Mathematical descriptions of the electromagnetic field

66: 4697: 8421: 5997: 4450: 204: 8137: 5710: 3162:, with the terms on the right side of the equation serving as the source functions for the wave. As with any wave equation, these equations lead to two types of solution: advanced potentials (which are related to the configuration of the sources at future points in time), and retarded potentials (which are related to the past configurations of the sources); the former are usually disregarded where the field is to analyzed from a causality perspective. 9478: 25: 4692:{\displaystyle \left({\boldsymbol {\nabla }}\cdot \mathbf {E} -{\frac {\rho }{\varepsilon _{0}}}\right)-c\left({\boldsymbol {\nabla }}\times \mathbf {B} -\mu _{0}\varepsilon _{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)+Ic\left({\boldsymbol {\nabla }}\cdot \mathbf {B} \right)=0} 8648: 6538: 2006: 8845: 6750: 128: 9229: 11044:—i.e., if they are valid in one inertial reference frame, then they are automatically valid in every other inertial reference frame. In fact, Maxwell's equations were crucial in the historical development of special relativity. However, in the usual formulation of Maxwell's equations, their consistency with special relativity is not obvious; it can only be proven by a laborious calculation. 10549: 10641:. 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 2514: 10375: 8416:{\displaystyle {\begin{aligned}\mathbf {F} \equiv &{\frac {1}{2}}F_{\mu \nu }\mathrm {d} x^{\mu }\wedge \mathrm {d} x^{\nu }\\=&E_{x}\mathrm {d} t\wedge \mathrm {d} x+E_{y}\mathrm {d} t\wedge \mathrm {d} y+E_{z}\mathrm {d} t\wedge \mathrm {d} z-B_{x}\mathrm {d} y\wedge \mathrm {d} z-B_{y}\mathrm {d} z\wedge \mathrm {d} x-B_{z}\mathrm {d} x\wedge \mathrm {d} y\end{aligned}}} 5992:{\displaystyle {\begin{aligned}\mathbf {F} &\equiv {\frac {1}{2}}F_{\mu \nu }\mathrm {d} x^{\mu }\wedge \mathrm {d} x^{\nu }\\&=B_{x}\mathrm {d} y\wedge \mathrm {d} z+B_{y}\mathrm {d} z\wedge \mathrm {d} x+B_{z}\mathrm {d} x\wedge \mathrm {d} y+E_{x}\mathrm {d} x\wedge \mathrm {d} t+E_{y}\mathrm {d} y\wedge \mathrm {d} t+E_{z}\mathrm {d} z\wedge \mathrm {d} t\end{aligned}}} 10211: 9658: 4444: 9994: 8426: 5516: 5205: 6321: 8659: 6982:, 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. 6564: 2810: 1821: 10382: 10218: 1536:, 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: 10042: 9483: 6316: 2956: 3103: 2331: 2522:. Secondly, solving for the magnetic vector potential is particularly difficult. This is the big disadvantage of this gauge. The third thing to note, and something that is not immediately obvious, is that the electric potential changes instantly everywhere in response to a change in conditions in one locality. 9473:{\displaystyle {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 }\,\!} 4355: 9845: 3691: 5348: 4958: 2530:, i.e. the impossibility of information, signals, or anything travelling faster than the speed of light. The resolution to this apparent problem lies in the fact that, as previously stated, no observers can measure the potentials; they measure the electric and magnetic fields. So, the combination of 2011:

These equations taken together are as powerful and complete as Maxwell's equations. Moreover, the problem has been reduced somewhat, as the electric and magnetic fields together had six components to solve for. In the potential formulation, there are only four components: the electric potential and

10659:

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,

6977:

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

3812: 2663: 2020:

These equations can be simplified by taking advantage of the fact that the electric and magnetic fields are physically meaningful quantities that can be measured; the potentials are not. There is a freedom to constrain the form of the potentials provided that this does not affect the resultant

1815: 1483:

with the permeability and permittivity of the linear material in question. For some materials that have 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 rapid field changes

10859: 4347: 10900:. Since the potentials are only defined up to gauge equivalence, we are free to impose additional equations on the potentials, as long as for every pair of potentials there is a gauge equivalent pair that satisfies the additional equations (i.e. if the gauge fixing equations define a 1649:(the homogeneous equations) turn out to be identically true for any potentials. This is because of the way the fields are expressed as gradients and curls of the scalar and vector potentials. The homogeneous equations in terms of these potentials involve the divergence of the curl 3435: 4446:

from which it is easily seen that Gauss's law is the scalar part, the Ampère–Maxwell law is the vector part, Faraday's law is the pseudovector part, and Gauss's law for magnetism is the pseudoscalar part of the equation. After expanding and rearranging, this can be written as

7852: 8643:{\displaystyle {{\star }\mathbf {F} }=-E_{x}\mathrm {d} y\wedge \mathrm {d} z-E_{y}\mathrm {d} z\wedge \mathrm {d} x-E_{z}\mathrm {d} x\wedge \mathrm {d} y-B_{x}\mathrm {d} t\wedge \mathrm {d} x-B_{y}\mathrm {d} t\wedge \mathrm {d} y-B_{z}\mathrm {d} t\wedge \mathrm {d} z.} 3329: 1417: 2654: 6533:{\displaystyle {\star }\mathbf {F} =-B_{x}\mathrm {d} x\wedge \mathrm {d} t-B_{y}\mathrm {d} y\wedge \mathrm {d} t-B_{z}\mathrm {d} z\wedge \mathrm {d} t+E_{x}\mathrm {d} y\wedge \mathrm {d} z+E_{y}\mathrm {d} z\wedge \mathrm {d} x+E_{z}\mathrm {d} x\wedge \mathrm {d} y} 2525:

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 violates causality in

2001:{\displaystyle \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} } 8840:{\displaystyle \mathbf {J} =-\rho \,\mathrm {d} x\wedge \mathrm {d} y\wedge \mathrm {d} z+j_{x}\mathrm {d} t\wedge \mathrm {d} y\wedge \mathrm {d} z+j_{y}\mathrm {d} t\wedge \mathrm {d} z\wedge \mathrm {d} x+j_{z}\mathrm {d} t\wedge \mathrm {d} x\wedge \mathrm {d} y} 6745:{\displaystyle \mathbf {J} =\rho \,\mathrm {d} x\wedge \mathrm {d} y\wedge \mathrm {d} z-j_{x}\mathrm {d} t\wedge \mathrm {d} y\wedge \mathrm {d} z-j_{y}\mathrm {d} t\wedge \mathrm {d} z\wedge \mathrm {d} x-j_{z}\mathrm {d} t\wedge \mathrm {d} x\wedge \mathrm {d} y.} 9218:. 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 6190: 10893:. Again by the Poincaré lemma (and under its assumptions), gauge freedom is the only source of indeterminacy, so the field formulation is equivalent to the potential formulation if we consider the potential equations as equations for gauge equivalence classes. 8958: 11020: 9835: 3546: 10704:. For example, the analysis of radio antennas makes full use of Maxwell's vector and scalar potentials to separate the variables, a common technique used in formulating the solutions of differential equations. The potentials can be introduced by using the 5341: 10544:{\displaystyle \mathrm {d} \mathbf {J} ={4\pi \over c}{j^{\alpha }}_{;\alpha }{\sqrt {-g}}\,\varepsilon _{\alpha \beta \gamma \delta }\mathrm {d} x^{\alpha }\wedge \mathrm {d} x^{\beta }\wedge \mathrm {d} x^{\gamma }\wedge \mathrm {d} x^{\delta }=0.} 6134: 8130: 7602: 2817: 2960: 10370:{\displaystyle \mathrm {d} {\star \mathbf {F} }={\frac {1}{6}}{F^{\alpha \beta }}_{;\alpha }{\sqrt {-g}}\,\varepsilon _{\beta \gamma \delta \alpha }\mathrm {d} x^{\gamma }\wedge \mathrm {d} x^{\delta }\wedge \mathrm {d} x^{\alpha }=\mathbf {J} ,} 9187: 7334: 1595: 9723:

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

10206:{\displaystyle \mathrm {d} \mathbf {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,} 11231:. Elements and operations of the algebra can generally be associated with geometric meaning. The members of the algebra may be decomposed by grade (as in the formalism of differential forms) and the (geometric) product of a vector with a 9653:{\displaystyle 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 }.\,} 4015: 2509:{\displaystyle \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)} 10904:

to the gauge action). The gauge-fixed potentials still have a gauge freedom under all gauge transformations that leave the gauge fixing equations invariant. Inspection of the potential equations suggests two natural choices. In the

3574: 1316: 7166: 7518: 6975: 3697: 4439:{\displaystyle {\boldsymbol {\nabla }}\mathbf {F} ={\boldsymbol {\nabla }}\cdot \mathbf {F} +{\boldsymbol {\nabla }}\wedge \mathbf {F} ={\boldsymbol {\nabla }}\cdot \mathbf {F} +I{\boldsymbol {\nabla }}\times \mathbf {F} } 3333: 2105: 9989:{\displaystyle \mathbf {J} ={4\pi \over c}\left({\frac {1}{6}}j^{\alpha }{\sqrt {-g}}\,\varepsilon _{\alpha \beta \gamma \delta }\mathrm {d} x^{\beta }\wedge \mathrm {d} x^{\gamma }\wedge \mathrm {d} x^{\delta }.\right)} 1726: 10777: 2327: 1206: 5511:{\displaystyle \gamma _{0}\nabla =\gamma _{0}\gamma ^{0}\partial _{0}+\gamma _{0}\gamma ^{k}\partial _{k}=\partial _{0}+\sigma ^{k}\partial _{k}={\frac {1}{c}}{\dfrac {\partial }{\partial t}}+{\boldsymbol {\nabla }},} 5200:{\displaystyle F=\mathbf {E} +Ic\mathbf {B} =E^{1}\gamma _{1}\gamma _{0}+E^{2}\gamma _{2}\gamma _{0}+E^{3}\gamma _{3}\gamma _{0}-c(B^{1}\gamma _{2}\gamma _{3}+B^{2}\gamma _{3}\gamma _{1}+B^{3}\gamma _{1}\gamma _{2}),} 3225: 2267: 2150: 9109: 4246: 11165:

Often, the time derivative in the Faraday–Maxwell equation motivates calling this equation "dynamical", which is somewhat misleading in the sense of the preceding analysis. This is rather an artifact of breaking

2805:{\displaystyle \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}}.} 1632: 4230: 2570: 4874: 8850: 6844: 2211: 7753: 4082: 10945: 9755: 3471: 1331: 7932: 5210: 4951: 1681: 11267:-vector component with the outer product. It is of algebraic convenience that the geometric product is invertible, while the inner and outer products are not. As such, powerful techniques such as 7233: 8142: 6033: 5715: 4175: 11279:-forms and there are corresponding operations. Maxwell's equations reduce to one equation in this formalism. This equation can be separated into parts as is done above for comparative reasons. 8032: 4811: 3165:

As pointed out above, the Lorenz gauge is no more valid than any other gauge since the potentials cannot be directly measured, however the Lorenz gauge has the advantage of the equations being

38: 1048:. 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 9699: 1022:

of nature. In this article, several approaches are discussed, although the equations are in terms of electric and magnetic fields, potentials, and charges with currents, generally speaking.

1710: 7261: 1539: 1250: 7199: 6803: 4761: 3909: 7696: 7401: 8005: 7084: 7035: 6311:{\displaystyle {\star }(\mathrm {d} x\wedge \mathrm {d} y)=-\mathrm {d} z\wedge \mathrm {d} t,\quad {\star }(\mathrm {d} x\wedge \mathrm {d} t)=\mathrm {d} y\wedge \mathrm {d} z,} 5572: 3921: 2545:

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.

7968: 1479:

When dealing with only nondispersive isotropic linear materials, Maxwell's equations are often modified to ignore bound charges by replacing the permeability and permittivity of

11159: 7168:

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:

7527: 7744: 7094:

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

4118: 4901: 7097: 6029: 9117: 7417: 7057: 7008: 5697: 2518:

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

44: 6920: 3132: 7638: 6881: 1526: 3156: 2951:{\displaystyle \nabla ^{2}\varphi '-\mu _{0}\varepsilon _{0}{\frac {\partial ^{2}\varphi '}{\partial t^{2}}}=-\Box ^{2}\varphi '=-{\frac {\rho }{\varepsilon _{0}}}} 3686:{\displaystyle \nabla ^{2}\mathbf {A} -{\frac {1}{c^{2}}}{\frac {\partial ^{2}\mathbf {A} }{\partial t^{2}}}=\mu _{0}e\psi ^{\dagger }{\boldsymbol {\alpha }}\psi } 3098:{\displaystyle \nabla ^{2}\mathbf {A} '-\mu _{0}\varepsilon _{0}{\frac {\partial ^{2}\mathbf {A} '}{\partial t^{2}}}=-\Box ^{2}\mathbf {A} '=-\mu _{0}\mathbf {J} } 2154:

This freedom can be used to simplify the potential formulation. Either of two such scalar functions is typically chosen: the Coulomb gauge and the Lorenz gauge.

1712:, which are always zero. The other two of Maxwell's equations (the inhomogeneous equations) are the ones that describe the dynamics in the potential formulation. 1113:. 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 2220: 2056: 149: 8965: 2274: 1265: 2109: 11035: 3807:{\displaystyle \nabla ^{2}\varphi -{\frac {1}{c^{2}}}{\frac {\partial ^{2}\varphi }{\partial t^{2}}}={\frac {1}{\varepsilon _{0}}}e\psi ^{\dagger }\psi } 734: 1599: 4187: 1810:{\displaystyle \nabla ^{2}\varphi +{\frac {\partial }{\partial t}}\left(\mathbf {\nabla } \cdot \mathbf {A} \right)=-{\frac {\rho }{\varepsilon _{0}}}} 11031: 746: 2012:

the three components of the vector potential. However, the equations are messier than Maxwell's equations using the electric and magnetic fields.

4024: 1163: 11275:. This formulation is as general as that of differential forms for manifolds with a metric tensor, as then these are naturally identified with 11271:

can be used. The derivatives that appear in Maxwell's equations are vectors and electromagnetic fields are represented by the Faraday bivector

10637:

In quantum mechanics, the connection itself is used to define the dynamics of the system. This formulation allows a natural description of the

4342:{\displaystyle \left({\frac {1}{c}}{\dfrac {\partial }{\partial t}}+{\boldsymbol {\nabla }}\right)\mathbf {F} =\mu _{0}c(c\rho -\mathbf {J} ).} 10755:) unobservable information. The non uniqueness of the potentials is well understood, however. For every scalar function of position and time 8010:

In this formulation, electromagnetism generalises immediately to any 4-dimensional oriented manifold or with small adaptations any manifold.

4910: 7203: 6138:

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 (

10648:

throughout the space-time region outside the tube, during the experiment. This means by definition that the connection ∇ is flat there.

3430:{\displaystyle \nabla ^{2}\varphi -{\frac {1}{c^{2}}}{\frac {\partial ^{2}\varphi }{\partial t^{2}}}=-{\frac {\rho }{\varepsilon _{0}}}} 11063:

field. The motion is exactly consistent in these two different reference frames, but it mathematically arises in quite different ways.

11022:

The Lorenz gauge condition has the advantage of being Lorentz invariant and leading to Lorentz-invariant equations for the potentials.

10676:

In advanced classical mechanics it is often useful, and in quantum mechanics frequently essential, to express Maxwell's equations in a

9200: 997: 766: 10854:{\displaystyle \varphi '=\varphi -{\frac {\partial \lambda }{\partial t}},\quad \mathbf {A} '=\mathbf {A} +\mathbf {\nabla } \lambda } 9665: 136: 7847:{\textstyle (g_{ij})=\left(\left\langle {\frac {\partial }{\partial x_{i}}},{\frac {\partial }{\partial x_{j}}}\right\rangle \right)} 1504:

Many times in the use and calculation of electric and magnetic fields, the approach used first computes an associated potential: the

11173:

by choosing a preferred time direction. To have physical degrees of freedom propagated by these field equations, one must include a

4816: 11075: 6809: 3324:{\displaystyle \nabla ^{2}\mathbf {A} -{\frac {1}{c^{2}}}{\frac {\partial ^{2}\mathbf {A} }{\partial t^{2}}}=-\mu _{0}\mathbf {J} } 1412:{\displaystyle \nabla \times \mathbf {B} =\mu _{0}\mathbf {J} +\mu _{0}\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}} 776: 11107:

Gauss's law for magnetism and the Faraday–Maxwell law can be grouped together since the equations are homogeneous, and be seen as

9222:. (Whether this is the appropriate generalization requires separate investigation.) The sourced and source-free equations become ( 7171: 601: 7857: 7339: 11616: 11597: 11567: 11433: 2175: 1652: 1492:), and possibly also field dependencies representing nonlinear and/or nonlocal material responses to large amplitude fields ( 616: 611: 238: 626: 11048: 4127: 4766: 2649:{\displaystyle \mathbf {\nabla } \cdot \mathbf {A} '=-\mu _{0}\varepsilon _{0}{\frac {\partial \varphi '}{\partial t}},} 11338: 7412: 1473: 11154:), coupling the field to matter. For the field formulation of Maxwell's equations in terms of a principle of extremal 8953:{\displaystyle {{\star }\mathbf {J} }=-\rho \,\mathrm {d} t+j_{x}\mathrm {d} x+j_{y}\mathrm {d} y+j_{z}\mathrm {d} z.} 11579: 228: 174: 109: 87: 52: 496: 80: 11015:{\displaystyle \mathbf {\nabla } \cdot \mathbf {A} +{\frac {1}{c^{2}}}{\frac {\partial \varphi }{\partial t}}=0\,.} 9830:{\displaystyle \mathbf {F} ={\frac {1}{2}}F_{\alpha \beta }\,\mathrm {d} x^{\alpha }\wedge \mathrm {d} x^{\beta }.} 3541:{\displaystyle \mathbf {J} =-e\psi ^{\dagger }{\boldsymbol {\alpha }}\psi \,\quad \rho =-e\psi ^{\dagger }\psi \,,} 2025:. Specifically for these equations, for any choice of a twice-differentiable scalar function of position and time 1686: 11059:

force. But in the frame of a conductor moving relative to the magnet, the conductor experiences a force due to an

5336:{\displaystyle J=J^{\mu }\gamma _{\mu }=c\rho \gamma _{0}+J^{k}\gamma _{k}=\gamma _{0}(c\rho -J^{k}\sigma _{k}).} 1221: 411: 6775: 4719: 4707: 3867: 11483: 10571: 7643: 6004: 3915: 990: 756: 233: 1641:

These relations can be substituted into Maxwell's equations to express the latter in terms of the potentials.

