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| {{for|an explanation and meanings of the index notation in this article see|Einstein notation|antisymmetric tensor}}
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| {{electromagnetism|cTopic=[[Covariant formulation of classical electromagnetism|Covariant formulation]]}}
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| 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>
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| :<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===
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| 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>
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| where ''c'' is the speed of light, and
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| :<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]].
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| In contravariant [[matrix (mathematics)|matrix]] form,
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| :<math>
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| \begin{bmatrix}
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| 0 & -E_x/c & -E_y/c & -E_z/c \\
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| E_x/c & 0 & -B_z & B_y \\
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| E_y/c & B_z & 0 & -B_x \\
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| E_z/c & -B_y & B_x & 0
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| \end{bmatrix} = F^{\mu\nu}.
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| </math> | |
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| The covariant form is given by [[Raising and lowering indices#Order-2|index lowering]], | |
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| :<math>
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| F_{\mu\nu} = \eta_{\mu\alpha}\eta_{\nu\beta}F^{\alpha\beta} = \begin{bmatrix}
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| 0 & E_x/c & E_y/c & E_z/c \\
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| -E_x/c & 0 & -B_z & B_y \\
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| -E_y/c & B_z & 0 & -B_x \\
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| -E_z/c & -B_y & B_x & 0 | |
| \end{bmatrix}.
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| </math>
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| 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
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| :<math>
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| F^{\mu}_{\nu} = \begin{bmatrix}
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| 0 & E_x/c & E_y/c & E_z/c \\
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| E_x/c & 0 & B_z & -B_y \\
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| E_y/c & -B_z & 0 & B_x \\
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| E_z/c & B_y & -B_x & 0
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| \end{bmatrix}.
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| </math>
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| 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.
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| ===Properties===
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| 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>
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| {{ordered list
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| |1='''[[antisymmetric|Antisymmetry]]:'''
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| :<math>F^{\mu\nu} \, = - F^{\nu\mu}</math>
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| (hence the name [[bivector]]).
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| |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>'').
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| |3='''Inner product:''' If one forms an inner product of the field strength tensor a [[Lorentz invariant]] is formed
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| :<math>F_{\mu\nu} F^{\mu\nu} = \ 2 \left( B^2 - \frac{E^2}{c^2} \right) </math>
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| meaning this number does not change from one [[frame of reference]] to another.
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| |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]]:
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| :<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>
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| 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>.
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| |5='''[[Determinant]]:'''
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| :<math> \det \left( F \right) = \frac{1}{c^2} \left( \bold B \cdot \bold E \right) ^{2} </math>
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| which is the square of the above invariant.
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| }}
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| ===Significance===
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| 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:
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| :<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>
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| and reduce to:
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| :<math>\partial_{\alpha} F^{\alpha\beta} = \mu_0 J^{\beta}</math>
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| where
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| :<math>J^{\alpha} = ( c\rho, \bold{J} ) </math>
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| 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:
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| :<math>\bold{\nabla} \cdot \bold{B} = 0,\quad \frac{ \partial \bold{B}}{ \partial t } + \bold{\nabla} \times \bold{E} = 0 </math>
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| which reduce to [[Bianchi identity]]:
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| :<math> \partial_\gamma F_{ \alpha \beta } + \partial_\alpha F_{ \beta \gamma } + \partial_\beta F_{ \gamma \alpha } = 0 </math> | |
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| 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:
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| :<math> \partial_{ [ \alpha } F_{ \beta \gamma ] } = 0 </math>
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| ==Relativity==
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| {{main|Maxwell's equations in curved spacetime}}
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| 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.
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| The second equation above leads to the [[continuity equation]]:
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| :<math>J^\alpha{}_{,\alpha} = 0</math>
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| implying [[conservation of charge]].
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| Maxwell's laws above can be generalised to [[curved spacetime]] by simply replacing [[partial derivative]]s with [[covariant derivative]]s:
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| :<math>F_{[\alpha\beta;\gamma]} = 0</math> and <math>F^{\alpha\beta}{}_{;\beta} \, = \mu_0 J^{\alpha}</math>
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| 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):
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| :<math>J^\alpha{}_{;\alpha} \, = 0</math>
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| ==Lagrangian formulation of classical electromagnetism (no charges and currents)==
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| {{see also|Classical field theory}}
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| 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]]:
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| :<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>
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| where
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| :<math>\mathrm{d}^4 x \;</math> is over space and time.
