Using this, we can re-write Maxwell's equations as:

Maxwell's equations (QED)

$\nabla^2 \mathbf A - \frac 1 {c^2} \frac{\partial^2 \mathbf A}{\partial t^2} = \mu_0 e \psi^{\dagger} \boldsymbol{\alpha} \psi$

$\nabla^2 \varphi - \frac 1 {c^2} \frac{\partial^2 \varphi}{\partial t^2} = \frac{1}{\varepsilon_0} e \psi^{\dagger} \psi$

which is the form used in quantum electrodynamics.

Geometric algebra (GA) formulation

Analagous to the tensor formulation, two objects: one for the field and one for the current, are introduced. In GA these are multivectors. The field multivector is

$F = \bold{E} + Ic\bold{B}$

and the current multivector is

$J = c \rho + \bold{J}.$

with the unit pseudoscalar I² = −1.

In geometric algebra, Maxwell's equations are reduced to a single equation,^[2]

Maxwell's equations (GA formulation)

$\left(\frac{1}{c}\dfrac{\partial }{\partial t} + \boldsymbol{\nabla}\right)F = \mu_0 c J$

The GA spatial gradient operator ∇ acts on a vector field, such that

$\boldsymbol{\nabla}\bold{F} = \boldsymbol{\nabla} \cdot \bold{F} + I \boldsymbol{\nabla} \times \bold{F},$

In spacetime algebra using the same geometric product the equation is simply

$\nabla F = \mu_0 c J,$

that is, the spacetime derivative of the electromagnetic field is its source. Here the (non-bold) spacetime gradient

$\nabla = \gamma^\mu \partial_\mu$

is a four vector, as is the current density

$J = \gamma_\mu J^\mu = \gamma_0 c \rho + J^k \gamma_k = (c \rho + \bold{J})\gamma_0.$

Relation to the vector calculus formalism

Starting from the spacetime algebra, a timelike direction $\gamma_0$ is selected, and we can deal with the 3D spatial algebra (equivalent to the Pauli algebra). So we need to expand

$\begin{align}\gamma_0 \left( \nabla F - c\mu_0J \right) = 0 \end{align}$

To do so first note that

$\begin{align}\gamma_0 \nabla &= \partial_t + \boldsymbol{\nabla} \end{align}$

where a bold font is used for the spatial gradient (no time components).

$\begin{align}\boldsymbol{\nabla} &= \sigma^k \partial_k\end{align}$

Similarly, multiplication of the four vector current density also has scalar and spatial vector components. With

$\mathbf{J} = J^k \sigma_k,\ c\rho = J^0$ this is

$\begin{align}\gamma_0 J &= c \rho - \mathbf{J}\end{align}$

One obtains

$\begin{align}\left( \frac{1}{c} \partial_t + \boldsymbol{\nabla} \right) (\mathbf{E} + I c \mathbf{B}) - \frac{\rho}{\epsilon_0} - c\mu_0\mathbf{J} = 0\end{align}$

Noting that the pseudoscalar I commutes with all spatial vectors, that

$I^2 = -1,\ \epsilon_0 \mu_0 c^2 = 1$

and

$\boldsymbol{\nabla} \mathbf{X} = \boldsymbol{\nabla} \cdot \mathbf{X} + I \boldsymbol{\nabla} \times \mathbf{X}$

for spatial vectors X, one can expand and regroup this yielding

$\left( \boldsymbol{\nabla} \cdot \mathbf{E} - \frac{\rho}{\epsilon_0} \right)- c \left( \boldsymbol{\nabla} \times \mathbf{B} - \mu_0 \epsilon_0 \frac{\partial {\mathbf{E}}}{\partial {t}} - \mu_0 \mathbf{J} \right)+ I \left( \boldsymbol{\nabla} \times \mathbf{E} + \frac{\partial {\mathbf{B}}}{\partial {t}} \right)+ I c \left( \boldsymbol{\nabla} \cdot \mathbf{B} \right)= 0$

We have scalar, vector, bivector, and trivector grades. Equating each to zero recovers all of Maxwell's equations in their traditional vector form.