11640: 11293: 10575: 6557: 1642: 1321: 771: 476: 6883:

is a linear transformation from the space of 2-forms to the space of (4 − 2)-forms defined by the metric in

6129:{\displaystyle \mathbf {A} =-\phi \,\mathrm {d} t+A_{x}\mathrm {d} x+A_{y}\mathrm {d} y+A_{z}\mathrm {d} z.} 11227:

generates through the introduction of a distributive, associative (but not commutative) product called the

8125:{\displaystyle \mathbf {A} =\phi \,\mathrm {d} t-A_{x}\mathrm {d} x-A_{y}\mathrm {d} y-A_{z}\mathrm {d} z.} 7977: 7597:{\textstyle \left\{{\frac {\partial }{\partial x_{1}}},\ldots ,{\frac {\partial }{\partial x_{n}}}\right\}} 1646: 1255: 636: 376: 243: 9999:

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

7062: 7013: 6914: 5532: 4352:

In three dimensions, the derivative has a special structure allowing the introduction of a cross product:

366: 10597:

and can be interpreted as a field strength. If the line bundle is trivial with flat reference connection

7937: 6184: 929: 804: 701: 676: 596: 4181:, but are usually not equated with them, as they are different objects with different interpretations. 1031: 429: 11366: 7701: 6143: 1131:

The behaviour of electric and magnetic fields, whether in cases of electrostatics, magnetostatics, or

11635: 11328: 11136: 10693: 10631: 9182:{\displaystyle {\star }(1)=\mathrm {d} t\wedge \mathrm {d} x\wedge \mathrm {d} y\wedge \mathrm {d} z} 8007:. Up to scaling, this is the only invariant tensor of this type that can be defined with the metric. 7329:{\displaystyle \mathbf {F} ={\frac {1}{2}}F_{pq}\mathbf {\theta } ^{p}\wedge \mathbf {\theta } ^{q}.} 6857:– a natural coordinate- and metric-independent differential operator acting on forms, and the (dual) 4090: 3858: 1529: 983: 944: 471: 461: 401: 396: 336: 4879: 3456:. If the matter field is taken so as to describe the interaction of electromagnetic fields with the 1590:{\displaystyle \mathbf {E} =-\mathbf {\nabla } \varphi -{\frac {\partial \mathbf {A} }{\partial t}}} 11450: 10652: 10638: 10579: 6009: 4177:, due to the fact that the basis used is orthonormal. These basis vectors share the algebra of the 1489: 481: 74: 8019: 7040: 6991: 5672: 914: 416: 11333: 11212: 10724:

in terms of the electric and magnetic potentials that then satisfy the homogeneous equations for

5608:, we implicitly take the sum over all values of the indices that can vary within the dimension. 3818: 3565: 1019: 794: 321: 311: 306: 141: 11381: 11066:

For this reason and others, it is often useful to rewrite Maxwell's equations in a way that is "

919: 889: 11457: 11421: 10708:

on the homogeneous equations to solve them in a universal way (this assumes that we consider a

10028: 6759: 5700: 3177: 3110: 2560: 2554: 1136: 1126: 1114: 741: 511: 286: 91: 7607: 6864: 1511: 11323: 11108: 7521: 5651: 4010:{\displaystyle \mathbf {F} =\mathbf {E} +Ic\mathbf {B} =E^{k}\sigma _{k}+IcB^{k}\sigma _{k},} 3830: 2519: 839: 526: 516: 466: 456: 10889:, and the freedom to select any pair of potentials in its gauge equivalence class is called 10861:

without changing the electric and magnetic field. Two pairs of gauge transformed potentials

1423: 11268: 11191: 11151: 11067: 10771: 10732:

as identities. Substitution gives the non-homogeneous Maxwell equations in potential form.

10716:). The potentials are defined as in the table above. Alternatively, these equations define 9219: 3452: 3141: 964: 864: 829: 581: 446: 346: 331: 266: 203: 8: 11131: 10890: 10032: 10024: 7408: 6854: 6000: 2163: 2022: 1485: 1110: 1102: 924: 904: 899: 706: 691: 576: 546: 441: 371: 6318:

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

5650:

are constant everywhere, Maxwell's equations simplify considerably once the language of

11318: 11167: 11041: 10713: 10681: 10627: 9732:

in every point of the open set where the coordinates are defined. Using this basis and

9702: 9215: 7971: 6888: 2527: 1505: 1311:{\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}} 799: 539: 341: 301: 7161:{\displaystyle C:\Lambda ^{2}\ni \mathbf {F} \mapsto \mathbf {G} \in \Lambda ^{(4-2)}} 3180:

of the electromagnetic fields proceeds by elevating the scalar and vector potentials;

621: 11612: 11593: 11575: 11563: 11560:

Electromagnetics, Microwave Circuit and Antenna Design for Communications Engineering

11429: 11413: 11308: 11303: 11228: 10752: 10623: 9720: 7513:{\displaystyle C_{pq}^{mn}={\frac {1}{2}}g^{ma}g^{nb}\varepsilon _{abpq}{\sqrt {-g}}} 5659: 5655: 5641: 5626: 5605: 5583: 4713: 4085: 3838: 3166: 2271:

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

1461: 1450: 859: 10705: 1036:

The most common description of the electromagnetic field uses two three-dimensional

11544: 11492: 11190:, and take into account the non-physical degrees of freedom that can be removed by 11155: 11074:

consistent with special relativity, even with just a glance at the equations—using

10697: 10685: 10655:, however, the connection depends on the magnetic field through the tube since the 10564: 8023: 7251: 5587: 3862: 3135: 1493: 1015: 959: 874: 834: 824: 711: 666: 649: 566: 501: 271: 195: 6970:{\displaystyle \mathrm {d} {\mathbf {J} }=\mathrm {d} ^{2}{\star }\mathbf {F} =0.} 11393: 10901: 7404: 6884: 3834: 1469: 1132: 894: 819: 814: 681: 556: 521: 381: 281: 11529: 10559:

An elegant and intuitive way to formulate Maxwell's equations is to use complex

934: 11401: 11378: 11298: 11288: 11052: 10583: 9733: 9223: 6139: 5601: 5593: 4904: 4178: 3846: 3555: 3457: 3197: 1472:, also a function of time and position. The equations take this form with the 1439: 1211: 1045: 1041: 854: 849: 671: 561: 486: 436: 386: 359: 316: 291: 261: 254: 11587: 11128:. Gauss's law for electricity and the Ampere–Maxwell law could be seen as the 10920:, which is mostly used in the case of magneto statics when we can neglect the 10554: 11629: 10939: 10906: 10011: 6892: 3159: 2169: 1152: 969: 954: 939: 879: 591: 506: 491: 406: 391: 296: 7403:

where the field coefficient functions and the constitutive coefficients are

11497: 11478: 11208: 11174: 11148: 11140: 10935: 10897: 4121: 4018: 3461: 1201:{\displaystyle \nabla \cdot \mathbf {E} ={\frac {\rho }{\varepsilon _{0}}}} 1037: 949: 844: 809: 751: 686: 606: 571: 451: 326: 2262:{\displaystyle \nabla ^{2}\lambda =-\mathbf {\nabla } \cdot \mathbf {A} .} 2100:{\displaystyle \varphi '=\varphi -{\frac {\partial \lambda }{\partial t}}} 11549: 11086: 10560: 10007: 9112: 7747: 5658:

is used. The electric and magnetic fields are now jointly described by a

3842: 869: 721: 551: 213: 11417: 11313: 11170: 9104:{\displaystyle {\mathbf {J} \wedge {\star }\mathbf {J} }=\,{\star }(1)} 6858: 6763: 6147: 2814:

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

2322:{\displaystyle \nabla ^{2}\varphi '=-{\frac {\rho }{\varepsilon _{0}}}} 1480: 586: 5703:) can be written as a 2-form in Minkowski space with metric signature 4907:

but their matrix representation is not needed. The derivative is now

11224: 11093: 11079: 9211: 5666: 2145:{\displaystyle \mathbf {A} '=\mathbf {A} +\mathbf {\nabla } \lambda } 909: 884: 696: 218: 6183:). For the basis cotangent vectors, the Hodge dual is given as (see 11259:-vector component can be identified with the inner product and the 10896:

The potential equations can be simplified using a procedure called

10709: 10656: 6979: 5597: 1120: 1101:) is non-zero, and is constant in time, the field is said to be an 661: 656: 276: 10743:

are consistent with given observable electric and magnetic fields

1627:{\displaystyle \mathbf {B} =\mathbf {\nabla } \times \mathbf {A} } 4225:{\displaystyle {\boldsymbol {\nabla }}=\sigma ^{k}\partial _{k},} 1109:) is non-zero and is constant in time, the field is said to be a 631: 6978:

timelike interval. As the exterior derivative is defined on any

2213:, which corresponds to the case of magnetostatics. In terms of 127: 10668:

Following are the reasons for using each of such formulations.

5617: 3449: 2041:

is a solution for a given system, then so is another potential

1636: 716: 223: 4869:{\displaystyle I=\gamma _{0}\gamma _{1}\gamma _{2}\gamma _{3}} 3829:

Analogous to the tensor formulation, two objects, one for the

11479:"On some recent interactions between mathematics and physics" 9714: 6839:{\displaystyle \mathrm {d} {\star }\mathbf {F} =\mathbf {J} } 10568: 10555:

Classical electrodynamics as the curvature of a line bundle

4077:{\displaystyle c\rho -\mathbf {J} =c\rho -J^{k}\sigma _{k}} 6150:

of this 2-form is needed. The Hodge star operator takes a

1442:, which can (and often does) depend on time and position, 11398:

Differential Forms with Applications to Physical Sciences

11025: 10035:

then shows that in this coordinate neighborhood we have:

7250:

When the fields are expressed as linear combinations (of

7089: 3911:, the field and current are represented by multivectors. 7927:{\displaystyle (g^{ij})=(\langle dx^{i},dx^{j}\rangle )} 6168:

is the number of dimensions. Here, it takes the 2-form (

7407:

for swapping of each one's indices. In particular, the

4946:{\displaystyle \nabla =\gamma ^{\mu }\partial _{\mu }.} 4234:

Maxwell's equations are reduced to the single equation

3172: 2206:{\displaystyle \mathbf {\nabla } \cdot \mathbf {A} '=0} 1676:{\displaystyle \nabla \cdot \nabla \times \mathbf {A} } 10589:= ∇, which is a two-form that automatically satisfies 7756: 7530: 7524:

with respect to a (not necessarily orthonormal) basis

6542: 3558:. Using this, we can re-write Maxwell's equations as: 3158:). These equations are inhomogeneous versions of the 1012:

mathematical descriptions of the electromagnetic field

11530:"Differential Forms and Electromagnetic Field Theory" 11384:(Am. J. Phys. 71 (2), February 2003, pp. 104–121) p26 10948: 10780: 10385: 10221: 10045: 9848: 9758: 9719:

The formulation of the Maxwell equations in terms of

9668: 9486: 9232: 9120: 8968: 8853: 8662: 8429: 8140: 8035: 7980: 7940: 7860: 7704: 7646: 7610: 7420: 7342: 7264: 7228:{\displaystyle \mathrm {d} \mathbf {G} =\mathbf {J} } 7206: 7174: 7100: 7065: 7043: 7016: 6994: 6923: 6867: 6812: 6778: 6567: 6324: 6193: 6036: 6012: 5713: 5675: 5535: 5481: 5351: 5213: 4961: 4913: 4882: 4819: 4769: 4722: 4453: 4358: 4266: 4249: 4190: 4130: 4093: 4027: 3924: 3870: 3700: 3577: 3474: 3336: 3228: 3144: 3113: 2963: 2820: 2666: 2573: 2334: 2277: 2223: 2178: 2112: 2059: 1824: 1729: 1689: 1655: 1602: 1542: 1514: 1334: 1268: 1224: 1166: 11076:

covariant and contravariant four-vectors and tensors

10023:. A small computation that uses the symmetry of the 6766:

is the current 3-form, express Maxwell's equations:

11412: 11036:

Covariant formulation of classical electromagnetism

10660:which leads to a shift in the diffraction pattern. 9705:

that characterizes the curvature of spacetime and ∇

5207:and the charge and current density become a vector 4708:

