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Display information for equation id:math.1802.38 on revision:1802
* Page found: Kovariante Schreibweise der Relativitätstheorie (eq math.1802.38)
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Hash: c7ae09ae90baa820aa55532aa6a58e02
TeX (original user input):
\begin{align}
& {{g}^{ik}}:={{\delta }^{ik}}={{\delta }^{i}}_{k}\quad f\ddot{u}r\ k=0 \\
& {{g}^{ik}}:={{\delta }^{ik}}=-{{\delta }^{i}}_{k}\quad f\ddot{u}r\ k=1,2,3 \\
& {{g}^{ik}}:={{\delta }^{ik}}=\left( \begin{matrix}
1 & {} & {} & {} \\
{} & -1 & {} & {} \\
{} & {} & -1 & {} \\
{} & {} & {} & -1 \\
\end{matrix} \right)={{g}_{ik}} \\
& {{g}^{ik}}{{a}_{k}}={{\delta }^{ik}}{{a}_{k}}={{a}_{i}}\quad f\ddot{u}r\ i=0\Rightarrow {{a}_{i}}={{a}^{i}} \\
& {{g}^{ik}}{{a}_{k}}={{\delta }^{ik}}{{a}_{k}}=-{{a}_{i}}\quad f\ddot{u}r\ i=1,2,3\Rightarrow -{{a}_{i}}={{a}^{i}} \\
\end{align}
TeX (checked):
{\begin{aligned}&{{g}^{ik}}:={{\delta }^{ik}}={{\delta }^{i}}_{k}\quad f{\ddot {u}}r\ k=0\\&{{g}^{ik}}:={{\delta }^{ik}}=-{{\delta }^{i}}_{k}\quad f{\ddot {u}}r\ k=1,2,3\\&{{g}^{ik}}:={{\delta }^{ik}}=\left({\begin{matrix}1&{}&{}&{}\\{}&-1&{}&{}\\{}&{}&-1&{}\\{}&{}&{}&-1\\\end{matrix}}\right)={{g}_{ik}}\\&{{g}^{ik}}{{a}_{k}}={{\delta }^{ik}}{{a}_{k}}={{a}_{i}}\quad f{\ddot {u}}r\ i=0\Rightarrow {{a}_{i}}={{a}^{i}}\\&{{g}^{ik}}{{a}_{k}}={{\delta }^{ik}}{{a}_{k}}=-{{a}_{i}}\quad f{\ddot {u}}r\ i=1,2,3\Rightarrow -{{a}_{i}}={{a}^{i}}\\\end{aligned}}
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<math class="mwe-math-element" xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow data-mjx-texclass="ORD"><mtable columnalign="right left right left right left right left right left right left" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true" rowspacing="3pt"><mtr><mtd></mtd><mtd><msup><mi>g</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>k</mi></mrow></mrow></msup><mi>:</mi><mo>=</mo><msup><mi>δ</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>k</mi></mrow></mrow></msup><mo>=</mo><msub><msup><mi>δ</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msup><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub><mspace width="1em"></mspace><mi>f</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>u</mi><mo>¨</mo></mover></mrow></mrow><mi>r</mi><mspace width="0.5em"/><mi>k</mi><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd></mtd><mtd><msup><mi>g</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>k</mi></mrow></mrow></msup><mi>:</mi><mo>=</mo><msup><mi>δ</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>k</mi></mrow></mrow></msup><mo>=</mo><mo>−</mo><msub><msup><mi>δ</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msup><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub><mspace width="1em"></mspace><mi>f</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>u</mi><mo>¨</mo></mover></mrow></mrow><mi>r</mi><mspace width="0.5em"/><mi>k</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn></mtd></mtr><mtr><mtd></mtd><mtd><msup><mi>g</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>k</mi></mrow></mrow></msup><mi>:</mi><mo>=</mo><msup><mi>δ</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>k</mi></mrow></mrow></msup><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>1</mn></mtd><mtd></mtd><mtd></mtd><mtd></mtd></mtr><mtr><mtd></mtd><mtd><mo>−</mo><mn>1</mn></mtd><mtd></mtd><mtd></mtd></mtr><mtr><mtd></mtd><mtd></mtd><mtd><mo>−</mo><mn>1</mn></mtd><mtd></mtd></mtr><mtr><mtd></mtd><mtd></mtd><mtd></mtd><mtd><mo>−</mo><mn>1</mn></mtd></mtr><mtr><mtd></mtd></mtr></mtable></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>=</mo><msub><mi>g</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>k</mi></mrow></mrow></msub></mtd></mtr><mtr><mtd></mtd><mtd><msup><mi>g</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>k</mi></mrow></mrow></msup><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub><mo>=</mo><msup><mi>δ</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>k</mi></mrow></mrow></msup><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub><mo>=</mo><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mspace width="1em"></mspace><mi>f</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>u</mi><mo>¨</mo></mover></mrow></mrow><mi>r</mi><mspace width="0.5em"/><mi>i</mi><mo>=</mo><mn>0</mn><mo>⇒</mo><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mo>=</mo><msup><mi>a</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msup></mtd></mtr><mtr><mtd></mtd><mtd><msup><mi>g</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>k</mi></mrow></mrow></msup><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub><mo>=</mo><msup><mi>δ</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>k</mi></mrow></mrow></msup><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub><mo>=</mo><mo>−</mo><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mspace width="1em"></mspace><mi>f</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>u</mi><mo>¨</mo></mover></mrow></mrow><mi>r</mi><mspace width="0.5em"/><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>⇒</mo><mo>−</mo><msub><mi>a</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mo>=</mo><msup><mi>a</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msup></mtd></mtr><mtr><mtd></mtd></mtr></mtable></mrow></mstyle></mrow></math>
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