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* Page found: Elektrodynamik Schöll (eq math.1199.788)

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\begin{align}
& \int_{\partial V}^{{}}{d\bar{f}}\left( \Phi \left( {\bar{r}} \right)\nabla \tilde{G}\left( \bar{r}-\bar{r}\acute{\ } \right)-\tilde{G}\left( \bar{r}-\bar{r}\acute{\ } \right)\nabla \Phi \left( {\bar{r}} \right) \right)=\int_{V}^{{}}{{{d}^{3}}r}\left( \Phi \left( {\bar{r}} \right)\Delta \tilde{G}\left( \bar{r}-\bar{r}\acute{\ } \right)-\tilde{G}\left( \bar{r}-\bar{r}\acute{\ } \right)\Delta \Phi \left( {\bar{r}} \right) \right) \\
& \Delta \tilde{G}\left( \bar{r}-\bar{r}\acute{\ } \right)=-\delta \left( \bar{r}-\bar{r}\acute{\ } \right)-{{k}^{2}}\tilde{G}\left( \bar{r}-\bar{r}\acute{\ } \right) \\
& \Delta \Phi \left( {\bar{r}} \right)=\frac{-\rho }{{{\varepsilon }_{0}}}-{{k}^{2}}\Phi \left( {\bar{r}} \right) \\
& \Rightarrow \int_{V}^{{}}{{{d}^{3}}r}\left( \Phi \left( {\bar{r}} \right)\Delta \tilde{G}\left( \bar{r}-\bar{r}\acute{\ } \right)-\tilde{G}\left( \bar{r}-\bar{r}\acute{\ } \right)\Delta \Phi \left( {\bar{r}} \right) \right)=-\Phi \left( \bar{r}\acute{\ } \right) \\
& \int_{\partial V}^{{}}{d\bar{f}}\left( \tilde{G}\left( \bar{r}-\bar{r}\acute{\ } \right)\nabla \Phi \left( {\bar{r}} \right)-\Phi \left( {\bar{r}} \right)\nabla \tilde{G}\left( \bar{r}-\bar{r}\acute{\ } \right) \right)=\Phi \left( \bar{r}\acute{\ } \right) \\
\end{align}

TeX (checked):

{\begin{aligned}&\int _{\partial V}^{}{d{\bar {f}}}\left(\Phi \left({\bar {r}}\right)\nabla {\tilde {G}}\left({\bar {r}}-{\bar {r}}{\acute {\ }}\right)-{\tilde {G}}\left({\bar {r}}-{\bar {r}}{\acute {\ }}\right)\nabla \Phi \left({\bar {r}}\right)\right)=\int _{V}^{}{{{d}^{3}}r}\left(\Phi \left({\bar {r}}\right)\Delta {\tilde {G}}\left({\bar {r}}-{\bar {r}}{\acute {\ }}\right)-{\tilde {G}}\left({\bar {r}}-{\bar {r}}{\acute {\ }}\right)\Delta \Phi \left({\bar {r}}\right)\right)\\&\Delta {\tilde {G}}\left({\bar {r}}-{\bar {r}}{\acute {\ }}\right)=-\delta \left({\bar {r}}-{\bar {r}}{\acute {\ }}\right)-{{k}^{2}}{\tilde {G}}\left({\bar {r}}-{\bar {r}}{\acute {\ }}\right)\\&\Delta \Phi \left({\bar {r}}\right)={\frac {-\rho }{{\varepsilon }_{0}}}-{{k}^{2}}\Phi \left({\bar {r}}\right)\\&\Rightarrow \int _{V}^{}{{{d}^{3}}r}\left(\Phi \left({\bar {r}}\right)\Delta {\tilde {G}}\left({\bar {r}}-{\bar {r}}{\acute {\ }}\right)-{\tilde {G}}\left({\bar {r}}-{\bar {r}}{\acute {\ }}\right)\Delta \Phi \left({\bar {r}}\right)\right)=-\Phi \left({\bar {r}}{\acute {\ }}\right)\\&\int _{\partial V}^{}{d{\bar {f}}}\left({\tilde {G}}\left({\bar {r}}-{\bar {r}}{\acute {\ }}\right)\nabla \Phi \left({\bar {r}}\right)-\Phi \left({\bar {r}}\right)\nabla {\tilde {G}}\left({\bar {r}}-{\bar {r}}{\acute {\ }}\right)\right)=\Phi \left({\bar {r}}{\acute {\ }}\right)\\\end{aligned}}

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Vdf¯(Φ(r¯)G~(r¯r¯´)G~(r¯r¯´)Φ(r¯))=Vd3r(Φ(r¯)ΔG~(r¯r¯´)G~(r¯r¯´)ΔΦ(r¯))ΔG~(r¯r¯´)=δ(r¯r¯´)k2G~(r¯r¯´)ΔΦ(r¯)=ρε0k2Φ(r¯)Vd3r(Φ(r¯)ΔG~(r¯r¯´)G~(r¯r¯´)ΔΦ(r¯))=Φ(r¯´)Vdf¯(G~(r¯r¯´)Φ(r¯)Φ(r¯)G~(r¯r¯´))=Φ(r¯´)
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