Spacetime algebra § Classical electromagnetism

4170:{\displaystyle I=\sigma _{1}\sigma _{2}\sigma _{3}} 11014: 10853: 10663: 10543: 10369: 10205: 9988: 9829: 9693: 9652: 9472: 9181: 9103: 8952: 8839: 8642: 8415: 8124: 7999: 7962: 7926: 7846: 7738: 7690: 7632: 7596: 7512: 7395: 7328: 7227: 7193: 7160: 7078: 7051: 7029: 7002: 6969: 6875: 6838: 6797: 6744: 6532: 6310: 6128: 6023: 5991: 5691: 5566: 5520:Maxwell's equations reduce to the single equation 5510: 5335: 5199: 4945: 4895: 4868: 4806:{\displaystyle \sigma _{k}=\gamma _{k}\gamma _{0}} 4805: 4755: 4691: 4438: 4341: 4224: 4169: 4112: 4076: 4009: 3903: 3806: 3685: 3540: 3429: 3323: 3150: 3126: 3097: 2950: 2804: 2648: 2508: 2321: 2261: 2205: 2144: 2099: 2000: 1809: 1704: 1675: 1626: 1589: 1520: 1411: 1310: 1244: 1200: 11032:Classical electromagnetism and special relativity 9469: 3824: 2500: 2474: 2468: 2467: 2391: 11627: 11382:"Reforming the Mathematical Language of Physics" 11102: 11040:Maxwell's equations are exactly consistent with 9694:{\displaystyle {\Gamma ^{\alpha }}_{\mu \beta }} 6172:) and gives another 2-form (in four dimensions, 3220:into the previous Lorenz gauge equations gives: 1121:Maxwell's equations in the vector field approach 8013: 2217:, this means that it must satisfy the equation 11574:(with worked problems in Warnick, Russer 2006 11218: 5577: 3468:, the current and charge densities have form: 11527: 3138:(some authors denote this by only the square 1705:{\displaystyle \nabla \times \nabla \varphi } 991: 11606: 11357:Introduction to Electrodynamics by Griffiths 11055:of the magnet, that conductor experiences a 8962:The current norm is now positive and equals 7918: 7886: 7685: 7647: 4903:s have the same algebraic properties of the 4107: 4094: 3852: 1637:Maxwell's equations in potential formulation 11585: 10751:, so the potentials seem to contain more, ( 4955:The Riemann–Silberstein becomes a bivector 4712:We can identify APS as a subalgebra of the 1499: 1245:{\displaystyle \nabla \cdot \mathbf {B} =0} 53:Learn how and when to remove these messages 11121:(a 2-form), which can be derived from the 9715:Formulation in terms of differential forms 9205: 7194:{\displaystyle \mathrm {d} \mathbf {F} =0} 6798:{\displaystyle \mathrm {d} \mathbf {F} =0} 6185:Hodge star operator § Four dimensions 4756:{\displaystyle C\ell _{1,3}(\mathbb {R} )} 3904:{\displaystyle C\ell _{3,0}(\mathbb {R} )} 3447:are the current and charge density of the 998: 984: 202: 11548: 11496: 11448: 11049:conductor moving in the field of a magnet 11008: 10446: 10285: 9907: 9790: 9649: 9468: 9086: 8875: 8677: 8047: 7691:{\displaystyle \{dx_{1},\ldots ,dx_{n}\}} 7336:the constitutive relation takes the form 6988:In much of the literature, the notations 6579: 6051: 4746: 3894: 3534: 3507: 2548: 1135:(electromagnetic fields), is governed by 1105:. Similarly, if only the magnetic field ( 175:Learn how and when to remove this message 110:Learn how and when to remove this message 11589:Foundations of Classical Electrodynamics 10671: 10031:) and the covariant constantness of the 9839:The current-vector infinitesimal 3-form 9210:Matter and energy generate curvature of 7396:{\displaystyle G_{pq}=C_{pq}^{mn}F_{mn}} 7239:still satisfies the continuity equation 1025: 152:of all important aspects of the article. 73:This article includes a list of general 11607:Doran, Chris; Lasenby, Anthony (2007). 11586:Hehl, Friedrich; Obukhov, Yuri (2003). 11513:On the electrodynamics of moving bodies 11476: 11223:This formulation uses the algebra that 9201:Maxwell's equations in curved spacetime 6913:satisfies the conservation of current ( 6887:(in four dimensions even by any metric 3676: 3500: 747:Electromagnetism and special relativity 11628: 11557: 11026:Manifestly covariant (tensor) approach 7090:Linear macroscopic influence of matter 148:Please consider expanding the lead to 11537:Progress in Electromagnetics Research 11528:Warnick, Karl; Russer, Peter (2014). 10770:, the potentials can be changed by a 8847:and the corresponding dual 1-form is 8134:The Faraday curvature 2-form becomes 8000:{\displaystyle \varepsilon _{1234}=1} 7086:is a 3-form called the dual current. 6762:, and the exterior derivative of its 2021:electric and magnetic fields, called 767:Maxwell equations in curved spacetime 9749:, corresponding to the field 2-form 8020:particle physicist's sign convention 7079:{\displaystyle {\star }\mathbf {J} } 7030:{\displaystyle {\star }\mathbf {J} } 5567:{\displaystyle \nabla F=\mu _{0}cJ.} 4701: 3914:The field multivector, known as the 3173:Extension to quantum electrodynamics 121: 59: 18: 10027:(i.e., the torsion-freeness of the 9192: 7963:{\displaystyle \varepsilon _{abpq}} 7411:that was used in the above case is 7059:is a 1-form called the current and 6891:to this metric). The fields are in 6543:Current 3-form, dual current 1-form 13: 11367:Quantum Electrodynamics, Mathworld 11339:List of electromagnetism equations 10993: 10985: 10950: 10844: 10809: 10801: 10521: 10503: 10485: 10467: 10387: 10342: 10324: 10306: 10223: 10180: 10162: 10144: 10118: 10092: 10066: 10047: 9964: 9946: 9928: 9810: 9792: 9672: 9624: 9598: 9572: 9546: 9520: 9494: 9432: 9429: 9426: 9391: 9379: 9376: 9373: 9328: 9290: 9262: 9172: 9161: 9150: 9139: 8940: 8919: 8898: 8877: 8830: 8819: 8808: 8787: 8776: 8765: 8744: 8733: 8722: 8701: 8690: 8679: 8630: 8619: 8598: 8587: 8566: 8555: 8534: 8523: 8502: 8491: 8470: 8459: 8402: 8391: 8370: 8359: 8338: 8327: 8306: 8295: 8274: 8263: 8242: 8231: 8197: 8179: 8112: 8091: 8070: 8049: 7819: 7815: 7794: 7790: 7573: 7569: 7542: 7538: 7208: 7176: 7137: 7108: 6941: 6925: 6814: 6780: 6732: 6721: 6710: 6689: 6678: 6667: 6646: 6635: 6624: 6603: 6592: 6581: 6523: 6512: 6491: 6480: 6459: 6448: 6427: 6416: 6395: 6384: 6363: 6352: 6298: 6287: 6273: 6262: 6242: 6231: 6214: 6203: 6116: 6095: 6074: 6053: 5978: 5967: 5946: 5935: 5914: 5903: 5882: 5871: 5850: 5839: 5818: 5807: 5772: 5754: 5536: 5487: 5483: 5458: 5435: 5422: 5389: 5362: 4931: 4914: 4635: 4623: 4559: 4547: 4272: 4268: 4210: 3752: 3738: 3702: 3633: 3617: 3579: 3388: 3374: 3338: 3284: 3268: 3230: 3032: 3011: 2965: 2885: 2866: 2822: 2790: 2782: 2744: 2721: 2707: 2668: 2634: 2621: 2575: 2559:A gauge that is often used is the 2491: 2478: 2464: 2404: 2382: 2336: 2279: 2244: 2225: 2180: 2135: 2088: 2080: 1963: 1955: 1920: 1910: 1885: 1869: 1831: 1767: 1752: 1748: 1731: 1696: 1690: 1662: 1656: 1612: 1578: 1568: 1555: 1528:, for the electric field, and the 1474:International System of Quantities 1400: 1390: 1335: 1299: 1289: 1269: 1225: 1167: 79:it lacks sufficient corresponding 14: 11652: 5600:are used. (To convert to SI, see 2164:Gauge fixing § Coulomb gauge 34:This article has multiple issues. 11609:Geometric Algebra for Physicists 11135:of the fields, obtained via the 10958: 10836: 10824: 10392: 10360: 10232: 10052: 9850: 9760: 8984: 8971: 8861: 8664: 8437: 8146: 8037: 7739:{\displaystyle V^{*}=T_{p}^{*}M} 7266: 7221: 7213: 7181: 7129: 7121: 7072: 7045: 7023: 6996: 6957: 6931: 6832: 6824: 6785: 6569: 6331: 6038: 6014: 5719: 5501: 4983: 4969: 4674: 4666: 4628: 4613: 4605: 4584: 4552: 4517: 4509: 4468: 4460: 4432: 4424: 4413: 4405: 4397: 4389: 4381: 4373: 4365: 4360: 4329: 4296: 4286: 4192: 4038: 3948: 3934: 3926: 3627: 3589: 3476: 3317: 3278: 3240: 3091: 3066: 3022: 2976: 2752: 2584: 2437: 2394: 2347: 2252: 2189: 2157: 2127: 2115: 2015: 1994: 1928: 1879: 1841: 1775: 1669: 1620: 1604: 1572: 1544: 1394: 1360: 1342: 1293: 1276: 1232: 1174: 126: 64: 23: 16:Formulations of electromagnetism 10821: 10664:Discussion and other approaches 10010:of the matrix representing the 9740:The antisymmetric field tensor 9728:that gives a basis of 1-forms d 8423:and the Maxwell tensor becomes 6252: 5611: 4113:{\displaystyle \{\sigma _{k}\}} 3508: 2563:. In this, the scalar function 140:may be too short to adequately 42:or discuss these issues on the 11562:(2nd ed.). Artech House. 11505: 11484:Canadian Mathematical Bulletin 11470: 11449:M. Murray (5 September 2008). 11442: 11406: 11387: 11371: 11360: 11351: 11143:, from the "interaction term" 10140: 10062: 9132: 9126: 9098: 9092: 9083: 9074: 9060: 9048: 9034: 9022: 9008: 8992: 7921: 7883: 7877: 7861: 7773: 7757: 7153: 7141: 7125: 6280: 6258: 6221: 6199: 6005:electromagnetic four-potential 5327: 5295: 5191: 5092: 4896:{\displaystyle \gamma _{\mu }} 4750: 4742: 4333: 4316: 4184:After defining the derivative 3898: 3890: 3825:Geometric algebra formulations 1014:that are used in the study of 150:provide an accessible overview 1: 11521: 11428:. W. H. Freeman. p. 81. 11294:Electromagnetic wave equation 11078:. This can be done using the 9711:is the covariant derivative. 6024:{\displaystyle \mathbf {A} :} 5669:manifold. The Faraday tensor 2172:is chosen in such a way that 1683:and the curl of the gradient 1137:Maxwell-Heaviside's equations 772:Relativistic electromagnetism 11451:"Line Bundles. Honours 1996" 8014:Alternative metric signature 7052:{\displaystyle \mathbf {J} } 7003:{\displaystyle \mathbf {J} } 5692:{\displaystyle F_{\mu \nu }} 3460:given by the four-component 1097:If only the electric field ( 7: 11282: 11219:Geometric calculus approach 11103:Differential forms approach 10582:∇ on the line bundle has a 5578:Differential forms approach 10: 11657: 11235:-vector decomposes into a 11029: 10735:Many different choices of 9198: 8029:, the potential 1-form is 5581: 4705: 3916:Riemann–Silberstein vector 3817:which is the form used in 2660:must satisfy the equation 2552: 2161: 1322:Faraday's law of induction 1124: 1032:Classical electromagnetism 1029: 497:Liénard–Wiechert potential 11611:. Cambridge Univ. Press. 11329:Electromagnetic radiation 10632:magnetic vector potential 10567:, on the fibers of which 9214:. This is the subject of 7235:where the current 3-form 3861:(APS), also known as the 3859:Algebra of physical space 3853:Algebra of physical space 3845:, which sometimes follow 3127:{\displaystyle \Box ^{2}} 1647:Gauss's law for magnetism 1530:magnetic vector potential 1256:Gauss's law for magnetism 1146: 762:Mathematical descriptions 472:Electromagnetic radiation 462:Electromagnetic induction 402:Magnetic vector potential 397:Magnetic scalar potential 11344: 11047:For example, consider a 10379:the continuity equation 7633:{\displaystyle V=T_{p}M} 6909:Since d = 0, the 3-form 6876:{\displaystyle {\star }} 1521:{\displaystyle \varphi } 1500:Potential field approach 1020:fundamental interactions 11511:Albert Einstein (1905) 11422:Wheeler, John Archibald 11334:Quantum electrodynamics 9206:Traditional formulation 7854:and its inverse matrix 6144:Ampère-Maxwell equation 3819:quantum electrodynamics 312:Electrostatic induction 307:Electrostatic discharge 94:more precise citations. 11558:Russer, Peter (2006). 11498:10.4153/CMB-1985-016-3 11458:University of Adelaide 11377:Oersted Medal Lecture 11160:electromagnetic tensor 11016: 10938:(named after the Dane 10855: 10545: 10371: 10207: 10029:Levi-Civita connection 9990: 9831: 9695: 9654: 9474: 9183: 9105: 8954: 8841: 8644: 8417: 8126: 8001: 7964: 7928: 7848: 7740: 7692: 7634: 7598: 7514: 7397: 7330: 7229: 7195: 7162: 7080: 7053: 7037:are switched, so that 7031: 7004: 6971: 6877: 6840: 6799: 6746: 6534: 6312: 6130: 6025: 5993: 5701:electromagnetic tensor 5693: 5568: 5512: 5345:Owing to the identity 5337: 5201: 4947: 4897: 4870: 4807: 4757: 4693: 4440: 4343: 4226: 4171: 4120:. Similarly, the unit 4114: 4078: 4011: 3905: 3808: 3687: 3542: 3431: 3325: 3178:Canonical quantization 3152: 3128: 3099: 2952: 2806: 2650: 2561:Lorenz gauge condition 2555:Lorenz gauge condition 2549:Lorenz gauge condition 2510: 2323: 2263: 2207: 2146: 2101: 2002: 1811: 1706: 1677: 1628: 1591: 1522: 1413: 1312: 1246: 1202: 742:Electromagnetic tensor 11324:Electromagnetic field 11152:covariant derivatives 11017: 10856: 10678:potential formulation 10672:Potential formulation 10565:principal U(1)-bundle 10546: 10372: 10208: 10039:the Bianchi identity 9991: 9832: 9696: 9655: 9475: 9220:covariant derivatives 9184: 9106: 8955: 8842: 8645: 8418: 8127: 8002: 7965: 7929: 7849: 7741: 7693: 7635: 7599: 7522:tensor index notation 7515: 7398: 7331: 7230: 7196: 7163: 7081: 7054: 7032: 7005: 6972: 6878: 6841: 6800: 6747: 6553:electric current form 6535: 6313: 6131: 6026: 5994: 5694: 5652:differential geometry 5569: 5513: 5338: 5202: 4948: 4898: 4871: 4808: 4758: 4706:Further information: 4694: 4441: 4344: 4227: 4172: 4115: 4079: 4012: 3906: 3837:, are introduced. In 3831:electromagnetic field 3809: 3688: 3543: 3432: 3326: 3153: 3151:{\displaystyle \Box } 3129: 3100: 2953: 2807: 2651: 2511: 2324: 2264: 2208: 2147: 2102: 2003: 1812: 1720:potential formulation 1707: 1678: 1629: 1592: 1523: 1470:current per unit area 1414: 1313: 1247: 1203: 1071:(electric field) and 1026:Vector field approach 735:Covariant formulation 527:Synchrotron radiation 467:Electromagnetic pulse 457:Electromagnetic field 11641:Mathematical physics 11550:10.2528/PIER14063009 11213:Faddeev–Popov ghosts 11192:gauge transformation 11147:(introduced through 11068:manifestly covariant 10946: 10778: 10772:gauge transformation 10653:Aharonov–Bohm effect 10639:Aharonov–Bohm effect 10383: 10219: 10215:the source equation 10043: 9846: 9756: 9666: 9484: 9230: 9118: 8966: 8851: 8660: 8427: 8138: 8033: 7978: 7938: 7858: 7754: 7702: 7644: 7608: 7528: 7418: 7340: 