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{{for|an explanation and meanings of the index notation in this article see|Einstein notation|antisymmetric tensor}}
== do not sound a hundred heart back into his head ==
{{electromagnetism|cTopic=[[Covariant formulation of classical electromagnetism|Covariant formulation]]}}


In [[electromagnetism]], the '''electromagnetic tensor''' or '''electromagnetic field tensor''' (sometimes called the '''field strength tensor''', '''Faraday tensor''' or '''Maxwell bivector''') is a mathematical object that describes the [[electromagnetic field]] of a physical system. The field tensor was first used after the 4-dimensional [[tensor]] formulation of [[special relativity]] was introduced by [[Hermann Minkowski]]. The tensor allows some physical laws to be written in a very concise form.
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SI units and the particle physicist's convention for the [[Metric signature|signature]] of Minkowski space <tt>(+,−,−,−)</tt>, will be used throughout this article.
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==Definition==
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The electromagnetic tensor, conventionally labelled ''F'', is defined as the [[Exterior_derivative#Exterior_derivative_of_a_k-form|exterior derivative]] of the [[electromagnetic four-potential]], ''A'', a differential 1-form:<ref>{{cite book | author=J.A. Wheeler, C. Misner, K.S. Thorne| title=[[Gravitation (book)|Gravitation]]| publisher=W.H. Freeman & Co| year=1973 | isbn=0-7167-0344-0}}</ref><ref>{{cite book | author=D.J. Griffiths| title=Introduction to Electrodynamics (3rd Edition)| publisher=Pearson Education, Dorling Kindersley| year=2007 | isbn=81-7758-293-3}}</ref>
== he is a funny clown look ==


:<math>F \ \stackrel{\mathrm{def}}{=}\ \mathrm{d}A.</math>
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Therefore ''F'' is a [[differential form|differential 2-form]]—that is, an antisymmetric rank-2 tensor field—on Minkowski space. In component form,
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:<math>F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu.</math>
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===Relationship with the Classical Fields===
== without long ==


The electromagnetic tensor is completely [[isomorphism|isomorphic]] to the electric and magnetic fields, though the electric and magnetic fields change with the choice of the reference frame, while the electromagnetic tensor does not. In general, the relationship is quite complicated, but in Cartesian coordinates, using the coordinate system's own reference frame, the relationship is very simple.
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:<math>E_i = c F^{i0},</math>
<ul>
where ''c'' is the speed of light, and
 
:<math>B_i = -\frac 1 2 \epsilon_{ijk} F^{jk},</math>
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where <math>\epsilon_{ijk}</math> is the [[Levi-Civita symbol]].
 
In contravariant [[matrix (mathematics)|matrix]] form,
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:<math>
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\begin{bmatrix}
 
0    & -E_x/c & -E_y/c & -E_z/c \\
  </ul>
E_x/c & 0      & -B_z  & B_y    \\
E_y/c & B_z    & 0      & -B_x  \\
E_z/c & -B_y  & B_x    & 0
\end{bmatrix} = F^{\mu\nu}.
</math>
 
The covariant form is given by [[Raising and lowering indices#Order-2|index lowering]],
 
:<math>
F_{\mu\nu} = \eta_{\mu\alpha}\eta_{\nu\beta}F^{\alpha\beta} = \begin{bmatrix}
0      & E_x/c  & E_y/c  & E_z/c \\
-E_x/c & 0      & -B_z  & B_y    \\
-E_y/c & B_z    & 0      & -B_x  \\
-E_z/c & -B_y  & B_x    & 0
\end{bmatrix}.
</math>
 
The mixed form appears in the [[Lorentz force]] equation when using the contravariant [[four-velocity]]: <math> \frac{d p^\mu}{d \tau} = q F^{\mu}_{\nu} u^\nu </math>, where
 
:<math>
F^{\mu}_{\nu} = \begin{bmatrix}
0      & E_x/c  & E_y/c  & E_z/c \\
E_x/c  & 0      & B_z    & -B_y    \\
E_y/c  & -B_z  & 0      & B_x  \\
E_z/c  & B_y    & -B_x  & 0
\end{bmatrix}.
</math>
 
From now on in this article, when the electric or magnetic fields are mentioned, a Cartesian coordinate system is being assumed, and the electric and magnetic fields are with respect to coordinate system's own reference frame, as in the equations above.
 
