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Display information for equation id:math.2174.25 on revision:2174
* Page found: Transformationsverhalten der Ströme und Felder (eq math.2174.25)
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\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}
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data-mjx-texclass="ORD"><mi>μ</mi></mrow></msub><mo data-mjx-texclass="CLOSE">}</mo></mrow><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mn>0</mn></mtd><mtd><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mi>c</mi></mrow></mfrac></mrow><msub><mi>E</mi><mrow data-mjx-texclass="ORD"><mi>x</mi></mrow></msub></mtd><mtd><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mi>c</mi></mrow></mfrac></mrow><msub><mi>E</mi><mrow data-mjx-texclass="ORD"><mi>y</mi></mrow></msub></mtd><mtd><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mi>c</mi></mrow></mfrac></mrow><msub><mi>E</mi><mrow data-mjx-texclass="ORD"><mi>z</mi></mrow></msub></mtd></mtr><mtr><mtd><mo>−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mi>c</mi></mrow></mfrac></mrow><msub><mi>E</mi><mrow data-mjx-texclass="ORD"><mi>x</mi></mrow></msub></mtd><mtd><mn>0</mn></mtd><mtd><mo>−</mo><msub><mi>B</mi><mrow data-mjx-texclass="ORD"><mi>z</mi></mrow></msub></mtd><mtd><msub><mi>B</mi><mrow data-mjx-texclass="ORD"><mi>y</mi></mrow></msub></mtd></mtr><mtr><mtd><mo>−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mi>c</mi></mrow></mfrac></mrow><msub><mi>E</mi><mrow data-mjx-texclass="ORD"><mi>y</mi></mrow></msub></mtd><mtd><msub><mi>B</mi><mrow data-mjx-texclass="ORD"><mi>z</mi></mrow></msub></mtd><mtd><mn>0</mn></mtd><mtd><mo>−</mo><msub><mi>B</mi><mrow data-mjx-texclass="ORD"><mi>x</mi></mrow></msub></mtd></mtr><mtr><mtd><mo>−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mi>c</mi></mrow></mfrac></mrow><msub><mi>E</mi><mrow data-mjx-texclass="ORD"><mi>z</mi></mrow></msub></mtd><mtd><mo>−</mo><msub><mi>B</mi><mrow data-mjx-texclass="ORD"><mi>y</mi></mrow></msub></mtd><mtd><msub><mi>B</mi><mrow data-mjx-texclass="ORD"><mi>x</mi></mrow></msub></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd></mtd></mtr></mtable></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd><msup><mi>F</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>μ</mi><mi>ν</mi></mrow></mrow></msup><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">{</mo><msup><mi>∂</mi><mrow data-mjx-texclass="ORD"><mi>μ</mi></mrow></msup><msup><mi mathvariant="normal">Φ</mi><mrow data-mjx-texclass="ORD"><mi>ν</mi></mrow></msup><mo>−</mo><msup><mi>∂</mi><mrow data-mjx-texclass="ORD"><mi>ν</mi></mrow></msup><msup><mi mathvariant="normal">Φ</mi><mrow data-mjx-texclass="ORD"><mi>μ</mi></mrow></msup><mo data-mjx-texclass="CLOSE">}</mo></mrow><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mn>0</mn></mtd><mtd><mo>−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mi>c</mi></mrow></mfrac></mrow><msub><mi>E</mi><mrow data-mjx-texclass="ORD"><mi>x</mi></mrow></msub></mtd><mtd><mo>−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mi>c</mi></mrow></mfrac></mrow><msub><mi>E</mi><mrow data-mjx-texclass="ORD"><mi>y</mi></mrow></msub></mtd><mtd><mo>−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mi>c</mi></mrow></mfrac></mrow><msub><mi>E</mi><mrow data-mjx-texclass="ORD"><mi>z</mi></mrow></msub></mtd></mtr><mtr><mtd><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mi>c</mi></mrow></mfrac></mrow><msub><mi>E</mi><mrow data-mjx-texclass="ORD"><mi>x</mi></mrow></msub></mtd><mtd><mn>0</mn></mtd><mtd><mo>−</mo><msub><mi>B</mi><mrow data-mjx-texclass="ORD"><mi>z</mi></mrow></msub></mtd><mtd><msub><mi>B</mi><mrow data-mjx-texclass="ORD"><mi>y</mi></mrow></msub></mtd></mtr><mtr><mtd><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mi>c</mi></mrow></mfrac></mrow><msub><mi>E</mi><mrow data-mjx-texclass="ORD"><mi>y</mi></mrow></msub></mtd><mtd><msub><mi>B</mi><mrow data-mjx-texclass="ORD"><mi>z</mi></mrow></msub></mtd><mtd><mn>0</mn></mtd><mtd><mo>−</mo><msub><mi>B</mi><mrow data-mjx-texclass="ORD"><mi>x</mi></mrow></msub></mtd></mtr><mtr><mtd><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mi>c</mi></mrow></mfrac></mrow><msub><mi>E</mi><mrow data-mjx-texclass="ORD"><mi>z</mi></mrow></msub></mtd><mtd><mo>−</mo><msub><mi>B</mi><mrow data-mjx-texclass="ORD"><mi>y</mi></mrow></msub></mtd><mtd><msub><mi>B</mi><mrow data-mjx-texclass="ORD"><mi>x</mi></mrow></msub></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd></mtd></mtr></mtable></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>⇔</mo><msup><mi>F</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>μ</mi><mi>ν</mi></mrow></mrow></msup><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">{</mo><msup><mi>∂</mi><mrow data-mjx-texclass="ORD"><mi>μ</mi></mrow></msup><msup><mi mathvariant="normal">Φ</mi><mrow data-mjx-texclass="ORD"><mi>ν</mi></mrow></msup><mo>−</mo><msup><mi>∂</mi><mrow data-mjx-texclass="ORD"><mi>ν</mi></mrow></msup><msup><mi mathvariant="normal">Φ</mi><mrow data-mjx-texclass="ORD"><mi>μ</mi></mrow></msup><mo data-mjx-texclass="CLOSE">}</mo></mrow><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mn>0</mn></mtd><mtd><mo>−</mo><msup><mi>E</mi><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msup></mtd><mtd><mo>−</mo><msup><mi>E</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></mtd><mtd><mo>−</mo><msup><mi>E</mi><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msup></mtd></mtr><mtr><mtd><msup><mi>E</mi><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msup></mtd><mtd><mn>0</mn></mtd><mtd><mo>−</mo><mi>c</mi><msup><mi>B</mi><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msup></mtd><mtd><mi>c</mi><msup><mi>B</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></mtd></mtr><mtr><mtd><msup><mi>E</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></mtd><mtd><mi>c</mi><msup><mi>B</mi><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msup></mtd><mtd><mn>0</mn></mtd><mtd><mo>−</mo><mi>c</mi><msup><mi>B</mi><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msup></mtd></mtr><mtr><mtd><msup><mi>E</mi><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msup></mtd><mtd><mo>−</mo><mi>c</mi><msup><mi>B</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></mtd><mtd><mi>c</mi><msup><mi>B</mi><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msup></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd></mtd></mtr></mtable></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd></mtr><mtr><mtd></mtd></mtr></mtable></mrow></mstyle></mrow></math>
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