7262: 7204: 7172: 7098: 7063: 7041: 7014: 6992: 6921: 6865: 6810: 6776: 6565: 6322: 6191: 6034: 6010: 5711: 5673: 5533: 5524:Maxwell's equations 5349: 5211: 4959: 4911: 4880: 4817: 4767: 4720: 4451: 4356: 4247: 4238:Maxwell's equations 4188: 4128: 4091: 4025: 3922: 3868: 3698: 3575: 3554:are the first three 3472: 3334: 3226: 3142: 3111: 2961: 2818: 2664: 2571: 2567:is chosen such that 2332: 2275: 2221: 2176: 2110: 2057: 1822: 1727: 1687: 1653: 1600: 1540: 1512: 1490:Green–Kubo relations 1332: 1266: 1222: 1164: 777:Stress–energy tensor 702:Reluctance (complex) 447:Displacement current 11132:equations of motion 10033:Hodge star operator 10025:Christoffel symbols 9111:with the canonical 8652:The current 3-form 7732: 7640:and its dual basis 7604:in a tangent space 7441: 7409:Hodge star operator 7379: 6915:continuity equation 6855:exterior derivative 6770:Maxwell's equations 6001:exterior derivative 5665:in a 4-dimensional 3562:Maxwell's equations 1716:Maxwell's equations 1486:dispersion (optics) 1148:Maxwell's equations 1127:Maxwell's equations 1115:Maxwell's equations 1111:magnetostatic field 1103:electrostatic field 692:Magnetomotive force 577:Electromotive force 547:Alternating current 482:Jefimenko equations 442:Cyclotron radiation 11414:Misner, Charles W. 11400:, pages 44 to 46, 11319:Near and far field 11042:special relativity 11012: 10851: 10714:contractible space 10694:magnetic potential 10682:electric potential 10628:electric potential 10541: 10367: 10203: 9986: 9827: 9734:cgs-Gaussian units 9721:differential forms 9703:Christoffel symbol 9691: 9650: 9470: 9224:cgs-Gaussian units 9216:general relativity 9179: 9101: 8950: 8837: 8640: 8413: 8411: 8122: 7997: 7972:Levi-Civita symbol 7960: 7924: 7844: 7736: 7718: 7688: 7630: 7594: 7510: 7421: 7413:obtained by taking 7393: 7359: 7326: 7225: 7191: 7158: 7076: 7049: 7027: 7000: 6967: 6873: 6836: 6795: 6742: 6530: 6308: 6126: 6021: 5989: 5987: 5689: 5656:differential forms 5594:cgs-Gaussian units 5564: 5508: 5495: 5333: 5197: 4943: 4893: 4866: 4803: 4753: 4689: 4436: 4339: 4280: 4222: 4167: 4110: 4074: 4007: 3901: 3804: 3683: 3538: 3427: 3321: 3196:), from fields to 3148: 3124: 3095: 2948: 2802: 2646: 2528:special relativity 2520:Poisson's equation 2506: 2319: 2259: 2203: 2142: 2097: 1998: 1807: 1702: 1673: 1624: 1587: 1518: 1506:electric potential 1424:Ampère-Maxwell law 1409: 1308: 1242: 1198: 1094:(magnetic field). 1018:, one of the four 1010:There are various 540:Electrical network 377:Gauss magnetic law 342:Static electricity 302:Electric potential 11618:978-0-521-71595-9 11599:978-0-8176-4222-8 11569:978-1-58053-907-4 11435:978-0-7167-0344-0 11309:Magnetic constant 11304:Electric constant 11269:Green's functions 11229:geometric product 11000: 10980: 10816: 10444: 10412: 10283: 10248: 9905: 9885: 9870: 9775: 9442: 9437: 9415: 9389: 9384: 9362: 9246: 8163: 7833: 7808: 7587: 7556: 7508: 7453: 7281: 7254:) of basis forms 7252:exterior products 6547:Here, the 3-form 5738: 5606:Einstein notation 5592:In what follows, 5584:Differential form 5527:(STA formulation) 5494: 5478: 4714:spacetime algebra 4702:Spacetime algebra 4644: 4568: 4490: 4279: 4263: 4241:(APS formulation) 4086:orthonormal basis 3839:geometric algebra 3786: 3766: 3732: 3647: 3611: 3425: 3402: 3368: 3298: 3262: 3167:Lorentz invariant 3046: 2946: 2899: 2797: 2735: 2641: 2498: 2418: 2317: 2095: 1970: 1950: 1899: 1863: 1805: 1759: 1585: 1462:magnetic constant 1451:electric constant 1430: 1429: 1407: 1306: 1196: 1008: 1007: 707:Reluctance (real) 677:Gyrator–capacitor 622:Resonant cavities 512:Maxwell equations 185: 184: 177: 167: 166: 120: 119: 112: 57: 11648: 11636:Electromagnetism 11622: 11603: 11573: 11554: 11552: 11534: 11515: 11509: 11503: 11502: 11500: 11477:R. Bott (1985). 11474: 11468: 11467: 11465: 11464: 11455: 11446: 11440: 11439: 11410: 11404: 11391: 11385: 11375: 11369: 11364: 11358: 11355: 11266: 11258: 11250: 11242: 11206: 11185: 11021: 11019: 11018: 11013: 11001: 10999: 10991: 10983: 10981: 10979: 10978: 10966: 10961: 10953: 10933: 10919: 10887:gauge equivalent 10884: 10872: 10860: 10858: 10857: 10852: 10847: 10839: 10831: 10827: 10817: 10815: 10807: 10799: 10788: 10769: 10698:vector potential 10686:scalar potential 10647: 10626:composed of the 10617: 10607: 10596: 10550: 10548: 10547: 10542: 10534: 10533: 10524: 10516: 10515: 10506: 10498: 10497: 10488: 10480: 10479: 10470: 10465: 10464: 10445: 10437: 10435: 10434: 10426: 10425: 10424: 10413: 10408: 10400: 10395: 10390: 10376: 10374: 10373: 10368: 10363: 10355: 10354: 10345: 10337: 10336: 10327: 10319: 10318: 10309: 10304: 10303: 10284: 10276: 10274: 10273: 10265: 10264: 10263: 10249: 10241: 10236: 10235: 10226: 10212: 10210: 10209: 10204: 10193: 10192: 10183: 10175: 10174: 10165: 10157: 10156: 10147: 10139: 10138: 10126: 10125: 10113: 10112: 10100: 10099: 10087: 10086: 10074: 10073: 10055: 10050: 10006:is as usual the 9995: 9993: 9992: 9987: 9985: 9981: 9977: 9976: 9967: 9959: 9958: 9949: 9941: 9940: 9931: 9926: 9925: 9906: 9898: 9896: 9895: 9886: 9878: 9871: 9866: 9858: 9853: 9836: 9834: 9833: 9828: 9823: 9822: 9813: 9805: 9804: 9795: 9789: 9788: 9776: 9768: 9763: 9700: 9698: 9697: 9692: 9690: 9689: 9681: 9680: 9679: 9659: 9657: 9656: 9651: 9645: 9644: 9632: 9631: 9619: 9618: 9606: 9605: 9593: 9592: 9580: 9579: 9567: 9566: 9554: 9553: 9541: 9540: 9528: 9527: 9515: 9514: 9502: 9501: 9479: 9477: 9476: 9471: 9467: 9466: 9458: 9457: 9456: 9440: 9439: 9438: 9436: 9435: 9423: 9418: 9413: 9412: 9411: 9399: 9398: 9387: 9386: 9385: 9383: 9382: 9370: 9365: 9360: 9359: 9358: 9346: 9345: 9337: 9336: 9335: 9321: 9320: 9308: 9307: 9299: 9298: 9297: 9283: 9282: 9270: 9269: 9257: 9256: 9247: 9242: 9234: 9193:Curved spacetime 9188: 9186: 9185: 9180: 9175: 9164: 9153: 9142: 9125: 9110: 9108: 9107: 9102: 9091: 9082: 9081: 9072: 9071: 9056: 9055: 9046: 9045: 9030: 9029: 9020: 9019: 9004: 9003: 8988: 8987: 8982: 8974: 8959: 8957: 8956: 8951: 8943: 8938: 8937: 8922: 8917: 8916: 8901: 8896: 8895: 8880: 8865: 8864: 8859: 8846: 8844: 8843: 8838: 8833: 8822: 8811: 8806: 8805: 8790: 8779: 8768: 8763: 8762: 8747: 8736: 8725: 8720: 8719: 8704: 8693: 8682: 8667: 8649: 8647: 8646: 8641: 8633: 8622: 8617: 8616: 8601: 8590: 8585: 8584: 8569: 8558: 8553: 8552: 8537: 8526: 8521: 8520: 8505: 8494: 8489: 8488: 8473: 8462: 8457: 8456: 8441: 8440: 8435: 8422: 8420: 8419: 8414: 8412: 8405: 8394: 8389: 8388: 8373: 8362: 8357: 8356: 8341: 8330: 8325: 8324: 8309: 8298: 8293: 8292: 8277: 8266: 8261: 8260: 8245: 8234: 8229: 8228: 8210: 8209: 8200: 8192: 8191: 8182: 8177: 8176: 8164: 8156: 8149: 8131: 8129: 8128: 8123: 8115: 8110: 8109: 8094: 8089: 8088: 8073: 8068: 8067: 8052: 8040: 8028: 8024:metric signature 8006: 8004: 8003: 7998: 7990: 7989: 7969: 7967: 7966: 7961: 7959: 7958: 7933: 7931: 7930: 7925: 7917: 7916: 7901: 7900: 7876: 7875: 7853: 7851: 7850: 7845: 7843: 7839: 7835: 7834: 7832: 7831: 7830: 7814: 7809: 7807: 7806: 7805: 7789: 7772: 7771: 7745: 7743: 7742: 7737: 7731: 7726: 7714: 7713: 7697: 7695: 7694: 7689: 7684: 7683: 7662: 7661: 7639: 7637: 7636: 7631: 7626: 7625: 7603: 7601: 7600: 7595: 7593: 7589: 7588: 7586: 7585: 7584: 7568: 7557: 7555: 7554: 7553: 7537: 7519: 7517: 7516: 7511: 7509: 7501: 7499: 7498: 7480: 7479: 7467: 7466: 7454: 7446: 7440: 7432: 7402: 7400: 7399: 7394: 7392: 7391: 7378: 7370: 7355: 7354: 7335: 7333: 7332: 7327: 7322: 7321: 7316: 7307: 7306: 7301: 7295: 7294: 7282: 7274: 7269: 7246: 7234: 7232: 7231: 7226: 7224: 7216: 7211: 7200: 7198: 7197: 7192: 7184: 7179: 7167: 7165: 7164: 7159: 7157: 7156: 7132: 7124: 7116: 7115: 7085: 7083: 7082: 7077: 7075: 7070: 7058: 7056: 7055: 7050: 7048: 7036: 7034: 7033: 7028: 7026: 7021: 7009: 7007: 7006: 7001: 6999: 6976: 6974: 6973: 6968: 6960: 6955: 6950: 6949: 6944: 6935: 6934: 6928: 6905: 6882: 6880: 6879: 6874: 6872: 6845: 6843: 6842: 6837: 6835: 6827: 6822: 6817: 6804: 6802: 6801: 6796: 6788: 6783: 6751: 6749: 6748: 6743: 6735: 6724: 6713: 6708: 6707: 6692: 6681: 6670: 6665: 6664: 6649: 6638: 6627: 6622: 6621: 6606: 6595: 6584: 6572: 6539: 6537: 6536: 6531: 6526: 6515: 6510: 6509: 6494: 6483: 6478: 6477: 6462: 6451: 6446: 6445: 6430: 6419: 6414: 6413: 6398: 6387: 6382: 6381: 6366: 6355: 6350: 6349: 6334: 6329: 6317: 6315: 6314: 6309: 6301: 6290: 6276: 6265: 6257: 6245: 6234: 6217: 6206: 6198: 6182: 6163: 6135: 6133: 6132: 6127: 6119: 6114: 6113: 6098: 6093: 6092: 6077: 6072: 6071: 6056: 6041: 6030: 6028: 6027: 6022: 6017: 5998: 5996: 5995: 5990: 5988: 5981: 5970: 5965: 5964: 5949: 5938: 5933: 5932: 5917: 5906: 5901: 5900: 5885: 5874: 5869: 5868: 5853: 5842: 5837: 5836: 5821: 5810: 5805: 5804: 5789: 5785: 5784: 5775: 5767: 5766: 5757: 5752: 5751: 5739: 5731: 5722: 5706: 5698: 5696: 5695: 5690: 5688: 5687: 5649: 5634: 5588:Exterior algebra 5573: 5571: 5570: 5565: 5554: 5553: 5517: 5515: 5514: 5509: 5504: 5496: 5493: 5482: 5479: 5471: 5466: 5465: 5456: 5455: 5443: 5442: 5430: 5429: 5420: 5419: 5410: 5409: 5397: 5396: 5387: 5386: 5377: 5376: 5361: 5360: 5342: 5340: 5339: 5334: 5326: 5325: 5316: 5315: 5294: 5293: 5281: 5280: 5271: 5270: 5258: 5257: 5239: 5238: 5229: 5228: 5206: 5204: 5203: 5198: 5190: 5189: 5180: 5179: 5170: 5169: 5157: 5156: 5147: 5146: 5137: 5136: 5124: 5123: 5114: 5113: 5104: 5103: 5085: 5084: 5075: 5074: 5065: 5064: 5052: 5051: 5042: 5041: 5032: 5031: 5019: 5018: 5009: 5008: 4999: 4998: 4986: 4972: 4952: 4950: 4949: 4944: 4939: 4938: 4929: 4928: 4902: 4900: 4899: 4894: 4892: 4891: 4875: 4873: 4872: 4867: 4865: 4864: 4855: 4854: 4845: 4844: 4835: 4834: 4812: 4810: 4809: 4804: 4802: 4801: 4792: 4791: 4779: 4778: 4762: 4760: 4759: 4754: 4749: 4741: 4740: 4698: 4696: 4695: 4690: 4682: 4678: 4677: 4669: 4650: 4646: 4645: 4643: 4642: 4633: 4632: 4631: 4621: 4616: 4608: 4592: 4588: 4587: 4582: 4581: 4569: 4567: 4566: 4557: 4556: 4555: 4545: 4543: 4542: 4533: 4532: 4520: 4512: 4496: 4492: 4491: 4489: 4488: 4476: 4471: 4463: 4445: 4443: 4442: 4437: 4435: 4427: 4416: 4408: 4400: 4392: 4384: 4376: 4368: 4363: 4348: 4346: 4345: 4340: 4332: 4312: 4311: 4299: 4294: 4290: 4289: 4281: 4278: 4267: 4264: 4256: 4231: 4229: 4228: 4223: 4218: 4217: 4208: 4207: 4195: 4176: 4174: 4173: 4168: 4166: 4165: 4156: 4155: 4146: 4145: 4119: 4117: 4116: 4111: 4106: 4105: 4083: 4081: 4080: 4075: 4073: 4072: 4063: 4062: 4041: 4016: 4014: 4013: 4008: 4003: 4002: 3993: 3992: 3974: 3973: 3964: 3963: 3951: 3937: 3929: 3910: 3908: 3907: 3902: 3897: 3889: 3888: 3863:Clifford algebra 3833:and one for the 3813: 3811: 3810: 3805: 3800: 3799: 3787: 3785: 3784: 3772: 3767: 3765: 3764: 3763: 3750: 3746: 3745: 3735: 3733: 3731: 3730: 3718: 3710: 3709: 3692: 3690: 3689: 3684: 3679: 3674: 3673: 3661: 3660: 3648: 3646: 3645: 3644: 3631: 3630: 3625: 3624: 3614: 3612: 3610: 3609: 3597: 3592: 3587: 3586: 3547: 3545: 3544: 3539: 3530: 3529: 3503: 3498: 3497: 3479: 3436: 3434: 3433: 3428: 3426: 3424: 3423: 3411: 3403: 3401: 3400: 3399: 3386: 3382: 3381: 3371: 3369: 3367: 3366: 3354: 3346: 3345: 3330: 3328: 3327: 3322: 3320: 3315: 3314: 3299: 3297: 3296: 3295: 3282: 3281: 3276: 3275: 3265: 3263: 3261: 3260: 3248: 3243: 3238: 3237: 3219: 3157: 3155: 3154: 3149: 3133: 3131: 3130: 3125: 3123: 3122: 3104: 3102: 3101: 3096: 3094: 3089: 3088: 3073: 3069: 3063: 3062: 3047: 3045: 3044: 3043: 3030: 3029: 3025: 3019: 3018: 3008: 3006: 3005: 2996: 2995: 2983: 2979: 2973: 2972: 2957: 2955: 2954: 2949: 2947: 2945: 2944: 2932: 2924: 2916: 2915: 2900: 2898: 2897: 2896: 2883: 2882: 2874: 2873: 2863: 2861: 2860: 2851: 2850: 2838: 2830: 2829: 2811: 2809: 2808: 2803: 2798: 2796: 2788: 2780: 2778: 2777: 2768: 2767: 2755: 2747: 2736: 2734: 2733: 2732: 2719: 2715: 2714: 2704: 2702: 2701: 2692: 2691: 2676: 2675: 2655: 2653: 2652: 2647: 2642: 2640: 2632: 2631: 2619: 2617: 2616: 2607: 2606: 2591: 2587: 2578: 2515: 2513: 2512: 2507: 2505: 2501: 2499: 2497: 2489: 2488: 2476: 2463: 2462: 2453: 2452: 2440: 2435: 2434: 2419: 2417: 2416: 2415: 2402: 2401: 2397: 2390: 2389: 2379: 2377: 2376: 2367: 2366: 2354: 2350: 2344: 2343: 2328: 2326: 2325: 2320: 2318: 2316: 2315: 2303: 2295: 2287: 2286: 2268: 2266: 2265: 2260: 2255: 2247: 2233: 2232: 2212: 2210: 2209: 2204: 2196: 2192: 2183: 2151: 2149: 2148: 2143: 2138: 2130: 2122: 2118: 2106: 2104: 2103: 2098: 2096: 2094: 2086: 2078: 2067: 2052: 2040: 2007: 2005: 2004: 1999: 1997: 1992: 1991: 1976: 1972: 1971: 1969: 1961: 1953: 1951: 1949: 1948: 1936: 1931: 1923: 1913: 1905: 1901: 1900: 1898: 1897: 1896: 1883: 1882: 1877: 1876: 1866: 1864: 1862: 1861: 1849: 1844: 1839: 1838: 1816: 1814: 1813: 1808: 1806: 1804: 1803: 1791: 1783: 1779: 1778: 1770: 1760: 1758: 1747: 1739: 1738: 1711: 1709: 1708: 1703: 1682: 1680: 1679: 1674: 1672: 1633: 1631: 1630: 1625: 1623: 1615: 1607: 1596: 1594: 1593: 1588: 1586: 1584: 1576: 1575: 1566: 1558: 1547: 1527: 1525: 1524: 1519: 1494:nonlinear optics 1418: 1416: 1415: 1410: 1408: 1406: 1398: 1397: 1388: 1386: 1385: 1376: 1375: 1363: 1358: 1357: 1345: 1317: 1315: 1314: 1309: 1307: 1305: 1297: 1296: 1287: 1279: 1251: 1249: 1248: 1243: 1235: 1207: 1205: 1204: 1199: 1197: 1195: 1194: 1182: 1177: 1144: 1143: 1093: 1070: 1016:electromagnetism 1000: 993: 986: 667:Electric machine 650:Magnetic circuit 612:Parallel circuit 602:Network analysis 567:Electric current 502:London equations 347:Triboelectricity 337:Potential energy 206: 196:Electromagnetism 187: 186: 180: 173: 162: 159: 153: 130: 122: 115: 108: 104: 101: 