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* Page found: Elektrostatik (eq math.1430.237)

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TeX (original user input):

\begin{align}
& \Phi (\bar{r}\acute{\ }){{\left. {} \right|}_{\bar{r}\acute{\ }\in S\beta }}=-{{\varepsilon }_{0}}\int_{\partial V}^{{}}{{}}d\bar{f}\cdot \Phi (\bar{r}){{\nabla }_{r}}G\left( \bar{r}-\bar{r}\acute{\ } \right){{\left. {} \right|}_{\bar{r}\acute{\ }\in S\beta }} \\
& =-{{\varepsilon }_{0}}\left[ \int_{\partial V}^{{}}{{}}d\bar{f}G\left( \bar{r}-\bar{r}\acute{\ } \right){{\left. {} \right|}_{\bar{r}\acute{\ }\in S\beta }}\cdot {{\nabla }_{r}}\Phi (\bar{r})+\int_{V}^{{}}{{{d}^{3}}r\left( \Phi (\bar{r}){{\left. {} \right|}_{\bar{r}\acute{\ }\in S\beta }}{{\Delta }_{r}}G\left( \bar{r}-\bar{r}\acute{\ } \right)-G\left( \bar{r}-\bar{r}\acute{\ } \right){{\left. {} \right|}_{\bar{r}\acute{\ }\in S\beta }}{{\Delta }_{r}}\Phi (\bar{r}) \right)} \right] \\
& G\left( \bar{r}-\bar{r}\acute{\ } \right){{\left. {} \right|}_{\bar{r}\acute{\ }\in S\beta }}=0 \\
& \Rightarrow \Phi (\bar{r}\acute{\ }){{\left. {} \right|}_{\bar{r}\acute{\ }\in S\beta }}=-{{\varepsilon }_{0}}\int_{V}^{{}}{{{d}^{3}}r\left( \Phi (\bar{r}){{\left. {} \right|}_{\bar{r}\acute{\ }\in S\beta }}{{\Delta }_{r}}G\left( \bar{r}-\bar{r}\acute{\ } \right) \right)=}\int_{V}^{{}}{{{d}^{3}}r\left( \Phi (\bar{r}){{\left. {} \right|}_{\bar{r}\acute{\ }\in S\beta }}\left( -\frac{1}{{{\varepsilon }_{0}}}\delta \left( \bar{r}-\bar{r}\acute{\ } \right) \right) \right)} \\
& =\int_{V}^{{}}{{{d}^{3}}r\left( \Phi (\bar{r}){{\left. {} \right|}_{\bar{r}\acute{\ }\in S\beta }}\delta \left( \bar{r}-\bar{r}\acute{\ } \right) \right)}=\Phi (\bar{r}\acute{\ }){{\left. {} \right|}_{\bar{r}\acute{\ }\in S\beta }}={{\Phi }_{\beta }} \\
\end{align}

TeX (checked):

{\begin{aligned}&\Phi ({\bar {r}}{\acute {\ }}){{\left.{}\right|}_{{\bar {r}}{\acute {\ }}\in S\beta }}=-{{\varepsilon }_{0}}\int _{\partial V}^{}{}d{\bar {f}}\cdot \Phi ({\bar {r}}){{\nabla }_{r}}G\left({\bar {r}}-{\bar {r}}{\acute {\ }}\right){{\left.{}\right|}_{{\bar {r}}{\acute {\ }}\in S\beta }}\\&=-{{\varepsilon }_{0}}\left[\int _{\partial V}^{}{}d{\bar {f}}G\left({\bar {r}}-{\bar {r}}{\acute {\ }}\right){{\left.{}\right|}_{{\bar {r}}{\acute {\ }}\in S\beta }}\cdot {{\nabla }_{r}}\Phi ({\bar {r}})+\int _{V}^{}{{{d}^{3}}r\left(\Phi ({\bar {r}}){{\left.{}\right|}_{{\bar {r}}{\acute {\ }}\in S\beta }}{{\Delta }_{r}}G\left({\bar {r}}-{\bar {r}}{\acute {\ }}\right)-G\left({\bar {r}}-{\bar {r}}{\acute {\ }}\right){{\left.{}\right|}_{{\bar {r}}{\acute {\ }}\in S\beta }}{{\Delta }_{r}}\Phi ({\bar {r}})\right)}\right]\\&G\left({\bar {r}}-{\bar {r}}{\acute {\ }}\right){{\left.{}\right|}_{{\bar {r}}{\acute {\ }}\in S\beta }}=0\\&\Rightarrow \Phi ({\bar {r}}{\acute {\ }}){{\left.{}\right|}_{{\bar {r}}{\acute {\ }}\in S\beta }}=-{{\varepsilon }_{0}}\int _{V}^{}{{{d}^{3}}r\left(\Phi ({\bar {r}}){{\left.{}\right|}_{{\bar {r}}{\acute {\ }}\in S\beta }}{{\Delta }_{r}}G\left({\bar {r}}-{\bar {r}}{\acute {\ }}\right)\right)=}\int _{V}^{}{{{d}^{3}}r\left(\Phi ({\bar {r}}){{\left.{}\right|}_{{\bar {r}}{\acute {\ }}\in S\beta }}\left(-{\frac {1}{{\varepsilon }_{0}}}\delta \left({\bar {r}}-{\bar {r}}{\acute {\ }}\right)\right)\right)}\\&=\int _{V}^{}{{{d}^{3}}r\left(\Phi ({\bar {r}}){{\left.{}\right|}_{{\bar {r}}{\acute {\ }}\in S\beta }}\delta \left({\bar {r}}-{\bar {r}}{\acute {\ }}\right)\right)}=\Phi ({\bar {r}}{\acute {\ }}){{\left.{}\right|}_{{\bar {r}}{\acute {\ }}\in S\beta }}={{\Phi }_{\beta }}\\\end{aligned}}

