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* Page found: Transformationsverhalten der Ströme und Felder (eq math.2174.51)

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\begin{align}
& {{\varepsilon }^{\kappa \lambda \mu \nu }}\acute{\ }={{U}^{\kappa }}_{\alpha }{{U}^{\lambda }}_{\beta }{{U}^{\mu }}_{\gamma }{{U}^{\nu }}_{\delta }{{\varepsilon }^{\alpha \beta \gamma \delta }} \\
& =\left| \begin{matrix}
{{U}^{\kappa }}_{0} & {{U}^{\kappa }}_{1} & {{U}^{\kappa }}_{2} & {{U}^{\kappa }}_{3}  \\
{{U}^{\lambda }}_{0} & {{U}^{\lambda }}_{1} & {{U}^{\lambda }}_{2} & {{U}^{\lambda }}_{3}  \\
{{U}^{\mu }}_{0} & {{U}^{\mu }}_{1} & {{U}^{\mu }}_{2} & {{U}^{\mu }}_{3}  \\
{{U}^{\nu }}_{0} & {{U}^{\nu }}_{1} & {{U}^{\nu }}_{2} & {{U}^{\nu }}_{3}  \\
\end{matrix} \right|=\left( \det U \right)\cdot {{\varepsilon }^{\kappa \lambda \mu \nu }} \\
& \left( \det U \right)=\pm 1 \\
\end{align}

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ε κ ⁢ λ ⁢ μ ⁢ ν ⁢ ´ = U κ α ⁢ U λ β ⁢ U μ γ ⁢ U ν δ ⁢ ε α ⁢ β ⁢ γ ⁢ δ = | U κ 0 U κ 1 U κ 2 U κ 3 U λ 0 U λ 1 U λ 2 U λ 3 U μ 0 U μ 1 U μ 2 U μ 3 U ν 0 U ν 1 U ν 2 U ν 3 | = ( det ⁡ U ) ⋅ ε κ ⁢ λ ⁢ μ ⁢ ν ( det ⁡ U ) = ± 1 missing-subexpression superscript 𝜀 𝜅 𝜆 𝜇 𝜈 ´ absent subscript superscript 𝑈 𝜅 𝛼 subscript superscript 𝑈 𝜆 𝛽 subscript superscript 𝑈 𝜇 𝛾 subscript superscript 𝑈 𝜈 𝛿 superscript 𝜀 𝛼 𝛽 𝛾 𝛿 missing-subexpression absent subscript superscript 𝑈 𝜅 0 subscript superscript 𝑈 𝜅 1 subscript superscript 𝑈 𝜅 2 subscript superscript 𝑈 𝜅 3 subscript superscript 𝑈 𝜆 0 subscript superscript 𝑈 𝜆 1 subscript superscript 𝑈 𝜆 2 subscript superscript 𝑈 𝜆 3 subscript superscript 𝑈 𝜇 0 subscript superscript 𝑈 𝜇 1 subscript superscript 𝑈 𝜇 2 subscript superscript 𝑈 𝜇 3 subscript superscript 𝑈 𝜈 0 subscript superscript 𝑈 𝜈 1 subscript superscript 𝑈 𝜈 2 subscript superscript 𝑈 𝜈 3 ⋅ 𝑈 superscript 𝜀 𝜅 𝜆 𝜇 𝜈 missing-subexpression 𝑈 plus-or-minus 1 {\displaystyle{\displaystyle\begin{aligned} &\displaystyle{{\varepsilon}^{% \kappa\lambda\mu\nu}}\acute{\ }={{U}^{\kappa}}_{\alpha}{{U}^{\lambda}}_{\beta}% {{U}^{\mu}}_{\gamma}{{U}^{\nu}}_{\delta}{{\varepsilon}^{\alpha\beta\gamma% \delta}}\\ &\displaystyle=\left|\begin{matrix}{{U}^{\kappa}}_{0}&{{U}^{\kappa}}_{1}&{{U}^% {\kappa}}_{2}&{{U}^{\kappa}}_{3}\\ {{U}^{\lambda}}_{0}&{{U}^{\lambda}}_{1}&{{U}^{\lambda}}_{2}&{{U}^{\lambda}}_{3% }\\ {{U}^{\mu}}_{0}&{{U}^{\mu}}_{1}&{{U}^{\mu}}_{2}&{{U}^{\mu}}_{3}\\ {{U}^{\nu}}_{0}&{{U}^{\nu}}_{1}&{{U}^{\nu}}_{2}&{{U}^{\nu}}_{3}\\ \end{matrix}\right|=\left(\det U\right)\cdot{{\varepsilon}^{\kappa\lambda\mu% \nu}}\\ &\displaystyle\left(\det U\right)=\pm 1\\ \end{aligned}}}

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εκλμν´=UκαUλβUμγUνδεαβγδ=|Uκ0Uκ1Uκ2Uκ3Uλ0Uλ1Uλ2Uλ3Uμ0Uμ1Uμ2Uμ3Uν0Uν1Uν2Uν3|=(det⁡U)⋅εκλμν(det⁡U)=±1

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