95: 90:this article by 81:inline citations 68: 67: 60: 49: 27: 26: 19: 11656: 11655: 11651: 11650: 11649: 11647: 11646: 11645: 11626: 11625: 11619: 11600: 11570: 11532: 11524: 11519: 11518: 11510: 11506: 11475: 11471: 11462: 11460: 11453: 11447: 11443: 11436: 11411: 11407: 11394:Harley Flanders 11392: 11388: 11376: 11372: 11365: 11361: 11356: 11352: 11347: 11285: 11260: 11252: 11244: 11236: 11221: 11194: 11177: 11114:expressing the 11105: 11038: 11030:Main articles: 11028: 10992: 10984: 10982: 10974: 10970: 10965: 10957: 10949: 10947: 10944: 10943: 10921: 10910: 10874: 10862: 10843: 10835: 10823: 10822: 10808: 10800: 10798: 10781: 10779: 10776: 10775: 10756: 10674: 10666: 10642: 10609: 10602: 10590: 10557: 10529: 10525: 10520: 10511: 10507: 10502: 10493: 10489: 10484: 10475: 10471: 10466: 10451: 10447: 10436: 10427: 10420: 10416: 10415: 10414: 10401: 10399: 10391: 10386: 10384: 10381: 10380: 10359: 10350: 10346: 10341: 10332: 10328: 10323: 10314: 10310: 10305: 10290: 10286: 10275: 10266: 10256: 10252: 10251: 10250: 10240: 10231: 10227: 10222: 10220: 10217: 10216: 10188: 10184: 10179: 10170: 10166: 10161: 10152: 10148: 10143: 10131: 10127: 10121: 10117: 10105: 10101: 10095: 10091: 10079: 10075: 10069: 10065: 10051: 10046: 10044: 10041: 10040: 10022: 9972: 9968: 9963: 9954: 9950: 9945: 9936: 9932: 9927: 9912: 9908: 9897: 9891: 9887: 9877: 9876: 9872: 9859: 9857: 9849: 9847: 9844: 9843: 9818: 9814: 9809: 9800: 9796: 9791: 9781: 9777: 9767: 9759: 9757: 9754: 9753: 9748: 9717: 9710: 9682: 9675: 9671: 9670: 9669: 9667: 9664: 9663: 9637: 9633: 9627: 9623: 9611: 9607: 9601: 9597: 9585: 9581: 9575: 9571: 9559: 9555: 9549: 9545: 9533: 9529: 9523: 9519: 9507: 9503: 9497: 9493: 9485: 9482: 9481: 9459: 9449: 9445: 9444: 9443: 9425: 9424: 9419: 9417: 9416: 9404: 9400: 9394: 9390: 9372: 9371: 9366: 9364: 9363: 9351: 9347: 9338: 9331: 9327: 9326: 9325: 9313: 9309: 9300: 9293: 9289: 9288: 9287: 9275: 9271: 9265: 9261: 9252: 9248: 9235: 9233: 9231: 9228: 9227: 9208: 9203: 9195: 9171: 9160: 9149: 9138: 9121: 9119: 9116: 9115: 9087: 9077: 9073: 9067: 9063: 9051: 9047: 9041: 9037: 9025: 9021: 9015: 9011: 8999: 8995: 8983: 8978: 8970: 8969: 8967: 8964: 8963: 8939: 8933: 8929: 8918: 8912: 8908: 8897: 8891: 8887: 8876: 8860: 8855: 8854: 8852: 8849: 8848: 8829: 8818: 8807: 8801: 8797: 8786: 8775: 8764: 8758: 8754: 8743: 8732: 8721: 8715: 8711: 8700: 8689: 8678: 8663: 8661: 8658: 8657: 8629: 8618: 8612: 8608: 8597: 8586: 8580: 8576: 8565: 8554: 8548: 8544: 8533: 8522: 8516: 8512: 8501: 8490: 8484: 8480: 8469: 8458: 8452: 8448: 8436: 8431: 8430: 8428: 8425: 8424: 8410: 8409: 8401: 8390: 8384: 8380: 8369: 8358: 8352: 8348: 8337: 8326: 8320: 8316: 8305: 8294: 8288: 8284: 8273: 8262: 8256: 8252: 8241: 8230: 8224: 8220: 8218: 8212: 8211: 8205: 8201: 8196: 8187: 8183: 8178: 8169: 8165: 8155: 8153: 8145: 8141: 8139: 8136: 8135: 8111: 8105: 8101: 8090: 8084: 8080: 8069: 8063: 8059: 8048: 8036: 8034: 8031: 8030: 8026: 8016: 7985: 7981: 7979: 7976: 7975: 7945: 7941: 7939: 7936: 7935: 7912: 7908: 7896: 7892: 7868: 7864: 7859: 7856: 7855: 7826: 7822: 7818: 7813: 7801: 7797: 7793: 7788: 7787: 7783: 7779: 7764: 7760: 7755: 7752: 7751: 7727: 7722: 7709: 7705: 7703: 7700: 7699: 7679: 7675: 7657: 7653: 7645: 7642: 7641: 7621: 7617: 7609: 7606: 7605: 7580: 7576: 7572: 7567: 7549: 7545: 7541: 7536: 7535: 7531: 7529: 7526: 7525: 7500: 7485: 7481: 7472: 7468: 7459: 7455: 7445: 7433: 7425: 7419: 7416: 7415: 7405:anticommutative 7384: 7380: 7371: 7363: 7347: 7343: 7341: 7338: 7337: 7317: 7312: 7311: 7302: 7297: 7296: 7287: 7283: 7273: 7265: 7263: 7260: 7259: 7240: 7220: 7212: 7207: 7205: 7202: 7201: 7180: 7175: 7173: 7170: 7169: 7140: 7136: 7128: 7120: 7111: 7107: 7099: 7096: 7095: 7092: 7071: 7066: 7064: 7061: 7060: 7044: 7042: 7039: 7038: 7022: 7017: 7015: 7012: 7011: 6995: 6993: 6990: 6989: 6956: 6951: 6945: 6940: 6939: 6930: 6929: 6924: 6922: 6919: 6918: 6903: 6896: 6885:Minkowski space 6868: 6866: 6863: 6862: 6847: 6831: 6823: 6818: 6813: 6811: 6808: 6807: 6784: 6779: 6777: 6774: 6773: 6731: 6720: 6709: 6703: 6699: 6688: 6677: 6666: 6660: 6656: 6645: 6634: 6623: 6617: 6613: 6602: 6591: 6580: 6568: 6566: 6563: 6562: 6545: 6522: 6511: 6505: 6501: 6490: 6479: 6473: 6469: 6458: 6447: 6441: 6437: 6426: 6415: 6409: 6405: 6394: 6383: 6377: 6373: 6362: 6351: 6345: 6341: 6330: 6325: 6323: 6320: 6319: 6297: 6286: 6272: 6261: 6253: 6241: 6230: 6213: 6202: 6194: 6192: 6189: 6188: 6173: 6155: 6115: 6109: 6105: 6094: 6088: 6084: 6073: 6067: 6063: 6052: 6037: 6035: 6032: 6031: 6013: 6011: 6008: 6007: 5986: 5985: 5977: 5966: 5960: 5956: 5945: 5934: 5928: 5924: 5913: 5902: 5896: 5892: 5881: 5870: 5864: 5860: 5849: 5838: 5832: 5828: 5817: 5806: 5800: 5796: 5787: 5786: 5780: 5776: 5771: 5762: 5758: 5753: 5744: 5740: 5730: 5723: 5718: 5714: 5712: 5709: 5708: 5704: 5680: 5676: 5674: 5671: 5670: 5647: 5636: 5632: 5621: 5614: 5590: 5580: 5575: 5549: 5545: 5534: 5531: 5530: 5500: 5486: 5480: 5470: 5461: 5457: 5451: 5447: 5438: 5434: 5425: 5421: 5415: 5411: 5405: 5401: 5392: 5388: 5382: 5378: 5372: 5368: 5356: 5352: 5350: 5347: 5346: 5321: 5317: 5311: 5307: 5289: 5285: 5276: 5272: 5266: 5262: 5253: 5249: 5234: 5230: 5224: 5220: 5212: 5209: 5208: 5185: 5181: 5175: 5171: 5165: 5161: 5152: 5148: 5142: 5138: 5132: 5128: 5119: 5115: 5109: 5105: 5099: 5095: 5080: 5076: 5070: 5066: 5060: 5056: 5047: 5043: 5037: 5033: 5027: 5023: 5014: 5010: 5004: 5000: 4994: 4990: 4982: 4968: 4960: 4957: 4956: 4934: 4930: 4924: 4920: 4912: 4909: 4908: 4887: 4883: 4881: 4878: 4877: 4860: 4856: 4850: 4846: 4840: 4836: 4830: 4826: 4818: 4815: 4814: 4797: 4793: 4787: 4783: 4774: 4770: 4768: 4765: 4764: 4745: 4730: 4726: 4721: 4718: 4717: 4710: 4704: 4673: 4665: 4664: 4660: 4638: 4634: 4627: 4626: 4622: 4620: 4612: 4604: 4603: 4599: 4583: 4577: 4573: 4562: 4558: 4551: 4550: 4546: 4544: 4538: 4534: 4528: 4524: 4516: 4508: 4507: 4503: 4484: 4480: 4475: 4467: 4459: 4458: 4454: 4452: 4449: 4448: 4431: 4423: 4412: 4404: 4396: 4388: 4380: 4372: 4364: 4359: 4357: 4354: 4353: 4350: 4328: 4307: 4303: 4295: 4285: 4271: 4265: 4255: 4254: 4250: 4248: 4245: 4244: 4213: 4209: 4203: 4199: 4191: 4189: 4186: 4185: 4161: 4157: 4151: 4147: 4141: 4137: 4129: 4126: 4125: 4101: 4097: 4092: 4089: 4088: 4068: 4064: 4058: 4054: 4037: 4026: 4023: 4022: 4021:multivector is 3998: 3994: 3988: 3984: 3969: 3965: 3959: 3955: 3947: 3933: 3925: 3923: 3920: 3919: 3893: 3878: 3874: 3869: 3866: 3865: 3855: 3841:(GA) these are 3835:current density 3827: 3815: 3795: 3791: 3780: 3776: 3771: 3759: 3755: 3751: 3741: 3737: 3736: 3734: 3726: 3722: 3717: 3705: 3701: 3699: 3696: 3695: 3675: 3669: 3665: 3656: 3652: 3640: 3636: 3632: 3626: 3620: 3616: 3615: 3613: 3605: 3601: 3596: 3588: 3582: 3578: 3576: 3573: 3572: 3525: 3521: 3499: 3493: 3489: 3475: 3473: 3470: 3469: 3419: 3415: 3410: 3395: 3391: 3387: 3377: 3373: 3372: 3370: 3362: 3358: 3353: 3341: 3337: 3335: 3332: 3331: 3316: 3310: 3306: 3291: 3287: 3283: 3277: 3271: 3267: 3266: 3264: 3256: 3252: 3247: 3239: 3233: 3229: 3227: 3224: 3223: 3218: 3212: 3201: 3200:. Substituting 3198:field operators 3175: 3143: 3140: 3139: 3118: 3114: 3112: 3109: 3108: 3090: 3084: 3080: 3065: 3064: 3058: 3054: 3039: 3035: 3031: 3021: 3020: 3014: 3010: 3009: 3007: 3001: 2997: 2991: 2987: 2975: 2974: 2968: 2964: 2962: 2959: 2958: 2940: 2936: 2931: 2917: 2911: 2907: 2892: 2888: 2884: 2875: 2869: 2865: 2864: 2862: 2856: 2852: 2846: 2842: 2831: 2825: 2821: 2819: 2816: 2815: 2789: 2781: 2779: 2773: 2769: 2763: 2759: 2751: 2743: 2728: 2724: 2720: 2710: 2706: 2705: 2703: 2697: 2693: 2687: 2683: 2671: 2667: 2665: 2662: 2661: 2633: 2624: 2620: 2618: 2612: 2608: 2602: 2598: 2583: 2582: 2574: 2572: 2569: 2568: 2557: 2551: 2490: 2481: 2477: 2475: 2473: 2469: 2458: 2454: 2448: 2444: 2436: 2430: 2426: 2411: 2407: 2403: 2393: 2392: 2385: 2381: 2380: 2378: 2372: 2368: 2362: 2358: 2346: 2345: 2339: 2335: 2333: 2330: 2329: 2311: 2307: 2302: 2288: 2282: 2278: 2276: 2273: 2272: 2251: 2243: 2228: 2224: 2222: 2219: 2218: 2188: 2187: 2179: 2177: 2174: 2173: 2166: 2160: 2134: 2126: 2114: 2113: 2111: 2108: 2107: 2087: 2079: 2077: 2060: 2058: 2055: 2054: 2042: 2030: 2018: 2009: 1993: 1987: 1983: 1962: 1954: 1952: 1944: 1940: 1935: 1927: 1919: 1918: 1914: 1909: 1892: 1888: 1884: 1878: 1872: 1868: 1867: 1865: 1857: 1853: 1848: 1840: 1834: 1830: 1829: 1825: 1823: 1820: 1819: 1799: 1795: 1790: 1774: 1766: 1765: 1761: 1751: 1746: 1734: 1730: 1728: 1725: 1724: 1688: 1685: 1684: 1668: 1654: 1651: 1650: 1639: 1619: 1611: 1603: 1601: 1598: 1597: 1577: 1571: 1567: 1565: 1554: 1543: 1541: 1538: 1537: 1513: 1510: 1509: 1502: 1459: 1448: 1399: 1393: 1389: 1387: 1381: 1377: 1371: 1367: 1359: 1353: 1349: 1341: 1333: 1330: 1329: 1298: 1292: 1288: 1286: 1275: 1267: 1264: 1263: 1231: 1223: 1220: 1219: 1190: 1186: 1181: 1173: 1165: 1162: 1161: 1133:electrodynamics 1129: 1123: 1072: 1049: 1034: 1028: 1004: 975: 974: 790: 782: 781: 737: 727: 726: 682:Induction motor 652: 642: 641: 557:Current density 542: 532: 531: 522:Poynting vector 432: 430:Electrodynamics 422: 421: 417:Right-hand rule 382:Magnetic dipole 372:Biot–Savart law 362: 352: 351: 287:Electric dipole 282:Electric charge 257: 181: 170: 169: 168: 163: 157: 154: 147: 135:This article's 131: 116: 105: 99: 96: 86:Please help to 85: 69: 65: 28: 24: 17: 12: 11: 5: 11654: 11644: 11643: 11638: 11624: 11623: 11617: 11604: 11598: 11592:. Birkhäuser. 11583: 11568: 11555: 11523: 11520: 11517: 11516: 11504: 11491:(2): 129–164. 11469: 11441: 11434: 11405: 11402:Academic Press 11386: 11379:David Hestenes 11370: 11359: 11349: 11348: 11346: 11343: 11342: 11341: 11336: 11331: 11326: 11321: 11316: 11311: 11306: 11301: 11299:Speed of light 11296: 11291: 11289:Ricci calculus 11284: 11281: 11243:-vector and a 11220: 11217: 11104: 11101: 11027: 11024: 11011: 11007: 11004: 10998: 10995: 10990: 10987: 10977: 10973: 10969: 10964: 10960: 10956: 10952: 10942:), we impose 10850: 10846: 10842: 10838: 10834: 10830: 10826: 10820: 10814: 10811: 10806: 10803: 10797: 10794: 10791: 10787: 10784: 10706:Poincaré lemma 10680:involving the 10673: 10670: 10665: 10662: 10572:acts regularly 10556: 10553: 10552: 10551: 10540: 10537: 10532: 10528: 10523: 10519: 10514: 10510: 10505: 10501: 10496: 10492: 10487: 10483: 10478: 10474: 10469: 10463: 10460: 10457: 10454: 10450: 10443: 10440: 10433: 10430: 10423: 10419: 10411: 10407: 10404: 10398: 10394: 10389: 10377: 10366: 10362: 10358: 10353: 10349: 10344: 10340: 10335: 10331: 10326: 10322: 10317: 10313: 10308: 10302: 10299: 10296: 10293: 10289: 10282: 10279: 10272: 10269: 10262: 10259: 10255: 10247: 10244: 10239: 10234: 10230: 10225: 10213: 10202: 10199: 10196: 10191: 10187: 10182: 10178: 10173: 10169: 10164: 10160: 10155: 10151: 10146: 10142: 10137: 10134: 10130: 10124: 10120: 10116: 10111: 10108: 10104: 10098: 10094: 10090: 10085: 10082: 10078: 10072: 10068: 10064: 10061: 10058: 10054: 10049: 10018: 9997: 9996: 9984: 9980: 9975: 9971: 9966: 9962: 9957: 9953: 9948: 9944: 9939: 9935: 9930: 9924: 9921: 9918: 9915: 9911: 9904: 9901: 9894: 9890: 9884: 9881: 9875: 9869: 9865: 9862: 9856: 9852: 9837: 9826: 9821: 9817: 9812: 9808: 9803: 9799: 9794: 9787: 9784: 9780: 9774: 9771: 9766: 9762: 9744: 9716: 9713: 9706: 9688: 9685: 9678: 9674: 9648: 9643: 9640: 9636: 9630: 9626: 9622: 9617: 9614: 9610: 9604: 9600: 9596: 9591: 9588: 9584: 9578: 9574: 9570: 9565: 9562: 9558: 9552: 9548: 9544: 9539: 9536: 9532: 9526: 9522: 9518: 9513: 9510: 9506: 9500: 9496: 9492: 9489: 9465: 9462: 9455: 9452: 9448: 9434: 9431: 9428: 9422: 9410: 9407: 9403: 9397: 9393: 9381: 9378: 9375: 9369: 9357: 9354: 9350: 9344: 9341: 9334: 9330: 9324: 9319: 9316: 9312: 9306: 9303: 9296: 9292: 9286: 9281: 9278: 9274: 9268: 9264: 9260: 9255: 9251: 9245: 9241: 9238: 9207: 9204: 9199:Main article: 9194: 9191: 9178: 9174: 9170: 9167: 9163: 9159: 9156: 9152: 9148: 9145: 9141: 9137: 9134: 9131: 9128: 9124: 9100: 9097: 9094: 9090: 9085: 9080: 9076: 9070: 9066: 9062: 9059: 9054: 9050: 9044: 9040: 9036: 9033: 9028: 9024: 9018: 9014: 9010: 9007: 9002: 8998: 8994: 8991: 8986: 8981: 8977: 8973: 8949: 8946: 8942: 8936: 8932: 8928: 8925: 8921: 8915: 8911: 8907: 8904: 8900: 8894: 8890: 8886: 8883: 8879: 8874: 8871: 8868: 8863: 8858: 8836: 8832: 8828: 8825: 8821: 8817: 8814: 8810: 8804: 8800: 8796: 8793: 8789: 8785: 8782: 8778: 8774: 8771: 8767: 8761: 8757: 8753: 8750: 8746: 8742: 8739: 8735: 8731: 8728: 8724: 8718: 8714: 8710: 8707: 8703: 8699: 8696: 8692: 8688: 8685: 8681: 8676: 8673: 8670: 8666: 8639: 8636: 8632: 8628: 8625: 8621: 8615: 8611: 8607: 8604: 8600: 8596: 8593: 8589: 8583: 8579: 8575: 8572: 8568: 8564: 8561: 8557: 8551: 8547: 8543: 8540: 8536: 8532: 8529: 8525: 8519: 8515: 8511: 8508: 8504: 8500: 8497: 8493: 8487: 8483: 8479: 8476: 8472: 8468: 8465: 8461: 8455: 8451: 8447: 8444: 8439: 8434: 8408: 8404: 8400: 8397: 8393: 8387: 8383: 8379: 8376: 8372: 8368: 8365: 8361: 8355: 8351: 8347: 8344: 8340: 8336: 8333: 8329: 8323: 8319: 8315: 8312: 8308: 8304: 8301: 8297: 8291: 8287: 8283: 8280: 8276: 8272: 8269: 8265: 8259: 8255: 8251: 8248: 8244: 8240: 8237: 8233: 8227: 8223: 8219: 8217: 8214: 8213: 8208: 8204: 8199: 8195: 8190: 8186: 8181: 8175: 8172: 8168: 8162: 8159: 8154: 8152: 8148: 8144: 8143: 8121: 8118: 8114: 8108: 8104: 8100: 8097: 8093: 8087: 8083: 8079: 8076: 8072: 8066: 8062: 8058: 8055: 8051: 8046: 8043: 8039: 8015: 8012: 7996: 7993: 7988: 7984: 7957: 