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\begin{matrix} Φ (\bar{r} \overset{´}{}) |_{\bar{r} \overset{´}{} \in S β} = - ε_{0} \int_{\partial V}^{} d \bar{f} \cdot Φ (\bar{r}) \nabla_{r} G (\bar{r} - \bar{r} \overset{´}{}) |_{\bar{r} \overset{´}{} \in S β} \\ = - ε_{0} [\int_{\partial V}^{} d \bar{f} G (\bar{r} - \bar{r} \overset{´}{}) |_{\bar{r} \overset{´}{} \in S β} \cdot \nabla_{r} Φ (\bar{r}) + \int_{V}^{} d^{3} r (Φ (\bar{r}) |_{\bar{r} \overset{´}{} \in S β} Δ_{r} G (\bar{r} - \bar{r} \overset{´}{}) - G (\bar{r} - \bar{r} \overset{´}{}) |_{\bar{r} \overset{´}{} \in S β} Δ_{r} Φ (\bar{r}))] \\ G (\bar{r} - \bar{r} \overset{´}{}) |_{\bar{r} \overset{´}{} \in S β} = 0 \\ \Rightarrow Φ (\bar{r} \overset{´}{}) |_{\bar{r} \overset{´}{} \in S β} = - ε_{0} \int_{V}^{} d^{3} r (Φ (\bar{r}) |_{\bar{r} \overset{´}{} \in S β} Δ_{r} G (\bar{r} - \bar{r} \overset{´}{})) = \int_{V}^{} d^{3} r (Φ (\bar{r}) |_{\bar{r} \overset{´}{} \in S β} (- \frac{1}{ε_{0}} δ (\bar{r} - \bar{r} \overset{´}{}))) \\ = \int_{V}^{} d^{3} r (Φ (\bar{r}) |_{\bar{r} \overset{´}{} \in S β} δ (\bar{r} - \bar{r} \overset{´}{})) = Φ (\bar{r} \overset{´}{}) |_{\bar{r} \overset{´}{} \in S β} = Φ_{β} \end{matrix}

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Identifiers

$Φ$

$\bar{r}$

$\overset{´}{}$

$\bar{r}$

$\overset{´}{}$

$S$

$β$

$ε_{0}$

$V$

$\bar{f}$

$Φ$

$\bar{r}$

$r$

$G$

$\bar{r}$

$\bar{r}$

$\overset{´}{}$

$\bar{r}$

$\overset{´}{}$

$S$

$β$

$ε_{0}$

$V$

$\bar{f}$

$G$

$\bar{r}$

$\bar{r}$

$\overset{´}{}$

$\bar{r}$

$\overset{´}{}$

$S$

$β$

$r$

$Φ$

$\bar{r}$

$V$

$r$

$Φ$

$\bar{r}$

$\bar{r}$

$\overset{´}{}$

$S$

$β$

$Δ_{r}$

$G$

$\bar{r}$

$\bar{r}$

$\overset{´}{}$

$G$

$\bar{r}$

$\bar{r}$

$\overset{´}{}$

$\bar{r}$

$\overset{´}{}$

$S$

$β$

$Δ_{r}$

$Φ$

$\bar{r}$

$G$

$\bar{r}$

$\bar{r}$

$\overset{´}{}$

$\bar{r}$

$\overset{´}{}$

$S$

$β$

$Φ$

$\bar{r}$

$\overset{´}{}$

$\bar{r}$

$\overset{´}{}$

$S$

$β$

$ε_{0}$

$V$

$r$

$Φ$

$\bar{r}$

$\bar{r}$

$\overset{´}{}$

$S$

$β$

$Δ_{r}$

$G$

$\bar{r}$

$\bar{r}$

$\overset{´}{}$

$V$

$r$

$Φ$

$\bar{r}$

$\bar{r}$

$\overset{´}{}$

$S$

$β$

$ε_{0}$

$δ$

$\bar{r}$

$\bar{r}$

$\overset{´}{}$

$V$

$r$

$Φ$

$\bar{r}$

$\bar{r}$

$\overset{´}{}$

$S$

$β$

$δ$

$\bar{r}$

$\bar{r}$

$\overset{´}{}$

$Φ$

$\bar{r}$

$\overset{´}{}$

$\bar{r}$

$\overset{´}{}$

$S$

$β$

$Φ_{β}$

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