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Display information for equation id:math.1411.75 on revision:1411
* Page found: Symmetrien und Erhaltungsgrößen (eq math.1411.75)
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\begin{align}
& \frac{\partial }{\partial \phi }{{r}_{ij}}=\frac{\partial }{\partial \phi }{{\left[ \left( {{{\bar{r}}}_{i}}-{{{\bar{r}}}_{j}} \right)\left( {{{\bar{r}}}_{i}}-{{{\bar{r}}}_{j}} \right) \right]}^{1/2}}=\frac{1}{{{r}_{ij}}}\left( {{{\bar{r}}}_{i}}-{{{\bar{r}}}_{j}} \right)\frac{\partial }{\partial \phi }\left( {{{\bar{r}}}_{i}}-{{{\bar{r}}}_{j}} \right)=\frac{{{{\bar{r}}}_{i}}-{{{\bar{r}}}_{j}}}{{{r}_{ij}}}\left( \frac{\partial }{\partial \phi }{{{\bar{r}}}_{i}}-\frac{\partial }{\partial \phi }{{{\bar{r}}}_{j}} \right) \\
& \frac{\partial }{\partial \phi }{{{\bar{r}}}_{i}}={{{\bar{r}}}_{i}}\times {{{\bar{e}}}_{k}} \\
& \Rightarrow \frac{{{{\bar{r}}}_{i}}-{{{\bar{r}}}_{j}}}{{{r}_{ij}}}\left( \frac{\partial }{\partial \phi }{{{\bar{r}}}_{i}}-\frac{\partial }{\partial \phi }{{{\bar{r}}}_{j}} \right)=\frac{{{{\bar{r}}}_{i}}-{{{\bar{r}}}_{j}}}{{{r}_{ij}}}\left[ \left( {{{\bar{r}}}_{i}}-{{{\bar{r}}}_{j}} \right)\times {{{\bar{e}}}_{k}} \right]=\frac{1}{{{r}_{ij}}}{{{\bar{e}}}_{k}}\left[ \left( {{{\bar{r}}}_{i}}-{{{\bar{r}}}_{j}} \right)\times \left( {{{\bar{r}}}_{i}}-{{{\bar{r}}}_{j}} \right) \right]=0 \\
\end{align}
TeX (checked):
{\begin{aligned}&{\frac {\partial }{\partial \phi }}{{r}_{ij}}={\frac {\partial }{\partial \phi }}{{\left[\left({{\bar {r}}_{i}}-{{\bar {r}}_{j}}\right)\left({{\bar {r}}_{i}}-{{\bar {r}}_{j}}\right)\right]}^{1/2}}={\frac {1}{{r}_{ij}}}\left({{\bar {r}}_{i}}-{{\bar {r}}_{j}}\right){\frac {\partial }{\partial \phi }}\left({{\bar {r}}_{i}}-{{\bar {r}}_{j}}\right)={\frac {{{\bar {r}}_{i}}-{{\bar {r}}_{j}}}{{r}_{ij}}}\left({\frac {\partial }{\partial \phi }}{{\bar {r}}_{i}}-{\frac {\partial }{\partial \phi }}{{\bar {r}}_{j}}\right)\\&{\frac {\partial }{\partial \phi }}{{\bar {r}}_{i}}={{\bar {r}}_{i}}\times {{\bar {e}}_{k}}\\&\Rightarrow {\frac {{{\bar {r}}_{i}}-{{\bar {r}}_{j}}}{{r}_{ij}}}\left({\frac {\partial }{\partial \phi }}{{\bar {r}}_{i}}-{\frac {\partial }{\partial \phi }}{{\bar {r}}_{j}}\right)={\frac {{{\bar {r}}_{i}}-{{\bar {r}}_{j}}}{{r}_{ij}}}\left[\left({{\bar {r}}_{i}}-{{\bar {r}}_{j}}\right)\times {{\bar {e}}_{k}}\right]={\frac {1}{{r}_{ij}}}{{\bar {e}}_{k}}\left[\left({{\bar {r}}_{i}}-{{\bar {r}}_{j}}\right)\times \left({{\bar {r}}_{i}}-{{\bar {r}}_{j}}\right)\right]=0\\\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><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>∂</mi></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>ϕ</mi></mrow></mrow></mfrac></mrow><msub><mi>r</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>j</mi></mrow></mrow></msub><mo>=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>∂</mi></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>ϕ</mi></mrow></mrow></mfrac></mrow><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mo>−</mo><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>j</mi></mrow></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mo>−</mo><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>j</mi></mrow></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo data-mjx-texclass="CLOSE">]</mo></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>1</mn><mo>/</mo><mn>2</mn></mrow></mrow></msup><mo>=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><msub><mi>r</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>j</mi></mrow></mrow></msub></mrow></mfrac></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mo>−</mo><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>j</mi></mrow></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>∂</mi></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>ϕ</mi></mrow></mrow></mfrac></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mo>−</mo><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>j</mi></mrow></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mo>−</mo><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>j</mi></mrow></msub></mrow></mrow><mrow data-mjx-texclass="ORD"><msub><mi>r</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>j</mi></mrow></mrow></msub></mrow></mfrac></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>∂</mi></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>ϕ</mi></mrow></mrow></mfrac></mrow><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mo>−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>∂</mi></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>ϕ</mi></mrow></mrow></mfrac></mrow><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>j</mi></mrow></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>∂</mi></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>ϕ</mi></mrow></mrow></mfrac></mrow><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mo>=</mo><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mo>×</mo><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>e</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub></mtd></mtr><mtr><mtd></mtd><mtd><mo>⇒</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mo>−</mo><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>j</mi></mrow></msub></mrow></mrow><mrow data-mjx-texclass="ORD"><msub><mi>r</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>j</mi></mrow></mrow></msub></mrow></mfrac></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>∂</mi></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>ϕ</mi></mrow></mrow></mfrac></mrow><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mo>−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>∂</mi></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>ϕ</mi></mrow></mrow></mfrac></mrow><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>j</mi></mrow></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mo>−</mo><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>j</mi></mrow></msub></mrow></mrow><mrow data-mjx-texclass="ORD"><msub><mi>r</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>j</mi></mrow></mrow></msub></mrow></mfrac></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mo>−</mo><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>j</mi></mrow></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>×</mo><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>e</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo>=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><msub><mi>r</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>j</mi></mrow></mrow></msub></mrow></mfrac></mrow><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>e</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mo>−</mo><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>j</mi></mrow></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>×</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mo>−</mo><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>j</mi></mrow></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd></mtd></mtr></mtable></mrow></mstyle></mrow></math>
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