7954: 7951: 7948: 7944: 7923: 7920: 7915: 7911: 7907: 7904: 7899: 7895: 7891: 7888: 7885: 7882: 7879: 7874: 7871: 7867: 7863: 7842: 7838: 7829: 7825: 7821: 7817: 7812: 7804: 7800: 7796: 7792: 7786: 7782: 7778: 7775: 7770: 7767: 7763: 7759: 7750:metric matrix 7735: 7730: 7725: 7721: 7717: 7712: 7708: 7687: 7682: 7678: 7674: 7671: 7668: 7665: 7660: 7656: 7652: 7649: 7629: 7624: 7620: 7616: 7613: 7592: 7583: 7579: 7575: 7571: 7566: 7563: 7560: 7552: 7548: 7544: 7540: 7534: 7507: 7504: 7497: 7494: 7491: 7488: 7484: 7478: 7475: 7471: 7465: 7462: 7458: 7452: 7449: 7444: 7439: 7436: 7431: 7428: 7424: 7390: 7387: 7383: 7377: 7374: 7369: 7366: 7362: 7358: 7353: 7350: 7346: 7325: 7320: 7315: 7310: 7305: 7300: 7293: 7290: 7286: 7280: 7277: 7272: 7268: 7223: 7219: 7215: 7210: 7190: 7187: 7183: 7178: 7155: 7152: 7149: 7146: 7143: 7139: 7135: 7131: 7127: 7123: 7119: 7114: 7110: 7106: 7103: 7091: 7088: 7074: 7069: 7047: 7025: 7020: 6998: 6966: 6963: 6959: 6954: 6948: 6943: 6938: 6933: 6927: 6901: 6871: 6834: 6830: 6826: 6821: 6816: 6794: 6791: 6787: 6782: 6768: 6741: 6738: 6734: 6730: 6727: 6723: 6719: 6716: 6712: 6706: 6702: 6698: 6695: 6691: 6687: 6684: 6680: 6676: 6673: 6669: 6663: 6659: 6655: 6652: 6648: 6644: 6641: 6637: 6633: 6630: 6626: 6620: 6616: 6612: 6609: 6605: 6601: 6598: 6594: 6590: 6587: 6583: 6578: 6575: 6571: 6558:current 3-form 6551:is called the 6544: 6541: 6529: 6525: 6521: 6518: 6514: 6508: 6504: 6500: 6497: 6493: 6489: 6486: 6482: 6476: 6472: 6468: 6465: 6461: 6457: 6454: 6450: 6444: 6440: 6436: 6433: 6429: 6425: 6422: 6418: 6412: 6408: 6404: 6401: 6397: 6393: 6390: 6386: 6380: 6376: 6372: 6369: 6365: 6361: 6358: 6354: 6348: 6344: 6340: 6337: 6333: 6328: 6307: 6304: 6300: 6296: 6293: 6289: 6285: 6282: 6279: 6275: 6271: 6268: 6264: 6260: 6256: 6251: 6248: 6244: 6240: 6237: 6233: 6229: 6226: 6223: 6220: 6216: 6212: 6209: 6205: 6201: 6197: 6164:)-form, where 6125: 6122: 6118: 6112: 6108: 6104: 6101: 6097: 6091: 6087: 6083: 6080: 6076: 6070: 6066: 6062: 6059: 6055: 6050: 6047: 6044: 6040: 6020: 6016: 5984: 5980: 5976: 5973: 5969: 5963: 5959: 5955: 5952: 5948: 5944: 5941: 5937: 5931: 5927: 5923: 5920: 5916: 5912: 5909: 5905: 5899: 5895: 5891: 5888: 5884: 5880: 5877: 5873: 5867: 5863: 5859: 5856: 5852: 5848: 5845: 5841: 5835: 5831: 5827: 5824: 5820: 5816: 5813: 5809: 5803: 5799: 5795: 5792: 5790: 5788: 5783: 5779: 5774: 5770: 5765: 5761: 5756: 5750: 5747: 5743: 5737: 5734: 5729: 5726: 5724: 5721: 5717: 5716: 5686: 5683: 5679: 5645: 5630: 5613: 5610: 5579: 5576: 5563: 5560: 5557: 5552: 5548: 5544: 5541: 5538: 5522: 5507: 5503: 5499: 5492: 5489: 5485: 5477: 5474: 5469: 5464: 5460: 5454: 5450: 5446: 5441: 5437: 5433: 5428: 5424: 5418: 5414: 5408: 5404: 5400: 5395: 5391: 5385: 5381: 5375: 5371: 5367: 5364: 5359: 5355: 5332: 5329: 5324: 5320: 5314: 5310: 5306: 5303: 5300: 5297: 5292: 5288: 5284: 5279: 5275: 5269: 5265: 5261: 5256: 5252: 5248: 5245: 5242: 5237: 5233: 5227: 5223: 5219: 5216: 5196: 5193: 5188: 5184: 5178: 5174: 5168: 5164: 5160: 5155: 5151: 5145: 5141: 5135: 5131: 5127: 5122: 5118: 5112: 5108: 5102: 5098: 5094: 5091: 5088: 5083: 5079: 5073: 5069: 5063: 5059: 5055: 5050: 5046: 5040: 5036: 5030: 5026: 5022: 5017: 5013: 5007: 5003: 4997: 4993: 4989: 4985: 4981: 4978: 4975: 4971: 4967: 4964: 4942: 4937: 4933: 4927: 4923: 4919: 4916: 4905:gamma matrices 4890: 4886: 4863: 4859: 4853: 4849: 4843: 4839: 4833: 4829: 4825: 4822: 4800: 4796: 4790: 4786: 4782: 4777: 4773: 4752: 4748: 4744: 4739: 4736: 4733: 4729: 4725: 4703: 4700: 4688: 4685: 4681: 4676: 4672: 4668: 4663: 4659: 4656: 4653: 4649: 4641: 4637: 4630: 4625: 4619: 4615: 4611: 4607: 4602: 4598: 4595: 4591: 4586: 4580: 4576: 4572: 4565: 4561: 4554: 4549: 4541: 4537: 4531: 4527: 4523: 4519: 4515: 4511: 4506: 4502: 4499: 4495: 4487: 4483: 4479: 4474: 4470: 4466: 4462: 4457: 4434: 4430: 4426: 4422: 4419: 4415: 4411: 4407: 4403: 4399: 4395: 4391: 4387: 4383: 4379: 4375: 4371: 4367: 4362: 4338: 4335: 4331: 4327: 4324: 4321: 4318: 4315: 4310: 4306: 4302: 4298: 4293: 4288: 4284: 4277: 4274: 4270: 4262: 4259: 4253: 4236: 4221: 4216: 4212: 4206: 4202: 4198: 4194: 4179:Pauli matrices 4164: 4160: 4154: 4150: 4144: 4140: 4136: 4133: 4109: 4104: 4100: 4096: 4071: 4067: 4061: 4057: 4053: 4050: 4047: 4044: 4040: 4036: 4033: 4030: 4006: 4001: 3997: 3991: 3987: 3983: 3980: 3977: 3972: 3968: 3962: 3958: 3954: 3950: 3946: 3943: 3940: 3936: 3932: 3928: 3900: 3896: 3892: 3887: 3884: 3881: 3877: 3873: 3854: 3851: 3847:Ricci calculus 3826: 3823: 3803: 3798: 3794: 3790: 3783: 3779: 3775: 3770: 3762: 3758: 3754: 3749: 3744: 3740: 3729: 3725: 3721: 3716: 3713: 3708: 3704: 3682: 3678: 3672: 3668: 3664: 3659: 3655: 3651: 3643: 3639: 3635: 3629: 3623: 3619: 3608: 3604: 3600: 3595: 3591: 3585: 3581: 3560: 3556:Dirac matrices 3537: 3533: 3528: 3524: 3520: 3517: 3514: 3511: 3506: 3502: 3496: 3492: 3488: 3485: 3482: 3478: 3458:Dirac electron 3422: 3418: 3414: 3409: 3406: 3398: 3394: 3390: 3385: 3380: 3376: 3365: 3361: 3357: 3352: 3349: 3344: 3340: 3319: 3313: 3309: 3305: 3302: 3294: 3290: 3286: 3280: 3274: 3270: 3259: 3255: 3251: 3246: 3242: 3236: 3232: 3216: 3210: 3174: 3171: 3147: 3134:is called the 3121: 3117: 3093: 3087: 3083: 3079: 3076: 3072: 3068: 3061: 3057: 3053: 3050: 3042: 3038: 3034: 3028: 3024: 3017: 3013: 3004: 3000: 2994: 2990: 2986: 2982: 2978: 2971: 2967: 2943: 2939: 2935: 2930: 2927: 2923: 2920: 2914: 2910: 2906: 2903: 2895: 2891: 2887: 2881: 2878: 2872: 2868: 2859: 2855: 2849: 2845: 2841: 2837: 2834: 2828: 2824: 2801: 2795: 2792: 2787: 2784: 2776: 2772: 2766: 2762: 2758: 2754: 2750: 2746: 2742: 2739: 2731: 2727: 2723: 2718: 2713: 2709: 2700: 2696: 2690: 2686: 2682: 2679: 2674: 2670: 2645: 2639: 2636: 2630: 2627: 2623: 2615: 2611: 2605: 2601: 2597: 2594: 2590: 2586: 2581: 2577: 2553:Main article: 2550: 2547: 2504: 2496: 2493: 2487: 2484: 2480: 2472: 2466: 2461: 2457: 2451: 2447: 2443: 2439: 2433: 2429: 2425: 2422: 2414: 2410: 2406: 2400: 2396: 2388: 2384: 2375: 2371: 2365: 2361: 2357: 2353: 2349: 2342: 2338: 2314: 2310: 2306: 2301: 2298: 2294: 2291: 2285: 2281: 2258: 2254: 2250: 2246: 2242: 2239: 2236: 2231: 2227: 2202: 2199: 2195: 2191: 2186: 2182: 2162:Main article: 2159: 2156: 2141: 2137: 2133: 2129: 2125: 2121: 2117: 2093: 2090: 2085: 2082: 2076: 2073: 2070: 2066: 2063: 2017: 2014: 1996: 1990: 1986: 1982: 1979: 1975: 1968: 1965: 1960: 1957: 1947: 1943: 1939: 1934: 1930: 1926: 1922: 1917: 1912: 1908: 1904: 1895: 1891: 1887: 1881: 1875: 1871: 1860: 1856: 1852: 1847: 1843: 1837: 1833: 1828: 1802: 1798: 1794: 1789: 1786: 1782: 1777: 1773: 1769: 1764: 1757: 1754: 1750: 1745: 1742: 1737: 1733: 1714: 1701: 1698: 1695: 1692: 1671: 1667: 1664: 1661: 1658: 1638: 1635: 1622: 1618: 1614: 1610: 1606: 1583: 1580: 1574: 1570: 1564: 1561: 1557: 1553: 1550: 1546: 1517: 1501: 1498: 1457: 1446: 1440:charge density 1432: 1431: 1428: 1427: 1420: 1405: 1402: 1396: 1392: 1384: 1380: 1374: 1370: 1366: 1362: 1356: 1352: 1348: 1344: 1340: 1337: 1326: 1325: 1318: 1304: 1301: 1295: 1291: 1285: 1282: 1278: 1274: 1271: 1260: 1259: 1252: 1241: 1238: 1234: 1230: 1227: 1216: 1215: 1208: 1193: 1189: 1185: 1180: 1176: 1172: 1169: 1158: 1157: 1125:Main article: 1122: 1119: 1046:magnetic field 1042:electric field 1030:Main article: 1027: 1024: 1006: 1005: 1003: 1002: 995: 988: 980: 977: 976: 973: 972: 967: 962: 957: 952: 947: 942: 937: 932: 927: 922: 917: 912: 907: 902: 897: 892: 887: 882: 877: 872: 867: 862: 857: 852: 847: 842: 837: 832: 827: 822: 817: 812: 807: 802: 797: 791: 788: 787: 784: 783: 780: 779: 774: 769: 764: 759: 757:Four-potential 754: 749: 744: 738: 733: 732: 729: 728: 725: 724: 719: 714: 709: 704: 699: 694: 689: 684: 679: 674: 672:Electric motor 669: 664: 659: 653: 648: 647: 644: 643: 640: 639: 634: 629: 627:Series circuit 624: 619: 614: 609: 604: 599: 597:Kirchhoff laws 594: 589: 584: 579: 574: 569: 564: 562:Direct current 559: 554: 549: 543: 538: 537: 534: 533: 530: 529: 524: 519: 517:Maxwell tensor 514: 509: 504: 499: 494: 489: 487:Larmor formula 484: 479: 474: 469: 464: 459: 454: 449: 444: 439: 437:Bremsstrahlung 433: 428: 427: 424: 423: 420: 419: 414: 409: 404: 399: 394: 389: 387:Magnetic field 384: 379: 374: 369: 363: 360:Magnetostatics 358: 357: 354: 353: 350: 349: 344: 339: 334: 329: 324: 319: 314: 309: 304: 299: 294: 292:Electric field 289: 284: 279: 274: 269: 264: 262:Charge density 258: 255:Electrostatics 253: 252: 249: 248: 247: 246: 241: 236: 231: 226: 221: 216: 208: 207: 199: 198: 192: 191: 190:Articles about 183: 182: 165: 164: 144:the key points 134: 132: 125: 118: 117: 72: 70: 63: 58: 32: 31: 29: 22: 15: 9: 6: 4: 3: 2: 11653: 11642: 11639: 11637: 11634: 11633: 11631: 11620: 11614: 11610: 11605: 11601: 11595: 11591: 11590: 11584: 11581: 11580:1-59693-096-9 11577: 11571: 11565: 11561: 11556: 11551: 11546: 11542: 11538: 11531: 11526: 11525: 11514: 11508: 11499: 11494: 11490: 11486: 11485: 11480: 11473: 11459: 11452: 11445: 11437: 11431: 11427: 11423: 11419: 11415: 11409: 11403: 11399: 11395: 11390: 11383: 11380: 11374: 11368: 11363: 11354: 11350: 11340: 11337: 11335: 11332: 11330: 11327: 11325: 11322: 11320: 11317: 11315: 11312: 11310: 11307: 11305: 11302: 11300: 11297: 11295: 11292: 11290: 11287: 11286: 11280: 11278: 11274: 11270: 11264: 11256: 11251:-vector. The 11248: 11240: 11234: 11230: 11226: 11216: 11214: 11210: 11205: 11201: 11197: 11193: 11189: 11184: 11180: 11176: 11172: 11169: 11163: 11161: 11157: 11153: 11150: 11146: 11142: 11139:principle of 11138: 11134: 11133: 11127: 11124: 11120: 11117: 11113: 11110: 11100: 11098: 11095: 11091: 11088: 11084: 11081: 11077: 11073: 11069: 11064: 11062: 11058: 11054: 11050: 11045: 11043: 11037: 11033: 11023: 11009: 11005: 11002: 10996: 10988: 10975: 10971: 10967: 10962: 10954: 10941: 10940:Ludvig Lorenz 10937: 10934:term. In the 10932: 10928: 10924: 10917: 10913: 10908: 10907:Coulomb gauge 10903: 10899: 10894: 10892: 10891:gauge freedom 10888: 10882: 10878: 10870: 10866: 10848: 10840: 10832: 10828: 10818: 10812: 10804: 10795: 10792: 10789: 10785: 10782: 10773: 10767: 10763: 10759: 10754: 10750: 10746: 10742: 10738: 10733: 10731: 10727: 10723: 10719: 10715: 10712:simple, e.g. 10711: 10710:topologically 10707: 10703: 10699: 10695: 10691: 10687: 10684:(also called 10683: 10679: 10669: 10661: 10658: 10654: 10651:In mentioned 10649: 10645: 10640: 10635: 10633: 10629: 10625: 10621: 10616: 10612: 10606: 10601:we can write 10600: 10594: 10588: 10585: 10581: 10577: 10573: 10570: 10566: 10562: 10538: 10535: 10530: 10526: 10517: 10512: 10508: 10499: 10494: 10490: 10481: 10476: 10472: 10461: 10458: 10455: 10452: 10448: 10441: 10438: 10431: 10428: 10421: 10417: 10409: 10405: 10402: 10396: 10378: 10364: 10356: 10351: 10347: 10338: 10333: 10329: 10320: 10315: 10311: 10300: 10297: 10294: 10291: 10287: 10280: 10277: 10270: 10267: 10260: 10257: 10253: 10245: 10242: 10237: 10228: 10214: 10200: 10197: 10194: 10189: 10185: 10176: 10171: 10167: 10158: 10153: 10149: 10135: 10132: 10128: 10122: 10114: 10109: 10106: 10102: 10096: 10088: 10083: 10080: 10076: 10070: 10059: 10056: 10038: 10037: 10036: 10034: 10030: 10026: 10021: 10017: 10013: 10012:metric tensor 10009: 10005: 10000: 9982: 9978: 9973: 9969: 9960: 9955: 9951: 9942: 9937: 9933: 9922: 9919: 9916: 9913: 9909: 9902: 9899: 9892: 9888: 9882: 9879: 9873: 9867: 9863: 9860: 9854: 9842: 9838: 9824: 9819: 9815: 9806: 9801: 9797: 9785: 9782: 9778: 9772: 9769: 9764: 9752: 9747: 9743: 9739: 9738: 9737: 9735: 9731: 9727: 9722: 9712: 9709: 9704: 9686: 9683: 9676: 9660: 9646: 9641: 9638: 9634: 9628: 9620: 9615: 9612: 9608: 9602: 9594: 9589: 9586: 9582: 9576: 9568: 9563: 9560: 9556: 9550: 9542: 9537: 9534: 9530: 9524: 9516: 9511: 9508: 9504: 9498: 9490: 9487: 9463: 9460: 9453: 9450: 9446: 9420: 9408: 9405: 9401: 9395: 9367: 9355: 9352: 9348: 9342: 9339: 9332: 9322: 9317: 9314: 9310: 9304: 9301: 9294: 9284: 9279: 9276: 9272: 9266: 9258: 9253: 9249: 9243: 9239: 9236: 9225: 9221: 9217: 9213: 9202: 9197: 9190: 9176: 9168: 9165: 9157: 9154: 9146: 9143: 9135: 9129: 9122: 9114: 9095: 9088: 9078: 9068: 9064: 9057: 9052: 9042: 9038: 9031: 9026: 9016: 9012: 9005: 9000: 8996: 8989: 8979: 8975: 8960: 8947: 8944: 8934: 8930: 8926: 8923: 8913: 8909: 8905: 8902: 8892: 8888: 8884: 8881: 8872: 8869: 8866: 8856: 8834: 8826: 8823: 8815: 8812: 8802: 8798: 8794: 8791: 8783: 8780: 8772: 8769: 8759: 8755: 8751: 8748: 8740: 8737: 8729: 8726: 8716: 8712: 8708: 8705: 8697: 8694: 8686: 8683: 8674: 8671: 8668: 8655: 8650: 8637: 8634: 8626: 8623: 8613: 8609: 8605: 8602: 8594: 8591: 8581: 8577: 8573: 8570: 8562: 8559: 8549: 8545: 8541: 8538: 8530: 8527: 8517: 8513: 8509: 8506: 8498: 8495: 8485: 8481: 8477: 8474: 8466: 8463: 8453: 8449: 8445: 8442: 8432: 8406: 8398: 8395: 8385: 8381: 8377: 8374: 8366: 8363: 8353: 8349: 8345: 8342: 8334: 8331: 8321: 8317: 8313: 8310: 8302: 8299: 8289: 8285: 8281: 8278: 8270: 8267: 8257: 8253: 8249: 8246: 8238: 8235: 8225: 8221: 8215: 8206: 8202: 8193: 8188: 8184: 8173: 8170: 8166: 8160: 8157: 8150: 8132: 8119: 8116: 8106: 8102: 8098: 8095: 8085: 8081: 8077: 8074: 8064: 8060: 8056: 8053: 8044: 8041: 8025: 8021: 8011: 8008: 7994: 7991: 7986: 7982: 7973: 7955: 7952: 7949: 7946: 7942: 7913: 7909: 7905: 7902: 7897: 7893: 7889: 7880: 7872: 7869: 7865: 7840: 7836: 7827: 7823: 7810: 7802: 7798: 7784: 7780: 7776: 7768: 7765: 7761: 7749: 7746:, having the 7733: 7728: 7723: 7719: 7715: 7710: 7706: 7680: 7676: 7672: 