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Transformationsverhalten der Ströme und Felder Eq: math.2174.25

Hash: 4a19cd8ef3d320cc16c175cf17c8ca87

TeX (original user input):

\begin{align}
& \left\{ {{F}_{\mu \nu }} \right\}=\left\{ {{\partial }_{\mu }}{{\Phi }_{\nu }}-{{\partial }_{\nu }}{{\Phi }_{\mu }} \right\}=\left( \begin{matrix}
0 & \frac{1}{c}{{E}_{x}} & \frac{1}{c}{{E}_{y}} & \frac{1}{c}{{E}_{z}}  \\
-\frac{1}{c}{{E}_{x}} & 0 & -{{B}_{z}} & {{B}_{y}}  \\
-\frac{1}{c}{{E}_{y}} & {{B}_{z}} & 0 & -{{B}_{x}}  \\
-\frac{1}{c}{{E}_{z}} & -{{B}_{y}} & {{B}_{x}} & 0  \\
\end{matrix} \right) \\
& {{F}^{\mu \nu }}=\left\{ {{\partial }^{\mu }}{{\Phi }^{\nu }}-{{\partial }^{\nu }}{{\Phi }^{\mu }} \right\}=\left( \begin{matrix}
0 & -\frac{1}{c}{{E}_{x}} & -\frac{1}{c}{{E}_{y}} & -\frac{1}{c}{{E}_{z}}  \\
\frac{1}{c}{{E}_{x}} & 0 & -{{B}_{z}} & {{B}_{y}}  \\
\frac{1}{c}{{E}_{y}} & {{B}_{z}} & 0 & -{{B}_{x}}  \\
\frac{1}{c}{{E}_{z}} & -{{B}_{y}} & {{B}_{x}} & 0  \\
\end{matrix} \right) \\
& \Leftrightarrow {{F}^{\mu \nu }}=\left\{ {{\partial }^{\mu }}{{\Phi }^{\nu }}-{{\partial }^{\nu }}{{\Phi }^{\mu }} \right\}=\left( \begin{matrix}
0 & -{{E}^{1}} & -{{E}^{2}} & -{{E}^{3}}  \\
{{E}^{1}} & 0 & -c{{B}^{3}} & c{{B}^{2}}  \\
{{E}^{2}} & c{{B}^{3}} & 0 & -c{{B}^{1}}  \\
{{E}^{3}} & -c{{B}^{2}} & c{{B}^{1}} & 0  \\
\end{matrix} \right) \\
\end{align}

TeX (checked):

{\begin{aligned}&\left\{{{F}_{\mu \nu }}\right\}=\left\{{{\partial }_{\mu }}{{\Phi }_{\nu }}-{{\partial }_{\nu }}{{\Phi }_{\mu }}\right\}=\left({\begin{matrix}0&{\frac {1}{c}}{{E}_{x}}&{\frac {1}{c}}{{E}_{y}}&{\frac {1}{c}}{{E}_{z}}\\-{\frac {1}{c}}{{E}_{x}}&0&-{{B}_{z}}&{{B}_{y}}\\-{\frac {1}{c}}{{E}_{y}}&{{B}_{z}}&0&-{{B}_{x}}\\-{\frac {1}{c}}{{E}_{z}}&-{{B}_{y}}&{{B}_{x}}&0\\\end{matrix}}\right)\\&{{F}^{\mu \nu }}=\left\{{{\partial }^{\mu }}{{\Phi }^{\nu }}-{{\partial }^{\nu }}{{\Phi }^{\mu }}\right\}=\left({\begin{matrix}0&-{\frac {1}{c}}{{E}_{x}}&-{\frac {1}{c}}{{E}_{y}}&-{\frac {1}{c}}{{E}_{z}}\\{\frac {1}{c}}{{E}_{x}}&0&-{{B}_{z}}&{{B}_{y}}\\{\frac {1}{c}}{{E}_{y}}&{{B}_{z}}&0&-{{B}_{x}}\\{\frac {1}{c}}{{E}_{z}}&-{{B}_{y}}&{{B}_{x}}&0\\\end{matrix}}\right)\\&\Leftrightarrow {{F}^{\mu \nu }}=\left\{{{\partial }^{\mu }}{{\Phi }^{\nu }}-{{\partial }^{\nu }}{{\Phi }^{\mu }}\right\}=\left({\begin{matrix}0&-{{E}^{1}}&-{{E}^{2}}&-{{E}^{3}}\\{{E}^{1}}&0&-c{{B}^{3}}&c{{B}^{2}}\\{{E}^{2}}&c{{B}^{3}}&0&-c{{B}^{1}}\\{{E}^{3}}&-c{{B}^{2}}&c{{B}^{1}}&0\\\end{matrix}}\right)\\\end{aligned}}

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MathML (8.501 KB / 666 B) :

\begin{aligned} {F_{μ ν}} = {\partial_{μ} Φ_{ν} - \partial_{ν} Φ_{μ}} = (\begin{matrix} 0 & \frac{1}{c} E_{x} & \frac{1}{c} E_{y} & \frac{1}{c} E_{z} \\ - \frac{1}{c} E_{x} & 0 & - B_{z} & B_{y} \\ - \frac{1}{c} E_{y} & B_{z} & 0 & - B_{x} \\ - \frac{1}{c} E_{z} & - B_{y} & B_{x} & 0 \end{matrix}) \\ F^{μ ν} = {\partial^{μ} Φ^{ν} - \partial^{ν} Φ^{μ}} = (\begin{matrix} 0 & - \frac{1}{c} E_{x} & - \frac{1}{c} E_{y} & - \frac{1}{c} E_{z} \\ \frac{1}{c} E_{x} & 0 & - B_{z} & B_{y} \\ \frac{1}{c} E_{y} & B_{z} & 0 & - B_{x} \\ \frac{1}{c} E_{z} & - B_{y} & B_{x} & 0 \end{matrix}) \\ \Leftrightarrow F^{μ ν} = {\partial^{μ} Φ^{ν} - \partial^{ν} Φ^{μ}} = (\begin{matrix} 0 & - E^{1} & - E^{2} & - E^{3} \\ E^{1} & 0 & - c B^{3} & c B^{2} \\ E^{2} & c B^{3} & 0 & - c B^{1} \\ E^{3} & - c B^{2} & c B^{1} & 0 \end{matrix}) \end{aligned}

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Identifiers

$F_{μ ν}$

$μ$

$Φ_{ν}$

$ν$

$Φ_{μ}$

$c$

$E_{x}$

$c$

$E_{y}$

$c$

$E_{z}$

$c$

$E_{x}$

$B_{z}$

$B_{y}$

$c$

$E_{y}$

$B_{z}$

$B_{x}$

$c$

$E_{z}$

$B_{y}$

$B_{x}$

$F$

$μ$

$ν$

$μ$

$Φ$

$ν$

$ν$

$Φ$

$μ$

$c$

$E_{x}$

$c$

$E_{y}$

$c$

$E_{z}$

$c$

$E_{x}$

$B_{z}$

$B_{y}$

$c$

$E_{y}$

$B_{z}$

$B_{x}$

$c$

$E_{z}$

$B_{y}$

$B_{x}$

$F$

$μ$

$ν$

$μ$

$Φ$

$ν$

$ν$

$Φ$

$μ$

$E$

$E$

$E$

$E$

$c$

$B$

$c$

$B$

$E$

$c$

$B$

$c$

$B$

$E$

$c$

$B$

$c$

$B$

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