Differential forms approach

See also: Differential form and Exterior algebra

Current 3-form, dual 1-form

In free space, where ε = ε₀ and μ = μ₀ are constant everywhere, Maxwell's equations simplify considerably once the language of differential geometry and differential forms is used. In what follows, cgs-Gaussian units, not SI units are used. (To convert to SI, see here.) The electric and magnetic fields are now jointly described by a 2-form F in a 4-dimensional spacetime manifold, The Maxwell tensor $F_{\mu\nu}$ can be written as a 2-form in Minkowski space as

$\begin{align} F \equiv & \frac{1}{2}F_{\mu\nu} dx^{\mu} \wedge dx^{\nu} \\ = & B_x dy \wedge dz + B_y dz \wedge dx + B_z dx \wedge dy + E_x dx \wedge dt + E_y dy \wedge dt + E_z dz \wedge dt \end{align}$

which, as the curvature form, is the exterior derivative of the electromagnetic four-potential,

$F = d A = ( \partial_{\mu} A_{\nu} ) dx^{\mu} \wedge dx^{\nu}$

The source free equations can be written by the action of the exterior derivative on this 2-form. But for the equations with source terms (Gauss law for electricity and the Ampere-Maxwell equation), it is required to define the Hodge dual of this 2-form. The Hodge 'star' dual takes a p-form to a d-p form, where d is the number of dimensions. Here, it takes the 2-form (F) and gives another 2-form (in d=4 dimensions, d-p = 4-2 = 2). For the basis cotangent vectors, the Hodge dual is given as (see here)

$\star dx \wedge dy = - dz \wedge dt ,\quad \star dx \wedge dt = dy \wedge dz,$

and so on. Using these relations, the dual of Maxwell 2-form is

$\star F = - B_x dx \wedge dt - B_y dy \wedge dt - B_z dz \wedge dt + E_x dy \wedge dz + E_y dz \wedge dx + E_z dx \wedge dy$

Maxwell's equations then reduce to the Bianchi identity and the source equation, respectively:^[3]

Maxwell's equations (Differential forms)

$\mathrm{d}\bold{F}=0$

$\mathrm{d}\, {\star\bold{F}}=\bold{J}$

where d denotes the exterior derivative — a natural coordinate and metric independent differential operator acting on forms, and the (dual) Hodge star operator $\star$ is a linear transformation from the space of 2-forms to the space of (4−2)-forms defined by the metric in Minkowski space (in four dimensions even by any metric conformal to this metric). The fields are in natural units where 1/4πε₀ = 1.

Here, the 3-form J is called the electric current form or current 3-form satisfying the continuity equation

$\mathrm{d}{\bold{J}}=0.$

The current 3-form can be integrated over a 3-dimensional space-time region. The physical interpretation of this integral is the charge in that region if it is spacelike, or the amount of charge that flows through a surface in a certain amount of time if that region is a spacelike surface cross a timelike interval. As the exterior derivative is defined on any manifold, the differential form version of the Bianchi identity makes sense for any 4-dimensional manifold, whereas the source equation is defined if the manifold is oriented and has a Lorentz metric. In particular the differential form version of the Maxwell equations are a convenient and intuitive formulation of the Maxwell equations in general relativity.

In a linear, macroscopic theory, the influence of matter on the electromagnetic field is described through more general linear transformation in the space of 2-forms. We call

$C:\Lambda^2\ni\bold{F}\mapsto \bold{G}\in\Lambda^{(4-2)}$

the constitutive transformation. The role of this transformation is comparable to the Hodge duality transformation. The Maxwell equations in the presence of matter then become:

$\mathrm{d}\bold{F} = 0$

$\mathrm{d}\bold{G} = \bold{J}$

where the current 3-form J still satisfies the continuity equation dJ = 0.