7669: 7666: 7663: 7658: 7654: 7650: 7627: 7622: 7618: 7614: 7611: 7590: 7581: 7577: 7564: 7561: 7558: 7550: 7546: 7532: 7523: 7505: 7502: 7495: 7492: 7489: 7486: 7482: 7476: 7473: 7469: 7463: 7460: 7456: 7450: 7447: 7442: 7437: 7434: 7429: 7426: 7422: 7414: 7410: 7406: 7388: 7385: 7381: 7375: 7372: 7367: 7364: 7360: 7356: 7351: 7348: 7344: 7323: 7318: 7313: 7308: 7303: 7298: 7291: 7288: 7284: 7278: 7275: 7270: 7257: 7253: 7248: 7244: 7238: 7217: 7188: 7185: 7150: 7147: 7144: 7133: 7117: 7112: 7104: 7101: 7087: 7067: 7018: 6987: 6983: 6981: 6964: 6961: 6952: 6946: 6936: 6916: 6912: 6907: 6900: 6894: 6893:natural units 6890: 6886: 6869: 6860: 6856: 6852: 6846: 6828: 6819: 6805: 6792: 6789: 6771: 6767: 6765: 6761: 6757: 6752: 6739: 6736: 6728: 6725: 6717: 6714: 6704: 6700: 6696: 6693: 6685: 6682: 6674: 6671: 6661: 6657: 6653: 6650: 6642: 6639: 6631: 6628: 6618: 6614: 6610: 6607: 6599: 6596: 6588: 6585: 6576: 6573: 6560: 6559: 6554: 6550: 6540: 6527: 6519: 6516: 6506: 6502: 6498: 6495: 6487: 6484: 6474: 6470: 6466: 6463: 6455: 6452: 6442: 6438: 6434: 6431: 6423: 6420: 6410: 6406: 6402: 6399: 6391: 6388: 6378: 6374: 6370: 6367: 6359: 6356: 6346: 6342: 6338: 6335: 6326: 6305: 6302: 6294: 6291: 6283: 6277: 6269: 6266: 6254: 6249: 6246: 6238: 6235: 6227: 6224: 6218: 6210: 6207: 6195: 6186: 6180: 6176: 6171: 6167: 6162: 6158: 6153: 6149: 6145: 6141: 6136: 6123: 6120: 6110: 6106: 6102: 6099: 6089: 6085: 6081: 6078: 6068: 6064: 6060: 6057: 6048: 6045: 6042: 6018: 6006: 6002: 5999:which is the 5982: 5974: 5971: 5961: 5957: 5953: 5950: 5942: 5939: 5929: 5925: 5921: 5918: 5910: 5907: 5897: 5893: 5889: 5886: 5878: 5875: 5865: 5861: 5857: 5854: 5846: 5843: 5833: 5829: 5825: 5822: 5814: 5811: 5801: 5797: 5793: 5791: 5781: 5777: 5768: 5763: 5759: 5748: 5745: 5741: 5735: 5732: 5727: 5725: 5702: 5684: 5681: 5677: 5668: 5664: 5661: 5657: 5653: 5648: 5644: 5639: 5633: 5629: 5624: 5619: 5609: 5607: 5603: 5599: 5595: 5589: 5585: 5574: 5561: 5558: 5555: 5550: 5546: 5542: 5539: 5528: 5525: 5521: 5518: 5505: 5497: 5490: 5475: 5472: 5467: 5462: 5452: 5448: 5444: 5439: 5431: 5426: 5416: 5412: 5406: 5402: 5398: 5393: 5383: 5379: 5373: 5369: 5365: 5357: 5353: 5343: 5330: 5322: 5318: 5312: 5308: 5304: 5301: 5298: 5290: 5286: 5282: 5277: 5273: 5267: 5263: 5259: 5254: 5250: 5246: 5243: 5240: 5235: 5231: 5225: 5221: 5217: 5214: 5194: 5186: 5182: 5176: 5172: 5166: 5162: 5158: 5153: 5149: 5143: 5139: 5133: 5129: 5125: 5120: 5116: 5110: 5106: 5100: 5096: 5089: 5086: 5081: 5077: 5071: 5067: 5061: 5057: 5053: 5048: 5044: 5038: 5034: 5028: 5024: 5020: 5015: 5011: 5005: 5001: 4995: 4991: 4987: 4979: 4976: 4973: 4965: 4962: 4953: 4940: 4935: 4925: 4921: 4917: 4906: 4888: 4884: 4861: 4857: 4851: 4847: 4841: 4837: 4831: 4827: 4823: 4820: 4798: 4794: 4788: 4784: 4780: 4775: 4771: 4737: 4734: 4731: 4727: 4723: 4715: 4709: 4699: 4686: 4683: 4679: 4670: 4661: 4657: 4654: 4651: 4647: 4639: 4617: 4609: 4600: 4596: 4593: 4589: 4578: 4574: 4570: 4563: 4539: 4535: 4529: 4525: 4521: 4513: 4504: 4500: 4497: 4493: 4485: 4481: 4477: 4472: 4464: 4455: 4428: 4420: 4417: 4409: 4401: 4393: 4385: 4377: 4369: 4349: 4336: 4325: 4322: 4319: 4313: 4308: 4304: 4300: 4291: 4282: 4275: 4260: 4257: 4251: 4242: 4239: 4235: 4232: 4219: 4214: 4204: 4200: 4196: 4182: 4180: 4162: 4158: 4152: 4148: 4142: 4138: 4134: 4131: 4123: 4102: 4098: 4087: 4069: 4065: 4059: 4055: 4051: 4048: 4045: 4042: 4034: 4031: 4028: 4020: 4004: 3999: 3995: 3989: 3985: 3981: 3978: 3975: 3970: 3966: 3960: 3956: 3952: 3944: 3941: 3938: 3930: 3917: 3912: 3885: 3882: 3879: 3875: 3871: 3864: 3860: 3850: 3848: 3844: 3840: 3836: 3832: 3822: 3820: 3814: 3801: 3796: 3792: 3788: 3781: 3777: 3773: 3768: 3760: 3756: 3747: 3742: 3727: 3723: 3719: 3714: 3711: 3706: 3693: 3680: 3670: 3666: 3662: 3657: 3653: 3649: 3641: 3637: 3621: 3606: 3602: 3598: 3593: 3583: 3569: 3568: 3563: 3559: 3557: 3553: 3552: 3535: 3531: 3526: 3522: 3518: 3515: 3512: 3509: 3504: 3494: 3490: 3486: 3483: 3480: 3467: 3463: 3459: 3455: 3454: 3451: 3446: 3442: 3437: 3420: 3416: 3412: 3407: 3404: 3396: 3392: 3383: 3378: 3363: 3359: 3355: 3350: 3347: 3342: 3311: 3307: 3303: 3300: 3292: 3288: 3272: 3257: 3253: 3249: 3244: 3234: 3221: 3215: 3209: 3205: 3199: 3195: 3191: 3187: 3183: 3179: 3170: 3168: 3163: 3161: 3160:wave equation 3145: 3137: 3136:d'Alembertian 3119: 3115: 3107:The operator 3105: 3085: 3081: 3077: 3074: 3070: 3059: 3055: 3051: 3048: 3040: 3036: 3026: 3015: 3002: 2998: 2992: 2988: 2984: 2980: 2969: 2941: 2937: 2933: 2928: 2925: 2921: 2918: 2912: 2908: 2904: 2901: 2893: 2889: 2879: 2876: 2870: 2857: 2853: 2847: 2843: 2839: 2835: 2832: 2826: 2812: 2799: 2793: 2785: 2774: 2770: 2764: 2760: 2756: 2748: 2740: 2737: 2729: 2725: 2716: 2711: 2698: 2694: 2688: 2684: 2680: 2677: 2672: 2659: 2656:meaning that 2643: 2637: 2628: 2625: 2613: 2609: 2603: 2599: 2595: 2592: 2588: 2579: 2566: 2562: 2556: 2546: 2544: 2540: 2536: 2533: 2529: 2523: 2521: 2516: 2502: 2494: 2485: 2482: 2470: 2459: 2455: 2449: 2445: 2441: 2431: 2427: 2423: 2420: 2412: 2408: 2398: 2386: 2373: 2369: 2363: 2359: 2355: 2351: 2340: 2312: 2308: 2304: 2299: 2296: 2292: 2289: 2283: 2269: 2256: 2248: 2240: 2237: 2234: 2229: 2216: 2200: 2197: 2193: 2184: 2171: 2170:Coulomb gauge 2165: 2158:Coulomb gauge 2155: 2152: 2139: 2131: 2123: 2119: 2091: 2083: 2074: 2071: 2068: 2064: 2061: 2050: 2046: 2038: 2034: 2028: 2024: 2023:gauge freedom 2016:Gauge freedom 2013: 2008: 1988: 1984: 1980: 1977: 1973: 1966: 1958: 1945: 1941: 1937: 1932: 1924: 1915: 1906: 1902: 1893: 1889: 1873: 1858: 1854: 1850: 1845: 1835: 1826: 1817: 1800: 1796: 1792: 1787: 1784: 1780: 1771: 1762: 1755: 1743: 1740: 1735: 1721: 1717: 1713: 1699: 1693: 1665: 1659: 1648: 1644: 1643:Faraday's law 1634: 1616: 1608: 1581: 1562: 1559: 1551: 1548: 1535: 1531: 1515: 1507: 1497: 1495: 1491: 1487: 1482: 1477: 1475: 1471: 1467: 1463: 1456: 1452: 1445: 1441: 1437: 1426: 1425: 1421: 1403: 1382: 1378: 1372: 1368: 1364: 1354: 1350: 1346: 1338: 1328: 1327: 1324: 1323: 1319: 1302: 1283: 1280: 1272: 1262: 1261: 1258: 1257: 1253: 1239: 1236: 1228: 1218: 1217: 1214: 1213: 1209: 1191: 1187: 1183: 1178: 1170: 1160: 1159: 1155: 1154: 1153:vector fields 1149: 1145: 1142: 1141: 1140: 1138: 1134: 1128: 1118: 1116: 1112: 1108: 1104: 1100: 1095: 1091: 1087: 1083: 1079: 1075: 1068: 1064: 1060: 1056: 1052: 1047: 1043: 1039: 1038:vector fields 1033: 1023: 1021: 1017: 1013: 1001: 996: 994: 989: 987: 982: 981: 979: 978: 971: 968: 966: 963: 961: 958: 956: 953: 951: 948: 946: 943: 941: 938: 936: 933: 931: 928: 926: 923: 921: 918: 916: 913: 911: 908: 906: 903: 901: 898: 896: 893: 891: 888: 886: 883: 881: 878: 876: 873: 871: 868: 866: 863: 861: 858: 856: 853: 851: 848: 846: 843: 841: 838: 836: 833: 831: 828: 826: 823: 821: 818: 816: 813: 811: 808: 806: 803: 801: 798: 796: 793: 792: 786: 785: 778: 775: 773: 770: 768: 765: 763: 760: 758: 755: 753: 750: 748: 745: 743: 740: 739: 736: 731: 730: 723: 720: 718: 715: 713: 710: 708: 705: 703: 700: 698: 695: 693: 690: 688: 685: 683: 680: 678: 675: 673: 670: 668: 665: 663: 660: 658: 655: 654: 651: 646: 645: 638: 635: 633: 630: 628: 625: 623: 620: 618: 615: 613: 610: 608: 605: 603: 600: 598: 595: 593: 592:Joule heating 590: 588: 585: 583: 580: 578: 575: 573: 570: 568: 565: 563: 560: 558: 555: 553: 550: 548: 545: 544: 541: 536: 535: 528: 525: 523: 520: 518: 515: 513: 510: 508: 507:Lorentz force 505: 503: 500: 498: 495: 493: 490: 488: 485: 483: 480: 478: 475: 473: 470: 468: 465: 463: 460: 458: 455: 453: 450: 448: 445: 443: 440: 438: 435: 434: 431: 426: 425: 418: 415: 413: 410: 408: 407:Magnetization 405: 403: 400: 398: 395: 393: 392:Magnetic flux 390: 388: 385: 383: 380: 378: 375: 373: 370: 368: 365: 364: 361: 356: 355: 348: 345: 343: 340: 338: 335: 333: 330: 328: 325: 323: 320: 318: 315: 313: 310: 308: 305: 303: 300: 298: 297:Electric flux 295: 293: 290: 288: 285: 283: 280: 278: 275: 273: 270: 268: 265: 263: 260: 259: 256: 251: 250: 245: 242: 240: 237: 235: 234:Computational 232: 230: 227: 225: 222: 220: 217: 215: 212: 211: 210: 209: 205: 201: 200: 197: 194: 193: 189: 188: 179: 176: 161: 151: 145: 143: 138: 133: 129: 124: 123: 114: 111: 103: 93: 89: 83: 82: 76: 71: 62: 61: 56: 54: 47: 46: 41: 40: 35: 30: 21: 20: 11608: 11588: 11559: 11540: 11536: 11512: 11507: 11488: 11482: 11472: 11461:. Retrieved 11444: 11425: 11408: 11397: 11389: 11373: 11362: 11353: 11276: 11272: 11262: 11254: 11246: 11238: 11232: 11222: 11209:gauge fixing 11203: 11199: 11195: 11187: 11182: 11178: 11175:kinetic term 11168:relativistic 11164: 11144: 11141:least action 11129: 11125: 11122: 11118: 11115: 11111: 11106: 11096: 11089: 11082: 11071: 11065: 11060: 11056: 11046: 11039: 10936:Lorenz gauge 10930: 10926: 10922: 10915: 10911: 10909:, we impose 10898:gauge fixing 10895: 10886: 10880: 10876: 10868: 10864: 10765: 10761: 10757: 10748: 10744: 10740: 10736: 10734: 10729: 10725: 10721: 10717: 10701: 10689: 10677: 10675: 10667: 10650: 10643: 10636: 10619: 10614: 10610: 10604: 10598: 10592: 10586: 10561:line bundles 10558: 10019: 10015: 10003: 10001: 9998: 9840: 9750: 9745: 9741: 9729: 9725: 9718: 9707: 9661: 9209: 9196: 8961: 8653: 8651: 8133: 8017: 8009: 7520:in terms of 7255: 7249: 7242: 7236: 7093: 6985: 6984: 6910: 6908: 6898: 6853:denotes the 6850: 6848: 6806: 6772: 6769: 6755: 6753: 6556: 6552: 6548: 6546: 6178: 6174: 6169: 6165: 6160: 6156: 6154:-form to a ( 6151: 6137: 5662: 5642: 5637: 5627: 5622: 5615: 5612:Field 2-form 5591: 5529: 5526: 5523: 5519: 5344: 4954: 4711: 4351: 4243: 4240: 4237: 4233: 4183: 4122:pseudoscalar 4019:four-current 3913: 3856: 3843:multivectors 3828: 3816: 3694: 3571: 3566: 3561: 3550: 3549: 3465: 3462:Dirac spinor 3448: 3444: 3440: 3438: 3222: 3213: 3207: 3203: 3193: 3189: 3185: 3181: 3176: 3164: 3106: 2813: 2657: 2564: 2558: 2542: 2538: 2534: 2531: 2524: 2517: 2270: 2214: 2167: 2153: 2048: 2044: 2036: 2032: 2026: 2019: 2010: 1818: 1723: 1719: 1715: 1640: 1533: 1503: 1478: 1465: 1454: 1443: 1435: 1433: 1422: 1320: 1254: 1210: 1151: 1147: 1130: 1106: 1098: 1096: 1089: 1085: 1081: 1077: 1073: 1066: 1062: 1058: 1054: 1050: 1035: 1011: 1009: 761: 752:Four-current 687:Linear motor 572:Electrolysis 452:Eddy current 412:Permeability 332:Polarization 327:Permittivity 171: 155: 139: 137:lead section 106: 97: 78: 50: 43: 37: 36:Please help 33: 11426:Gravitation 11418:Thorne, Kip 11207:. See also 11123:4-potential 11092:, with the 11087:4-potential 10885:are called 10753:classically 10008:determinant 9113:volume form 6760:closed form 6181:= 4 − 2 = 2 6140:Gauss's law 4763:, defining 1212:Gauss's law 1040:called the 722:Transformer 552:Capacitance 477:Faraday law 272:Coulomb law 214:Electricity 92:introducing 11630:Categories 11543:: 83–112. 11522:References 11463:2010-11-19 11314:Free space 11171:covariance 11137:Lagrangian 11130:dynamical 11112:identities 10692:, and the 10580:connection 9736:we define 6859:Hodge star 6764:Hodge dual 6148:Hodge dual 5618:free space 5582:See also: 2053:given by: 1481:free space 789:Scientists 637:Waveguides 617:Resistance 587:Inductance 367:Ampère law 158:March 2024 100:March 2024 75:references 39:improve it 11225:spacetime 11109:geometric 11094:4-current 11085:, or the 11080:EM tensor 11072:obviously 11051:. In the 10994:∂ 10989:φ 10986:∂ 10955:⋅ 10951:∇ 10849:λ 10845:∇ 10810:∂ 10805:λ 10802:∂ 10796:− 10793:φ 10783:φ 10584:curvature 10576:principal 10531:δ 10518:∧ 10513:γ 10500:∧ 10495:β 10482:∧ 10477:α 10462:δ 10459:γ 10456:β 10453:α 10449:ε 10439:− 10432:α 10422:α 10406:π 10352:α 10339:∧ 10334:δ 10321:∧ 10316:γ 10301:α 10298:δ 10295:γ 10292:β 10288:ε 10278:− 10271:α 10261:β 10258:α 10229:⋆ 10190:γ 10177:∧ 10172:β 10159:∧ 10154:α 10136:γ 10133:β 10123:α 10119:∂ 10110:α 10107:γ 10097:β 10093:∂ 10084:β 10081:α 10071:γ 10067:∂ 9974:δ 9961:∧ 9956:γ 9943:∧ 9938:β 9923:δ 9920:γ 9917:β 9914:α 9910:ε 9900:− 9893:α 9864:π 9820:β 9807:∧ 9802:α 9786:β 9783:α 9687:β 9684:μ 9677:α 9673:Γ 9642:γ 9639:β 9629:α 9625:∇ 9616:α 9613:γ 9603:β 9599:∇ 9590:β 9587:α 9577:γ 9573:∇ 9564:γ 9561:β 9551:α 9547:∂ 9538:α 9535:γ 9525:β 9521:∂ 9512:β 9509:α 9499:γ 9495:∂ 9464:α 9454:β 9451:α 9409:β 9406:α 9396:α 9392:∇ 9356:μ 9353:α 9343:α 9340:μ 9333:β 9329:Γ 9318:β 9315:μ 9305:α 9302:μ 9295:α 9291:Γ 9280:β 9277:α 9267:α 9263:∂ 9254:β 9240:π 9212:spacetime 9169:∧ 9158:∧ 9147:∧ 9123:⋆ 9089:⋆ 8997:ρ 8980:⋆ 8976:∧ 8873:ρ 8870:− 8857:⋆ 8827:∧ 8816:∧ 8784:∧ 8773:∧ 8741:∧ 8730:∧ 8698:∧ 8687:∧ 8675:ρ 8672:− 8627:∧ 8606:− 8595:∧ 8574:− 8563:∧ 8542:− 8531:∧ 8510:− 8499:∧ 8478:− 8467:∧ 8446:− 8433:⋆ 8399:∧ 8378:− 8367:∧ 8346:− 8335:∧ 8314:− 8303:∧ 8271:∧ 8239:∧ 8207:ν 8194:∧ 8189:μ 8174:ν 8171:μ 8151:≡ 8099:− 8078:− 8057:− 8045:ϕ 8027:(+ − − −) 7983:ε 7943:ε 7919:⟩ 7887:⟨ 7820:∂ 7816:∂ 7795:∂ 7791:∂ 7729:∗ 7711:∗ 7667:… 7574:∂ 7570:∂ 7562:… 7543:∂ 7539:∂ 7503:− 7483:ε 7314:θ 7309:∧ 7299:θ 7148:− 7138:Λ 7134:∈ 7126:↦ 7118:∋ 7109:Λ 7068:⋆ 7019:⋆ 6953:⋆ 6889:conformal 6870:⋆ 6861:operator 6820:⋆ 6729:∧ 6718:∧ 6697:− 6686:∧ 6675:∧ 6654:− 6643:∧ 6632:∧ 6611:− 6600:∧ 6589:∧ 6577:ρ 6520:∧ 6488:∧ 6456:∧ 6424:∧ 6403:− 6392:∧ 6371:− 6360:∧ 6339:− 6327:⋆ 6295:∧ 6270:∧ 6255:⋆ 6239:∧ 6228:− 6211:∧ 6196:⋆ 6049:ϕ 6046:− 5975:∧ 5943:∧ 5911:∧ 5879:∧ 5847:∧ 5815:∧ 5782:ν 5769:∧ 5764:μ 5749:ν 5746:μ 5728:≡ 5705:(− + + +) 5685:ν 5682:μ 5667:spacetime 5547:μ 5537:∇ 5502:∇ 5488:∂ 5484:∂ 5459:∂ 5449:σ 5436:∂ 5423:∂ 5413:γ 5403:γ 5390:∂ 5380:γ 5370:γ 5363:∇ 5354:γ 5319:σ 