===Properties===
 
The matrix form of the field tensor yields the following properties:<ref>{{cite book | author=J.A. Wheeler, C. Misner, K.S. Thorne| title=[[Gravitation (book)|Gravitation]]| publisher=W.H. Freeman & Co| year=1973 | isbn=0-7167-0344-0}}</ref>
 
{{ordered list
|1='''[[antisymmetric|Antisymmetry]]:'''
 
:<math>F^{\mu\nu} \, = - F^{\nu\mu}</math>  
 
(hence the name [[bivector]]).
 
|2='''Six independent components:''' In Cartesian coordinates, these are simply the three spatial components of the electric field (''E<sub>x</sub>, E<sub>y</sub>, E<sub>z</sub>'') and magnetic field (''B<sub>x</sub>, B<sub>y</sub>, B<sub>z</sub>'').
 
|3='''Inner product:''' If one forms an inner product of the field strength tensor a [[Lorentz invariant]] is formed
 
:<math>F_{\mu\nu} F^{\mu\nu} = \ 2 \left( B^2 - \frac{E^2}{c^2} \right) </math>
 
meaning this number does not change from one [[frame of reference]] to another.
|4='''[[Pseudoscalar]] invariant:''' The product of the tensor <math>\scriptstyle (F^{\mu\nu})</math> with its '''[[Hodge dual|dual tensor]]''' <math>\scriptstyle (G^{\mu\nu})</math> gives the [[Lorentz invariant]]:
 
:<math> G_{\gamma\delta}F^{\gamma\delta}=\frac{1}{2}\epsilon_{\alpha\beta\gamma\delta}F^{\alpha\beta} F^{\gamma\delta} = -\frac{4}{c} \left( \bold B \cdot \bold E \right)  \,</math>
 
where <math>\epsilon_{\alpha\beta\gamma\delta} </math> is the rank-4 [[Levi-Civita symbol]]. The sign for the above depends on the convention used for the Levi-Civita symbol. The convention used here is <math> \epsilon_{0123} = +1 </math>.
 
|5='''[[Determinant]]:'''
 
:<math> \det \left( F \right) = \frac{1}{c^2} \left( \bold B \cdot \bold E \right) ^{2} </math>
 
which is the square of the above invariant.
}}
 
===Significance===
 
This tensor simplifies and reduces [[Maxwell's equations]] as four vector calculus equations into two tensor field equations. In [[electrostatic]]s and [[electrodynamic]]s, [[Gauss's law]] and [[Ampère's circuital law]] are respectively:
 
:<math>\bold{\nabla} \cdot \bold{E} = \frac{\rho}{\epsilon_0},\quad \bold{\nabla} \times \bold{B} - \frac{1}{c^2} \frac{ \partial \bold{E}}{\partial t} = \mu_0 \bold{J} </math>
 
and reduce to:
 
:<math>\partial_{\alpha} F^{\alpha\beta} = \mu_0 J^{\beta}</math>
 
where
 
:<math>J^{\alpha} = ( c\rho, \bold{J} ) </math>
 
is the [[4-current]]. In [[magnetostatic]]s and magnetodynamics, [[Gauss's law for magnetism]] and [[Faraday's law of induction|Maxwell–Faraday equation]] are respectively:
 
:<math>\bold{\nabla} \cdot \bold{B} = 0,\quad \frac{ \partial \bold{B}}{ \partial t } + \bold{\nabla} \times \bold{E} = 0 </math>
 
which reduce to [[Bianchi identity]]:
 
:<math> \partial_\gamma F_{ \alpha \beta } + \partial_\alpha F_{ \beta \gamma } + \partial_\beta F_{ \gamma \alpha } = 0 </math>
 
or using the [[Ricci calculus#Symmetric and antisymmetric parts|index notation with square brackets]]{{ref|antisymmetric|[note 1]}} for the antisymmetric part of the tensor:
 
:<math> \partial_{ [ \alpha } F_{ \beta \gamma ] } = 0 </math>
 
==Relativity==
 
{{main|Maxwell's equations in curved spacetime}}
 
The field tensor derives its name from the fact that the electromagnetic field is found to obey the [[tensor transformation law]], this general property of (non-gravitational) physical laws being recognised after the advent of [[special relativity]]. This theory stipulated that all the (non-gravitational) laws of physics should take the same form in all coordinate systems - this led to the introduction of [[tensor]]s. The tensor formalism also leads to a mathematically simpler presentation of physical laws.  
 
