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* Page found: Elektromagnetische Wellen (eq math.1439.227)

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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¯)=−ρε0−k2Φ(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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  •  ´
  • Φ
  • r¯
  • V
  • r
  • Φ
  • r¯
  • Δ
  • G~
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  •  ´
  • G~
  • r¯
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  • Δ
  • G~
  • r¯
  • r¯
  •  ´
  • δ
  • r¯
  • r¯
  •  ´
  • k
  • G~
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  •  ´
  • Δ
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