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| This means the [[Lagrangian]] density is
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| :<math>\begin{align}
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| \mathcal{L} & = -\frac{1}{4\mu_0} F_{\mu\nu} F^{\mu\nu} \\
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| & = - \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) \\
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| & = -\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)\\
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| \end{align}</math>
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| The two middle terms are the same, so the Lagrangian density is
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| :<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>
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| Substituting this into the [[Euler-Lagrange equation]] of motion for a field:
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| :<math> \partial_\mu \left( \frac{\partial \mathcal{L}}{\partial ( \partial_\mu A_\nu )} \right) - \frac{\partial \mathcal{L}}{\partial A_\nu} = 0 </math>
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| The second term is zero because the Lagrangian in this case only contains derivatives. So the Euler-Lagrange equation becomes:
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| :<math> \partial_\mu \left( \partial^\mu A^\nu - \partial^\nu A^\mu \right) = 0. \,</math>
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| The quantity in parentheses above is just the field tensor, so this finally simplifies to
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| :<math> \partial_\mu F^{\mu \nu} = 0 </math>
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| That equation is another way of writing the two homogeneous [[Maxwell's equations]], making the substitutions:
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| :<math>~E^i/c = -F^{0 i} \,</math>
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| :<math>\epsilon^{ijk} B_k = -F^{ij} \,</math>
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| where ''i, j, k'' take the values 1, 2, and 3.
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| When there are sources, the Lagrangian needs an extra term to account for the coupling between charges (currents) and the electromagnetic field:
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| <math> J^\mu A_\mu </math>.
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| In that case the [[Euler-Lagrange equation]] yields the inhomogeneous [[Maxwell's equations]]:
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| <math> \partial_\mu F^{\mu \nu} = \mu_0 J^\nu </math>.
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| ===Quantum electrodynamics and field theory===
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| {{main|Quantum electrodynamics|quantum field theory}}
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| 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>  to incorporate the creation and annihilation of photons (and electrons).
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| 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.
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| ==Notes==
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| {{Reflist|group="note"}}
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| {{ordered list
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| |1={{note|antisymmetric}} By definition,
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| :<math> T_{[abc]} = \frac{1}{3!}(T_{abc} + T_{bca} + T_{cab} - T_{acb} - T_{bac} - T_{cba})</math>
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| So if
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| :<math> \partial_\gamma F_{ \alpha \beta } + \partial_\alpha F_{ \beta \gamma } + \partial_\beta F_{ \gamma \alpha } = 0</math> | |
| then
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| :<math>\begin{align}
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| 0 & = \begin{matrix} \frac{2}{6} \end{matrix} ( \partial_\gamma F_{ \alpha \beta } + \partial_\alpha F_{ \beta \gamma } + \partial_\beta F_{ \gamma \alpha }) \\
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| & = \begin{matrix} \frac{1}{6} \end{matrix} \{ \partial_\gamma (2F_{ \alpha \beta }) + \partial_\alpha (2F_{ \beta \gamma }) + \partial_\beta (2F_{ \gamma \alpha }) \} \\
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| & = \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}) \} \\
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| & = \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} ) \\
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| & = \partial_{[ \gamma} F_{ \alpha \beta ]}
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| \end{align}</math>
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| }}
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| ==See also==
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| * [[Classification of electromagnetic fields]]
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| * [[Covariant formulation of classical electromagnetism]]
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| * [[Electromagnetic stress–energy tensor]]
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| * [[Gluon field strength tensor]]
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| * [[Ricci calculus]]
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| * [[Riemann–Silberstein vector]]
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| ==References==
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| {{reflist}}
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| *{{cite book | author=Brau, Charles A. | title=Modern Problems in Classical Electrodynamics | publisher=Oxford University Press | year=2004 | isbn=0-19-514665-4}}
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| *{{cite book | author=Jackson, John D. | title=Classical Electrodynamics | publisher=John Wiley & Sons, Inc. | year=1999 | isbn=0-471-30932-X}}
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| *{{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}}
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| {{tensors}}
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| [[Category:Electromagnetism]]
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| [[Category:Minkowski spacetime]]
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| [[Category:Theory of relativity]]
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| [[Category:Tensors]]
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| [[Category:Tensors in general relativity]]
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