The second equation above leads to the [[continuity equation]]:
 
:<math>J^\alpha{}_{,\alpha} = 0</math>
 
implying [[conservation of charge]].
 
Maxwell's laws above can be generalised to [[curved spacetime]] by simply replacing [[partial derivative]]s with [[covariant derivative]]s:
 
:<math>F_{[\alpha\beta;\gamma]} = 0</math> and  <math>F^{\alpha\beta}{}_{;\beta} \, = \mu_0 J^{\alpha}</math>
 
where the semi-colon represents a covariant derivative, as opposed to a partial derivative. These equations are sometimes referred to as the [[Maxwell's equations in curved spacetime|curved space Maxwell equations]]. Again, the second equation implies charge conservation (in curved spacetime):
 
:<math>J^\alpha{}_{;\alpha} \, = 0</math>
 
==Lagrangian formulation of classical electromagnetism (no charges and currents)==
 
{{see also|Classical field theory}}
 
When there are no electric charges (''ρ'' = 0) and no electric currents ('''J''' = '''0'''), [[Classical electromagnetism]] and [[Maxwell's equations]] can be derived from the [[action (physics)|action]]:
 
:<math>\mathcal{S} = \int \left( -\begin{matrix} \frac{1}{4 \mu_0} \end{matrix} F_{\mu\nu} F^{\mu\nu} \right) \mathrm{d}^4 x \,</math>
 
where
 
:<math>\mathrm{d}^4 x \;</math> &nbsp; is over space and time.
 
This means the [[Lagrangian]] density is
 
:<math>\begin{align}
\mathcal{L} & = -\frac{1}{4\mu_0} F_{\mu\nu} F^{\mu\nu} \\
& = - \frac{1}{4\mu_0} \left( \partial_\mu A_\nu - \partial_\nu A_\mu \right) \left( \partial^\mu A^\nu - \partial^\nu A^\mu \right) \\
& = -\frac{1}{4\mu_0} \left( \partial_\mu A_\nu \partial^\mu A^\nu - \partial_\nu A_\mu \partial^\mu A^\nu - \partial_\mu A_\nu \partial^\nu A^\mu + \partial_\nu A_\mu \partial^\nu A^\mu \right)\\
\end{align}</math>
 
The two middle terms are the same, so the Lagrangian density is
 
:<math>\mathcal{L} = - \frac{1}{2\mu_0} \left( \partial_\mu A_\nu \partial^\mu A^\nu - \partial_\nu A_\mu \partial^\mu A^\nu \right).</math>
 
Substituting this into the [[Euler-Lagrange equation]] of motion for a field:
 
:<math> \partial_\mu \left( \frac{\partial \mathcal{L}}{\partial ( \partial_\mu A_\nu )} \right) - \frac{\partial \mathcal{L}}{\partial A_\nu} = 0 </math>
 
The second term is zero because the Lagrangian in this case only contains derivatives. So the Euler-Lagrange equation becomes:
 
:<math> \partial_\mu \left( \partial^\mu A^\nu - \partial^\nu A^\mu \right) = 0. \,</math>
 
The quantity in parentheses above is just the field tensor, so this finally simplifies to
 
:<math> \partial_\mu F^{\mu \nu} = 0 </math>
 
That equation is another way of writing the two homogeneous [[Maxwell's equations]], making the substitutions:
 
:<math>~E^i/c = -F^{0 i} \,</math>
:<math>\epsilon^{ijk} B_k = -F^{ij} \,</math>
 
where ''i, j, k'' take the values 1, 2, and 3.
 
When there are sources, the Lagrangian needs an extra term to account for the coupling between charges (currents) and the electromagnetic field:
 
<math> J^\mu A_\mu </math>.
 
In that case the [[Euler-Lagrange equation]] yields the inhomogeneous [[Maxwell's equations]]:
 
<math> \partial_\mu F^{\mu \nu} = \mu_0 J^\nu </math>.
 