When the fields are expressed as linear combinations (of exterior products) of basis forms θ^p,

$\bold{F} = \frac{1}{2}F_{pq}\bold{\theta}^p\wedge\bold{\theta}^q.$

the constitutive relation takes the form

$G_{pq} = C_{pq}^{mn}F_{mn}$

where the field coefficient functions are antisymmetric in the indices and the constitutive coefficients are antisymmetric in the corresponding pairs. In particular, the Hodge duality transformation leading to the vacuum equations discussed above are obtained by taking

$C_{pq}^{mn} = \frac{1}{2}g^{ma}g^{nb} \epsilon_{abpq} \sqrt{-g}$

which up to scaling is the only invariant tensor of this type that can be defined with the metric.

In this formulation, electromagnetism generalises immediately to any 4-dimensional oriented manifold or with small adaptations any manifold, requiring not even a metric.

Current 1-form, dual 3-form

Also, the source 4-vector J can be written as a 1-form

$J \equiv J_{\mu} = -\rho dt + j_x dx + j_y dy + j_z dz$

and the dual 3-form

$\star J = \rho dx \wedge dy \wedge dz - j_x dt \wedge dy \wedge dz - j_y dt \wedge dz \wedge dx - j_z dt \wedge dx \wedge dy$

In terms of these forms, the Maxwell equations are

$dF = 0, \quad d {\star F} = \star J$

The conservation of current (Continuity equation) simply follows from the property of exterior derivative that d² = 0 (Exterior derivative).

$d {\star J} = d^2 {\star F} = 0$

which is the conservation of current.

Thus the expression of Maxwell's equations in terms of differential forms leads to a further notational and conceptual simplification. Whereas Maxwell's Equations could be written as two tensor equations instead of eight scalar equations, from which the propagation of electromagnetic disturbances and the continuity equation could be derived with a little effort, using differential forms leads to an even simpler derivation of these results.

Curved spacetime

Main article: Maxwell's equations in curved spacetime

Traditional formulation

Matter and energy generate curvature of spacetime. This is the subject of general relativity. Curvature of spacetime affects electrodynamics. An electromagnetic field having energy and momentum also generates curvature in spacetime. Maxwell's equations in curved spacetime can be obtained by replacing the derivatives in the equations in flat spacetime with covariant derivatives. (Whether this is the appropriate generalization requires separate investigation.) The sourced and source-free equations become (cgs-Gaussian units):

${ 4 \pi \over c }j^{\beta} = \partial_{\alpha} F^{\alpha\beta} + {\Gamma^{\alpha}}_{\mu\alpha} F^{\mu\beta} + {\Gamma^{\beta}}_{\mu\alpha} F^{\alpha \mu} \ \stackrel{\mathrm{def}}{=}\ \nabla_{\alpha} F^{\alpha\beta} \ \stackrel{\mathrm{def}}{=}\ {F^{\alpha\beta}}_{;\alpha} \, \!$

and

$0 = \partial_{\gamma} F_{\alpha\beta} + \partial_{\beta} F_{\gamma\alpha} + \partial_{\alpha} F_{\beta\gamma} = \nabla_{\gamma} F_{\alpha\beta} + \nabla_{\beta} F_{\gamma\alpha} + \nabla_{\alpha} F_{\beta\gamma}.\,$

Here,

${\Gamma^{\alpha}}_{\mu\beta} \!$

is a Christoffel symbol that characterizes the curvature of spacetime and ∇_α is the covariant derivative.

Formulation in terms of differential forms

The formulation of the Maxwell equations in terms of differential forms can be used without change in general relativity. The equivalence of the more traditional general relativistic formulation using the covariant derivative with the differential form formulation can be seen as follows. Choose local coordinates x^α which gives a basis of 1-forms dx^α in every point of the open set where the coordinates are defined. Using this basis and cgs-Gaussian units we define

The antisymmetric infinitesimal field tensor F_αβ, corresponding to the field 2-form F

$\bold{F} = \frac{1}{2}F_{\alpha\beta} \,\mathrm{d}\,x^{\alpha} \wedge \mathrm{d}\,x^{\beta}.$

The current-vector infinitesimal 3-form J

$\bold{J} = {4 \pi \over c } \left ( \frac{1}{6} j^{\alpha} \sqrt{-g} \, \epsilon_{\alpha\beta\gamma\delta} \mathrm{d}\,x^{\beta} \wedge \mathrm{d}\,x^{\gamma} \wedge \mathrm{d}\,x^{\delta}. \right)$ ^{[note 1]}

