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Display information for equation id:math.2077.11 on revision:2077

* Page found: Poisson- Gleichung und Greensche Funktion (eq math.2077.11)

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Poisson- Gleichung und Greensche Funktion Eq: math.2077.11

Hash: f1176d804f3ff504ad1c60fcc1d84e69

TeX (original user input):

\begin{align}
& {{\Delta }_{r}}\frac{1}{|\bar{r}-\bar{r}\acute{\ }|}={{\nabla }_{S}}\left( {{\nabla }_{S}}\frac{1}{s} \right)=-{{\nabla }_{S}}\frac{1}{{{s}^{2}}}\frac{{\bar{s}}}{s}=-\frac{1}{{{s}^{3}}}{{\nabla }_{S}}\bar{s}-\bar{s}{{\nabla }_{S}}\frac{1}{{{s}^{3}}} \\
& {{\nabla }_{S}}\bar{s}=3 \\
& \Rightarrow {{\Delta }_{r}}\frac{1}{|\bar{r}-\bar{r}\acute{\ }|}=-\frac{1}{{{s}^{3}}}{{\nabla }_{S}}\bar{s}-\bar{s}{{\nabla }_{S}}\frac{1}{{{s}^{3}}}=-\frac{3}{{{s}^{3}}}+\frac{1}{{{s}^{3}}}=0 \\
\end{align}

TeX (checked):

{\begin{aligned}&{{\Delta }_{r}}{\frac {1}{|{\bar {r}}-{\bar {r}}{\acute {\ }}|}}={{\nabla }_{S}}\left({{\nabla }_{S}}{\frac {1}{s}}\right)=-{{\nabla }_{S}}{\frac {1}{{s}^{2}}}{\frac {\bar {s}}{s}}=-{\frac {1}{{s}^{3}}}{{\nabla }_{S}}{\bar {s}}-{\bar {s}}{{\nabla }_{S}}{\frac {1}{{s}^{3}}}\\&{{\nabla }_{S}}{\bar {s}}=3\\&\Rightarrow {{\Delta }_{r}}{\frac {1}{|{\bar {r}}-{\bar {r}}{\acute {\ }}|}}=-{\frac {1}{{s}^{3}}}{{\nabla }_{S}}{\bar {s}}-{\bar {s}}{{\nabla }_{S}}{\frac {1}{{s}^{3}}}=-{\frac {3}{{s}^{3}}}+{\frac {1}{{s}^{3}}}=0\\\end{aligned}}

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\begin{aligned} Δ_{r} \frac{1}{| \bar{r} - \bar{r} \overset{´}{} |} = \nabla_{S} (\nabla_{S} \frac{1}{s}) = - \nabla_{S} \frac{1}{s^{2}} \frac{\bar{s}}{s} = - \frac{1}{s^{3}} \nabla_{S} \bar{s} - \bar{s} \nabla_{S} \frac{1}{s^{3}} \\ \nabla_{S} \bar{s} = 3 \\ \Rightarrow Δ_{r} \frac{1}{| \bar{r} - \bar{r} \overset{´}{} |} = - \frac{1}{s^{3}} \nabla_{S} \bar{s} - \bar{s} \nabla_{S} \frac{1}{s^{3}} = - \frac{3}{s^{3}} + \frac{1}{s^{3}} = 0 \end{aligned}

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Identifiers

$Δ_{r}$

$\bar{r}$

$\bar{r}$

$\overset{´}{}$

$S$

$S$

$s$

$S$

$s$

$\bar{s}$

$s$

$s$

$S$

$\bar{s}$

$\bar{s}$

$S$

$s$

$S$

$\bar{s}$

$Δ_{r}$

$\bar{r}$

$\bar{r}$

$\overset{´}{}$

$s$

$S$

$\bar{s}$

$\bar{s}$

$S$

$s$

$s$

$s$

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