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Display information for equation id:math.1199.238 on revision:1199
* Page found: Elektrodynamik Schöll (eq math.1199.238)
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\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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data-mjx-texclass="ORD"><mover><mspace width="0.5em"/><mo data-mjx-pseudoscript="true">´</mo></mover></mrow></mrow><mo>∈</mo><mi>S</mi><mi>β</mi></mrow></mrow></msub><mi>δ</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mo>−</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mspace width="0.5em"/><mo data-mjx-pseudoscript="true">´</mo></mover></mrow></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow></mrow><mo>=</mo><mi mathvariant="normal">Φ</mi><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mspace width="0.5em"/><mo data-mjx-pseudoscript="true">´</mo></mover></mrow></mrow><mo stretchy="false">)</mo><msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN"></mo><mo data-mjx-texclass="CLOSE">|</mo></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mspace width="0.5em"/><mo data-mjx-pseudoscript="true">´</mo></mover></mrow></mrow><mo>∈</mo><mi>S</mi><mi>β</mi></mrow></mrow></msub><mo>=</mo><msub><mi mathvariant="normal">Φ</mi><mrow data-mjx-texclass="ORD"><mi>β</mi></mrow></msub></mtd></mtr><mtr><mtd></mtd></mtr></mtable></mrow></mstyle></mrow></math>
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