===Quantum electrodynamics and field theory===
 
{{main|Quantum electrodynamics|quantum field theory}}
 
The [[Lagrangian]] of [[quantum electrodynamics]] extends beyond the classical Lagrangian established in relativity, from <math>\mathcal{L}=\bar\psi(i\hbar c \, \gamma^\alpha D_\alpha - mc^2)\psi -\frac{1}{4 \mu_0}F_{\alpha\beta}F^{\alpha\beta},</math> &ensp;to incorporate the creation and annihilation of photons (and electrons).
 
In [[quantum field theory]] it is used as the template for the gauge field strength tensor. By being employed in addition to the local interaction Lagrangian it reprises its usual role in QED.
 
==Notes==
 
{{Reflist|group="note"}}
 
{{ordered list
|1={{note|antisymmetric}} By definition,
 
:<math> T_{[abc]} = \frac{1}{3!}(T_{abc} + T_{bca} + T_{cab} - T_{acb} - T_{bac} - T_{cba})</math>
So if
:<math> \partial_\gamma F_{ \alpha \beta } + \partial_\alpha F_{ \beta \gamma } + \partial_\beta F_{ \gamma \alpha } = 0</math>
then
:<math>\begin{align}
0 & = \begin{matrix} \frac{2}{6} \end{matrix} ( \partial_\gamma F_{ \alpha \beta } + \partial_\alpha F_{ \beta \gamma } + \partial_\beta F_{ \gamma \alpha }) \\
& = \begin{matrix} \frac{1}{6} \end{matrix} \{ \partial_\gamma (2F_{ \alpha \beta }) + \partial_\alpha (2F_{ \beta \gamma }) + \partial_\beta (2F_{ \gamma \alpha }) \} \\
& = \begin{matrix} \frac{1}{6} \end{matrix} \{ \partial_\gamma (F_{ \alpha \beta } - F_{ \beta \alpha}) + \partial_\alpha (F_{ \beta \gamma } - F_{ \gamma \beta}) + \partial_\beta (F_{ \gamma \alpha } - F_{ \alpha \gamma}) \} \\
& = \begin{matrix} \frac{1}{6} \end{matrix} ( \partial_\gamma F_{ \alpha \beta } + \partial_\alpha F_{ \beta \gamma } + \partial_\beta F_{ \gamma \alpha } - \partial_\gamma F_{ \beta \alpha} - \partial_\alpha F_{ \gamma \beta} - \partial_\beta F_{ \alpha \gamma} ) \\
  & = \partial_{[ \gamma} F_{ \alpha \beta ]}
\end{align}</math>
}}
 
==See also==
* [[Classification of electromagnetic fields]]
* [[Covariant formulation of classical electromagnetism]]
* [[Electromagnetic stress–energy tensor]]
* [[Gluon field strength tensor]]
* [[Ricci calculus]]
* [[Riemann–Silberstein vector]]
 
==References==
{{reflist}}
*{{cite book | author=Brau, Charles A. | title=Modern Problems in Classical Electrodynamics | publisher=Oxford University Press | year=2004 | isbn=0-19-514665-4}}
*{{cite book | author=Jackson, John D. | title=Classical Electrodynamics | publisher=John Wiley & Sons, Inc. | year=1999 | isbn=0-471-30932-X}}
*{{cite book | author=Peskin, Michael E.; Schroeder, Daniel V. | title=An Introduction to Quantum Field Theory | publisher=Perseus Publishing | year=1995 | isbn=0-201-50397-2}}
 
{{tensors}}
 
[[Category:Electromagnetism]]
[[Category:Minkowski spacetime]]
[[Category:Theory of relativity]]
[[Category:Tensors]]
[[Category:Tensors in general relativity]]

Revision as of 19:23, 27 February 2014

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He proudly said, Hey, do not sound a hundred heart back into his head, only to find all gathered behind him, and Yu Feng chuckled and asked: '? into the back door of how it feels.'

'yo, Caoge have this hobby.' mouse Road with, Li Mei, biting his lower lip angry, afraid to talk to each group of hooligans.

Caoya Jie cheap cheap smile, a touch of handsome hair, black jack knock an Enter key, a remote execution of the program, brush brush lit screen, a synchronized to ケイトスペード 財布 ゴールド a monitoring unit here, mouse stunned with: 'Oh, This feeling is the back door into the great. 相关的主题文章：

'hate it ...... people ignore you.'