Here g is as usual the determinant of the metric tensor g_αβ. A small computation that uses the symmetry of the Christoffel symbols (i.e., the torsion-freeness of the Levi-Civita connection) and the covariant constantness of the Hodge star operator then shows that in this coordinate neighborhood we have:

the Bianchi identity

$\mathrm{d}\bold{F} = 2(\partial_{\gamma} F_{\alpha\beta} + \partial_{\beta} F_{\gamma\alpha} + \partial_{\alpha} F_{\beta\gamma})\mathrm{d}\,x^{\alpha}\wedge \mathrm{d}\,x^{\beta} \wedge \mathrm{d}\,x^{\gamma} = 0,$

the source equation

$\mathrm{d}\,{\star \bold{F}} = \frac{1}{6}{F^{\alpha\beta}}_{;\alpha}\sqrt{-g} \, \epsilon_{\beta\gamma\delta\eta}\mathrm{d}\,x^{\gamma} \wedge \mathrm{d}\,x^{\delta} \wedge \mathrm{d}\,x^{\eta} = \bold{J},$

the continuity equation

$\mathrm{d}\bold{J} = { 4 \pi \over c } {j^{\alpha}}_{;\alpha} \sqrt{-g} \, \epsilon_{\alpha\beta\gamma\delta}\mathrm{d}\,x^{\alpha}\wedge \mathrm{d}\,x^{\beta} \wedge \mathrm{d}\,x^{\gamma} \wedge \mathrm{d}\,x^{\delta} = 0.$

Classical electrodynamics as the curvature of a line bundle

An elegant and intuitive way to formulate Maxwell's equations is to use complex line bundles or principal bundles with fibre U(1). The connection ∇ on the line bundle has a curvature F = ∇² which is a two-form that automatically satisfies dF = 0 and can be interpreted as a field-strength. If the line bundle is trivial with flat reference connection d we can write ∇ = d + A and F = dA with A the 1-form composed of the electric potential and the magnetic vector potential.

In quantum mechanics, the connection itself is used to define the dynamics of the system. This formulation allows a natural description of the Aharonov–Bohm effect. In this experiment, a static magnetic field runs through a long magnetic wire (e.g., an iron wire magnetized longitudinally). Outside of this wire the magnetic induction is zero, in contrast to the vector potential, which essentially depends on the magnetic flux through the cross-section of the wire and does not vanish outside. Since there is no electric field either, the Maxwell tensor F = 0 throughout the space-time region outside the tube, during the experiment. This means by definition that the connection ∇ is flat there.

However, as mentioned, the connection depends on the magnetic field through the tube since the holonomy along a non-contractible curve encircling the tube is the magnetic flux through the tube in the proper units. This can be detected quantum-mechanically with a double-slit electron diffraction experiment on an electron wave traveling around the tube. The holonomy corresponds to an extra phase shift, which leads to a shift in the diffraction pattern.^[4]^[5]

References and notes

^ Quantum Electrodynamics, Mathworld [1]
^ Oersted Medal Lecture David Hestenes "Reforming the Mathematical Language of Physics" (Am. J. Phys. 71 (2), February 2003, pp. 104–121) Online:http://geocalc.clas.asu.edu/html/Oersted-ReformingTheLanguage.html p26
^ Misner, Charles W.; Thorne, Kip; Wheeler, John Archibald (1973). Gravitation. W. H. Freeman. p. 81. ISBN 0-7167-0344-0.
^ M. Murray (5 September 2008). "Line Bundles. Honours 1996". University of Adelaide. http://www.maths.adelaide.edu.au/michael.murray/line_bundles.pdf. Retrieved 2010-11-19.
^ R. Bott (1985). "On some recent interactions between mathematics and physics". Canadian Mathematical Bulletin 28 (2): 129–164. doi:10.4153/CMB-1985-016-3.

^ The epsilon tensor contracted with the differential 3-form produces 6 times the number of terms required.

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