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Display information for equation id:math.2077.19 on revision:2077

* Page found: Poisson- Gleichung und Greensche Funktion (eq math.2077.19)

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Poisson- Gleichung und Greensche Funktion Eq: math.2077.19

Hash: 0f5579b2ddcea579f2e04fc060a6f8e6

TeX (original user input):

\begin{align}
& \int_{V}^{{}}{{{d}^{3}}}r\Delta \Phi (\bar{r})=\frac{1}{4\pi {{\varepsilon }_{0}}}\int_{{{R}^{3}}}^{{}}{{{d}^{3}}r\acute{\ }}\rho \left( \bar{r}\acute{\ } \right)\oint\limits_{\partial V}{d\bar{f}\cdot {{\nabla }_{r}}}\frac{1}{|\bar{r}-\bar{r}\acute{\ }|} \\
& {{\nabla }_{r}}\frac{1}{|\bar{r}-\bar{r}\acute{\ }|}=-\frac{\left( \bar{r}-\bar{r}\acute{\ } \right)}{|\bar{r}-\bar{r}\acute{\ }{{|}^{3}}} \\
\end{align}

TeX (checked):

{\begin{aligned}&\int _{V}^{}{{d}^{3}}r\Delta \Phi ({\bar {r}})={\frac {1}{4\pi {{\varepsilon }_{0}}}}\int _{{R}^{3}}^{}{{{d}^{3}}r{\acute {\ }}}\rho \left({\bar {r}}{\acute {\ }}\right)\oint \limits _{\partial V}{d{\bar {f}}\cdot {{\nabla }_{r}}}{\frac {1}{|{\bar {r}}-{\bar {r}}{\acute {\ }}|}}\\&{{\nabla }_{r}}{\frac {1}{|{\bar {r}}-{\bar {r}}{\acute {\ }}|}}=-{\frac {\left({\bar {r}}-{\bar {r}}{\acute {\ }}\right)}{|{\bar {r}}-{\bar {r}}{\acute {\ }}{{|}^{3}}}}\\\end{aligned}}

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\begin{aligned} \int_{V}^{} d^{3} r Δ Φ (\bar{r}) = \frac{1}{4 π ε_{0}} \int_{R^{3}}^{} d^{3} r \overset{´}{} ρ (\bar{r} \overset{´}{}) \oint_{\partial V} d \bar{f} \cdot \nabla_{r} \frac{1}{| \bar{r} - \bar{r} \overset{´}{} |} \\ \nabla_{r} \frac{1}{| \bar{r} - \bar{r} \overset{´}{} |} = - \frac{(\bar{r} - \bar{r} \overset{´}{})}{| \bar{r} - \bar{r} \overset{´}{} |^{3}} \end{aligned}

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Identifiers

$V$

$r$

$Δ$

$Φ$

$\bar{r}$

$π$

$ε_{0}$

$R$

$r$

$\overset{´}{}$

$ρ$

$\bar{r}$

$\overset{´}{}$

$V$

$d$

$\bar{f}$

$r$

$\bar{r}$

$\bar{r}$

$\overset{´}{}$

$r$

$\bar{r}$

$\bar{r}$

$\overset{´}{}$

$\bar{r}$

$\bar{r}$

$\overset{´}{}$

$\bar{r}$

$\bar{r}$

$\overset{´}{}$

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