Focus, stretched out from the window head, looked one bedroom skirt, very proud of 姚曼兰 chest, 姚曼兰 a ケイトスペード時計人気 shame, ケイトスペード バッグ 激安 embarrassed, said:. 'I know you're the line'

'That night we try ...... or else?' I ケイトスペード 財布 値段 sin good direct, tongue licked his lips.

'Ah, and then talk to you, my face would not want to, it will not subtle point ah.' Yaoman Lan embarrassed authentic, sideways dodging.

'including ... including ...... mouth can, I do not mind.' I stretch Shenbozi ケイトスペード アウトレット バッグ crime, bad smile.

have le to Yaoman Lan also hear this experienced man to the ears of the smell of urine, then deputy director Yin Yin I stared at his face, chances are an obscenity mouth, 'contain' forward is like? After stomping her embarrassed authentic: 'hate it ...... people kate spade バッグ ignore you.'

said ran away, occasionally looking back Chou Chou, I can still see the face of the crime that immorality laugh.

what men have sex she does not accidental, but in fact done this degree so she surprises today to plug a small gift, bluntly pretend played. 相关的主题文章：

he is a funny clown look

Wear more, I do not know sin hides, he is a funny clown look, or the cute guy roles, but months for the first time so close to the contact, the sense of excitement is always just lingering child ...... yes ah, so burning weather, he did not even An Jialu nose also clearly visible on the beads of sweat that crystal clear look, which kate spade マザーズバッグ seems reflected his own shadow Ai

'Hey, God spoke all gone?' An Jialu found.

'ah, that there is, I attentively listening.' I sin stall road.

'Should not you say?' An Jialu smile, when a smile, greasy face on a small ケイトスペード リボン バッグ shallow dimples, ケイトスペード 人気バッグ good fresh to say. More than sin straining sip sip ケイトスペードニューヨーク 財布 straight swallow the saliva flow want to go, hard to ask: '? You let me say what.'

'such great pains to put forward a picnic, but also to engage in false 財布 ケイトスペード Yifeng hand ...... not just to eat it?' An Jialu wise to look at the seemingly do not have a motive more than sin.

Damn, misunderstood, but just kind of misunderstanding, I sin smiled, straights 相关的主题文章：

with jubilation stomping

Han jian your ケイトスペード 財布 店舗 style sao it?

knocking, knock, rattled seemingly disordered voices, Xiaomeng Qi did not understand that this is doing, Luo Jialong also holding chopsticks knock on, and Wang Shen repair バッグ ケイトスペード unexpectedly laughed, ケイトスペード 財布 通販 joined the ranks of the bowl knock knock dish, knocking knock with, even Xiong Jianfei and mouse seems to have cheered up a ding ...... Dangdang, with jubilation stomping, then I say crime lead singing:

na kate spade ハンドバッグ brother, my brother, the most pro is you.

mortgage, debt, the weight I do not play.

official, public grain, tired I straight pant.

flies are hard to force, who put who despise.

say all laugh, laugh Xiaomeng Qi direct spray, it seems that this is a summary of married life, and this one also opened a smooth like-minded people, who will バッグ ケイトスペード take more than a question of sin, 骆家龙 is connected to the:

na brother, my brother, the most pro is you.

what conscientious, what dedication, it is for the play are special.
What
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without long

A light? 'Yuan Liang Tao.

than sin took a deep breath, smiled, he knew, struggling in this state of mind is what ケイトスペード ハンドバッグ kind of taste, he thought, seemed to try to figure out the person's credibility, he seems a long ケイトスペード バッグ 新作 while from the other side complex and clear eyes found the things kate spade マザーズバッグ they need, opening with: '? Well, I ask you, if you have a chance to catch Wu Xiaolei, you will ケイトスペード トートバッグ do it.'

'That, of course, we are not doing it?' Yuan Liang Tao.

'If this thing to break your bottom kate spade マザーズバッグ line, you will do it? For example, I really put his parents in isolation, without long, according to the normal procedure to go on the trip.' I sin road.

Yuan Liang thought, nodded:. 'If necessary, you can do so ...... this unfinished case, the burden is on them, live every day in fear of the taste and feel good.'

'Well, we do this together, I am here to have a detailed plan, is looking for people to talk about ...... you are mentally prepared, you may want to touch your bottom line, you sure you want between us 相关的主题文章：

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