5305:− 5302:ρ 5287:γ 5274:γ 5251:γ 5247:ρ 5236:μ 5232:γ 5226:μ 5183:γ 5173:γ 5150:γ 5140:γ 5117:γ 5107:γ 5087:− 5078:γ 5068:γ 5045:γ 5035:γ 5012:γ 5002:γ 4936:μ 4932:∂ 4926:μ 4922:γ 4915:∇ 4889:μ 4885:γ 4858:γ 4848:γ 4838:γ 4828:γ 4795:γ 4785:γ 4772:σ 4728:ℓ 4671:⋅ 4667:∇ 4636:∂ 4624:∂ 4610:× 4606:∇ 4575:μ 4571:− 4560:∂ 4548:∂ 4536:ε 4526:μ 4522:− 4514:× 4510:∇ 4498:− 4482:ε 4478:ρ 4473:− 4465:⋅ 4461:∇ 4429:× 4425:∇ 4410:⋅ 4406:∇ 4394:∧ 4390:∇ 4378:⋅ 4374:∇ 4361:∇ 4326:− 4323:ρ 4305:μ 4287:∇ 4273:∂ 4269:∂ 4211:∂ 4201:σ 4193:∇ 4159:σ 4149:σ 4139:σ 4099:σ 4084:using an 4066:σ 4052:− 4049:ρ 4035:− 4032:ρ 3996:σ 3967:σ 3876:ℓ 3802:ψ 3797:† 3793:ψ 3778:ε 3753:∂ 3748:φ 3739:∂ 3715:− 3712:φ 3703:∇ 3681:ψ 3677:α 3671:† 3667:ψ 3654:μ 3634:∂ 3618:∂ 3594:− 3580:∇ 3532:ψ 3527:† 3523:ψ 3516:− 3510:ρ 3505:ψ 3501:α 3495:† 3491:ψ 3484:− 3417:ε 3413:ρ 3408:− 3389:∂ 3384:φ 3375:∂ 3351:− 3348:φ 3339:∇ 3308:μ 3304:− 3285:∂ 3269:∂ 3245:− 3231:∇ 3146:◻ 3116:◻ 3082:μ 3078:− 3056:◻ 3052:− 3033:∂ 3012:∂ 2999:ε 2989:μ 2985:− 2966:∇ 2938:ε 2934:ρ 2929:− 2919:φ 2909:◻ 2905:− 2886:∂ 2877:φ 2867:∂ 2854:ε 2844:μ 2840:− 2833:φ 2823:∇ 2791:∂ 2786:φ 2783:∂ 2771:ε 2761:μ 2757:− 2749:⋅ 2745:∇ 2741:− 2722:∂ 2717:λ 2708:∂ 2695:ε 2685:μ 2681:− 2678:λ 2669:∇ 2635:∂ 2626:φ 2622:∂ 2610:ε 2600:μ 2596:− 2580:⋅ 2576:∇ 2492:∂ 2483:φ 2479:∂ 2465:∇ 2456:ε 2446:μ 2428:μ 2424:− 2405:∂ 2383:∂ 2370:ε 2360:μ 2356:− 2337:∇ 2309:ε 2305:ρ 2300:− 2290:φ 2280:∇ 2249:⋅ 2245:∇ 2241:− 2235:λ 2226:∇ 2185:⋅ 2181:∇ 2140:λ 2136:∇ 2089:∂ 2084:λ 2081:∂ 2075:− 2072:φ 2062:φ 1985:μ 1981:− 1964:∂ 1959:φ 1956:∂ 1925:⋅ 1921:∇ 1911:∇ 1907:− 1886:∂ 1870:∂ 1846:− 1832:∇ 1797:ε 1793:ρ 1788:− 1772:⋅ 1768:∇ 1753:∂ 1749:∂ 1741:φ 1732:∇ 1700:φ 1697:∇ 1694:× 1691:∇ 1666:× 1663:∇ 1660:⋅ 1657:∇ 1617:× 1613:∇ 1579:∂ 1569:∂ 1563:− 1560:φ 1556:∇ 1552:− 1516:φ 1401:∂ 1391:∂ 1379:ε 1369:μ 1351:μ 1339:× 1336:∇ 1300:∂ 1290:∂ 1284:− 1273:× 1270:∇ 1229:⋅ 1226:∇ 1188:ε 1184:ρ 1171:⋅ 1168:∇ 945:Steinmetz 875:Kirchhoff 860:Jefimenko 855:Hopkinson 840:Helmholtz 835:Heaviside 697:Permeance 582:Impedance 322:Insulator 317:Gauss law 267:Conductor 244:Phenomena 239:Textbooks 219:Magnetism 142:summarize 45:talk page 11424:(1973). 11283:See also 11061:electric 11057:magnetic 10914:⋅ 10829:′ 10786:′ 10657:holonomy 10630:and the 10603:∇ = d + 8022:for the 7837:⟩ 7785:⟨ 6980:manifold 6142:and the 5620:, where 5598:SI units 4017:and the 3071:′ 3027:′ 2981:′ 2922:′ 2880:′ 2836:′ 2629:′ 2589:′ 2486:′ 2399:′ 2352:′ 2293:′ 2194:′ 2120:′ 2065:′ 1044:and the 970:Wiechert 925:Poynting 815:Einstein 662:DC motor 657:AC motor 492:Lenz law 277:Electret 11396:(1963) 11070:"—i.e. 8018:In the 7970:is the 6146:), the 6003:of the 4876:. The 3857:In the 1468:is the 1460:is the 1449:is the 1438:is the 955:Thomson 930:Ritchie 920:Poisson 905:Neumann 900:Maxwell 895:Lorentz 890:Liénard 820:Faraday 805:Coulomb 632:Voltage 607:Ohm law 229:History 88:improve 11615: 11596: 11578: 11566: 11432: 11158:, see 11156:action 10624:1-form 10574:. The 9662:Here, 9441: 9414: 9388: 9361: 7934:, and 6895:where 5660:2-form 5604:.) By 5596:, not 4716:(STA) 3548:where 3464:field 3450:matter 3439:Here, 1464:, and 1434:where 1419: 940:Singer 935:Savart 915:Ørsted 880:Larmor 870:Kelvin 825:Fizeau 795:Ampère 717:Stator 224:Optics 77:, but 11533:(PDF) 11454:(PDF) 11345:Notes 11149:gauge 11116:field 11053:frame 10902:slice 10618:with 10578:U(1)- 10563:or a 10002:Here 9701:is a 7974:with 6986:Note: 6904:) = 1 6849:Here 6758:is a 6754:That 3918:, is 3453:field 2537:and ∂ 2029:, if 965:Weber 960:Volta 950:Tesla 865:Joule 850:Hertz 845:Henry 830:Gauss 712:Rotor 11613:ISBN 11594:ISBN 11576:ISBN 11564:ISBN 11430:ISBN 11265:+ 1) 11257:− 1) 11249:+ 1) 11241:− 1) 11211:and 11186:for 11034:and 10873:and 10774:as 10747:and 10739:and 10728:and 10720:and 10622:the 10608:and 10569:U(1) 9480:and 7987:1234 7748:gram 7010:and 6897:1/(4 5654:and 5635:and 5602:here 5586:and 4813:and 3443:and 2168:The 1645:and 885:Lenz 810:Davy 800:Biot 11545:doi 11541:148 11493:doi 11202:− d 10918:= 0 10879:′, 10696:(a 10646:= 0 10613:= d 10595:= 0 9226:): 8656:is 7698:in 7245:= 0 6917:): 6555:or 5707:as 5616:In 4124:is 3567:QED 3188:), 2047:′, 1496:). 910:Ohm 11632:: 11539:. 11535:. 11489:28 11487:. 11481:. 11456:. 11420:; 11416:; 11215:. 11198:↦ 11162:. 11145:AJ 11099:. 10929:/∂ 10883:′) 10867:, 10764:, 10700:) 10688:) 10634:. 10539:0. 10020:αβ 10014:, 9746:αβ 9189:. 7258:, 7247:. 6965:0. 6906:. 6899:πε 6561:: 6187:) 6177:− 6159:− 5640:= 5625:= 3849:. 3821:. 3570:) 3206:= 3202:1/ 3169:. 2541:/∂ 2051:′) 2035:, 1722:) 1532:, 1508:, 1488:, 1476:. 1453:, 1156:) 1139:: 1117:. 1088:, 1084:, 1080:, 1065:, 1061:, 1057:, 48:. 11621:. 11602:. 11582:) 11572:. 11553:. 11547:: 11501:. 11495:: 11466:. 11438:. 11277:r 11273:F 11263:k 11261:( 11255:k 11253:( 11247:k 11245:( 11239:k 11237:( 11233:k 11204:α 11200:A 11196:A 11188:A 11183:F 11181:⋆ 11179:F 11126:A 11119:F 11097:J 11090:A 11083:F 11010:. 11006:0 11003:= 10997:t 10976:2 10972:c 10968:1 10963:+ 10959:A 10931:t 10927:A 10925:∂ 10923:c 10916:A 10912:∇ 10881:A 10877:φ 10875:( 10871:) 10869:A 10865:φ 10863:( 10841:+ 10837:A 10833:= 10825:A 10819:, 10813:t 10790:= 10768:) 10766:t 10762:x 10760:( 10758:λ 10749:B 10745:E 10741:φ 10737:A 10730:B 10726:E 10722:B 10718:E 10702:A 10690:φ 10644:F 10620:A 10615:A 10611:F 10605:A 10599:d 10593:F 10591:d 10587:F 10536:= 10527:x 10522:d 10509:x 10504:d 10491:x 10486:d 10473:x 10468:d 10442:g 10429:; 10418:j 10410:c 10403:4 10397:= 10393:J 10388:d 10365:, 10361:J 10357:= 10348:x 10343:d 10330:x 10325:d 10312:x 10307:d 10281:g 10268:; 10254:F 10246:6 10243:1 10238:= 10233:F 10224:d 10201:, 10198:0 10195:= 10186:x 10181:d 10168:x 10163:d 10150:x 10145:d 10141:) 10129:F 10115:+ 10103:F 10089:+ 10077:F 10063:( 10060:2 10057:= 10053:F 10048:d 10016:g 10004:g 9983:) 9979:. 9970:x 9965:d 9952:x 9947:d 9934:x 9929:d 9903:g 9889:j 9883:6 9880:1 9874:( 9868:c 9861:4 9855:= 9851:J 9841:J 9825:. 9816:x 9811:d 9798:x 9793:d 9779:F 9773:2 9770:1 9765:= 9761:F 9751:F 9742:F 9730:x 9726:x 9708:α 9647:. 9635:F 9621:+ 9609:F 9595:+ 9583:F 9569:= 9557:F 9543:+ 9531:F 9517:+ 9505:F 9491:= 9488:0 9461:; 9447:F 9433:f 9430:e 9427:d 9421:= 9402:F 9380:f 9377:e 9374:d 9368:= 9349:F 9323:+ 9311:F 9285:+ 9273:F 9259:= 9250:j 9244:c 9237:4 9177:z 9173:d 9166:y 9162:d 9155:x 9151:d 9144:t 9140:d 9136:= 9133:) 9130:1 9127:( 9099:) 9096:1 9093:( 9084:] 9079:2 9075:) 9069:z 9065:j 9061:( 9058:+ 9053:2 9049:) 9043:y 9039:j 9035:( 9032:+ 9027:2 9023:) 9017:x 9013:j 9009:( 9006:+ 9001:2 8993:[ 8990:= 8985:J 8972:J 8948:. 8945:z 8941:d 8935:z 8931:j 8927:+ 8924:y 8920:d 8914:y 8910:j 8906:+ 8903:x 8899:d 8893:x 8889:j 8885:+ 8882:t 8878:d 8867:= 8862:J 8835:y 8831:d 8824:x 8820:d 8813:t 8809:d 8803:z 8799:j 8795:+ 8792:x 8788:d 8781:z 8777:d 8770:t 8766:d 8760:y 8756:j 8752:+ 8749:z 8745:d 8738:y 8734:d 8727:t 8723:d 8717:x 8713:j 8709:+ 8706:z 8702:d 8695:y 8691:d 8684:x 8680:d 8669:= 8665:J 8654:J 8638:. 8635:z 8631:d 8624:t 8620:d 8614:z 8610:B 8603:y 8599:d 8592:t 8588:d 8582:y 8578:B 8571:x 8567:d 8560:t 8556:d 8550:x 8546:B 8539:y 8535:d 8528:x 8524:d 8518:z 8514:E 8507:x 8503:d 8496:z 8492:d 8486:y 8482:E 8475:z 8471:d 8464:y 8460:d 8454:x 8450:E 8443:= 8438:F 8407:y 8403:d 8396:x 8392:d 8386:z 8382:B 8375:x 8371:d 8364:z 8360:d 8354:y 8350:B 8343:z 8339:d 8332:y 8328:d 8322:x 8318:B 8311:z 8307:d 8300:t 8296:d 8290:z 8286:E 8282:+ 8279:y 8275:d 8268:t 8264:d 8258:y 8254:E 8250:+ 8247:x 8243:d 8236:t 8232:d 8226:x 8222:E 8216:= 8203:x 8198:d 8185:x 8180:d 8167:F 8161:2 8158:1 8147:F 8120:. 8117:z 8113:d 8107:z 8103:A 8096:y 8092:d 8086:y 8082:A 8075:x 8071:d 8065:x 8061:A 8054:t 8050:d 8042:= 8038:A 7995:1 7992:= 7956:q 7953:p 7950:b 7947:a 7922:) 7914:j 7910:x 7906:d 7903:, 7898:i 7894:x 7890:d 7884:( 7881:= 7878:) 7873:j 7870:i 7866:g 7862:( 7841:) 7828:j 7824:x 7811:, 7803:i 7799:x 7781:( 7777:= 7774:) 7769:j 7766:i 7762:g 7758:( 7734:M 7724:p 7720:T 7716:= 7707:V 7686:} 7681:n 7677:x 7673:d 7670:, 7664:, 7659:1 7655:x 7651:d 7648:{ 7628:M 7623:p 7619:T 7615:= 7612:V 7591:} 7582:n 7578:x 7565:, 7559:, 7551:1 7547:x 7533:{ 7506:g 7496:q 7493:p 7490:b 7487:a 7477:b 7474:n 7470:g 7464:a 7461:m 7457:g 7451:2 7448:1 7443:= 7438:n 7435:m 7430:q 7427:p 7423:C 7389:n 7386:m 7382:F 7376:n 7373:m 7368:q 7365:p 7361:C 7357:= 7352:q 7349:p 7345:G 7324:. 7319:q 7304:p 7292:q 7289:p 7285:F 7279:2 7276:1 7271:= 7267:F 7256:θ 7243:J 7241:d 7237:J 7222:J 7218:= 7214:G 7209:d 7189:0 7186:= 7182:F 7177:d 7154:) 7151:2 7145:4 7142:( 7130:G 7122:F 7113:2 7105:: 7102:C 7073:J 7046:J 7024:J 6997:J 6962:= 6958:F 6947:2 6942:d 6937:= 6932:J 6926:d 6911:J 6902:0 6851:d 6833:J 6829:= 6825:F 6815:d 6793:0 6790:= 6786:F 6781:d 6756:F 6740:. 6737:y 6733:d 6726:x 6722:d 6715:t 6711:d 6705:z 6701:j 6694:x 6690:d 6683:z 6679:d 6672:t 6668:d 6662:y 6658:j 6651:z 6647:d 6640:y 6636:d 6629:t 6625:d 6619:x 6615:j 6608:z 6604:d 6597:y 6593:d 6586:x 6582:d 6574:= 6570:J 6549:J 6528:y 6524:d 6517:x 6513:d 6507:z 6503:E 6499:+ 6496:x 6492:d 6485:z 6481:d 6475:y 6471:E 6467:+ 6464:z 6460:d 6453:y 6449:d 6443:x 6439:E 6435:+ 6432:t 6428:d 6421:z 6417:d 6411:z 6407:B 6400:t 6396:d 6389:y 6385:d 6379:y 6375:B 6368:t 6364:d 6357:x 6353:d 6347:x 6343:B 6336:= 6332:F 6306:, 6303:z 6299:d 6292:y 6288:d 6284:= 6281:) 6278:t 6274:d 6267:x 6263:d 6259:( 6250:, 6247:t 6243:d 6236:z 6232:d 6225:= 6222:) 6219:y 6215:d 6208:x 6204:d 6200:( 6179:p 6175:n 6170:F 6166:n 6161:p 6157:n 6152:p 6124:. 6121:z 6117:d 6111:z 6107:A 6103:+ 6100:y 6096:d 6090:y 6086:A 6082:+ 6079:x 6075:d 6069:x 6065:A 6061:+ 6058:t 6054:d 6043:= 6039:A 6019:: 6015:A 5983:t 5979:d 5972:z 5968:d 5962:z 5958:E 5954:+ 5951:t 5947:d 5940:y 5936:d 5930:y 5926:E 5922:+ 5919:t 5915:d 5908:x 5904:d 5898:x 5894:E 5890:+ 5887:y 5883:d 5876:x 5872:d 5866:z 5862:B 5858:+ 5855:x 5851:d 5844:z 5840:d 5834:y 5830:B 5826:+ 5823:z 5819:d 5812:y 5808:d 5802:x 5798:B 5794:= 5778:x 5773:d 5760:x 5755:d 5742:F 5736:2 5733:1 5720:F 5699:( 5678:F 5663:F 5646:0 5643:μ 5638:μ 5631:0 5628:ε 5623:ε 5562:. 5559:J 5556:c 5551:0 5543:= 5540:F 5506:, 5498:+ 5491:t 5476:c 5473:1 5468:= 5463:k 5453:k 5445:+ 5440:0 5432:= 5427:k 5417:k 5407:0 5399:+ 5394:0 5384:0 5374:0 5366:= 5358:0 5331:. 5328:) 5323:k 5313:k 5309:J 5299:c 5296:( 5291:0 5283:= 5278:k 5268:k 5264:J 5260:+ 5255:0 5244:c 5241:= 5222:J 5218:= 5215:J 5195:, 5192:) 5187:2 5177:1 5167:3 5163:B 5159:+ 5154:1 5144:3 5134:2 5130:B 5126:+ 5121:3 5111:2 5101:1 5097:B 5093:( 5090:c 5082:0 5072:3 5062:3 5058:E 5054:+ 5049:0 5039:2 5029:2 5025:E 5021:+ 5016:0 5006:1 4996:1 4992:E 4988:= 4984:B 4980:c 4977:I 4974:+ 4970:E 4966:= 4963:F 4941:. 4918:= 4862:3 4852:2 4842:1 4832:0 4824:= 4821:I 4799:0 4789:k 4781:= 4776:k 4751:) 4747:R 4743:( 4738:3 4735:, 4732:1 4724:C 4687:0 4684:= 4680:) 4675:B 4662:( 4658:c 4655:I 4652:+ 4648:) 4640:t 4629:B 4618:+ 4614:E 4601:( 4597:I 4594:+ 4590:) 4585:J 4579:0 4564:t 4553:E 4540:0 4530:0 4518:B 4505:( 4501:c 4494:) 4486:0 4469:E 4456:( 4433:F 4421:I 4418:+ 4414:F 4402:= 4398:F 4386:+ 4382:F 4370:= 4366:F 4337:. 4334:) 4330:J 4320:c 4317:( 4314:c 4309:0 4301:= 4297:F 4292:) 4283:+ 4276:t 4261:c 4258:1 4252:( 4220:, 4215:k 4205:k 4197:= 4163:3 4153:2 4143:1 4135:= 4132:I 4108:} 4103:k 4095:{ 4070:k 4060:k 4056:J 4046:c 4043:= 4039:J 4029:c 4005:, 4000:k 3990:k 3986:B 3982:c 3979:I 3976:+ 3971:k 3961:k 3957:E 3953:= 3949:B 3945:c 3942:I 3939:+ 3935:E 3931:= 3927:F 3899:) 3895:R 3891:( 3886:0 3883:, 3880:3 3872:C 3789:e 3782:0 3774:1 3769:= 3761:2 3757:t 3743:2 3728:2 3724:c 3720:1 3707:2 3663:e 3658:0 3650:= 3642:2 3638:t 3628:A 3622:2 3607:2 3603:c 3599:1 3590:A 3584:2 3564:( 3551:α 3536:, 3519:e 3513:= 3487:e 3481:= 3477:J 3466:ψ 3445:ρ 3441:J 3421:0 3405:= 3397:2 3393:t 3379:2 3364:2 3360:c 3356:1 3343:2 3318:J 3312:0 3301:= 3293:2 3289:t 3279:A 3273:2 3258:2 3254:c 3250:1 3241:A 3235:2 3217:0 3214:μ 3211:0 3208:ε 3204:c 3194:x 3192:( 3190:A 3186:x 3184:( 3182:φ 3120:2 3092:J 3086:0 3075:= 3067:A 3060:2 3049:= 3041:2 3037:t 3023:A 3016:2 3003:0 2993:0 2977:A 2970:2 2942:0 2926:= 2913:2 2902:= 2894:2 2890:t 2871:2 2858:0 2848:0 2827:2 2800:. 2794:t 2775:0 2765:0 2753:A 2738:= 2730:2 2726:t 2712:2 2699:0 2689:0 2673:2 2658:λ 2644:, 2638:t 2614:0 2604:0 2593:= 2585:A 2565:λ 2543:t 2539:A 2535:φ 2532:∇ 2503:) 2495:t 2471:( 2460:0 2450:0 2442:+ 2438:J 2432:0 2421:= 2413:2 2409:t 2395:A 2387:2 2374:0 2364:0 2348:A 2341:2 2313:0 2297:= 2284:2 2257:. 2253:A 2238:= 2230:2 2215:λ 2201:0 2198:= 2190:A 2132:+ 2128:A 2124:= 2116:A 2092:t 2069:= 2049:A 2045:φ 2043:( 2039:) 2037:A 2033:φ 2031:( 2027:λ 1995:J 1989:0 1978:= 1974:) 1967:t 1946:2 1942:c 1938:1 1933:+ 1929:A 1916:( 1903:) 1894:2 1890:t 1880:A 1874:2 1859:2 1855:c 1851:1 1842:A 1836:2 1827:( 1801:0 1785:= 1781:) 1776:A 1763:( 1756:t 1744:+ 1736:2 1718:( 1670:A 1621:A 1609:= 1605:B 1582:t 1573:A 1549:= 1545:E 1534:A 1484:( 1466:J 1458:0 1455:μ 1447:0 1444:ε 1436:ρ 1404:t 1395:E 1383:0 1373:0 1365:+ 1361:J 1355:0 1347:= 1343:B 1303:t 1294:B 1281:= 1277:E 1240:0 1237:= 1233:B 1192:0 1179:= 1175:E 1150:( 1107:B 1099:E 1092:) 1090:t 1086:z 1082:y 1078:x 1076:( 1074:B 1069:) 1067:t 1063:z 1059:y 1055:x 1053:( 1051:E 999:e 992:t 985:v 178:) 172:( 160:) 156:( 146:. 113:) 107:( 102:) 98:( 84:. 55:) 51:(

Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.

Knowledge

Mathematical descriptions of the electromagnetic field

Index