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* Page found: Elektrodynamik Schöll (eq math.1199.885)

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\begin{align}
& \Phi \left( \bar{r},t \right)=\frac{1}{\Delta V}\int_{\Delta V}^{{}}{{}}{{d}^{3}}s{{\Phi }_{m}}\left( \bar{r}+\bar{s},t \right) \\
& =-\frac{1}{4\pi {{\varepsilon }_{0}}}\frac{1}{\Delta V}\int_{\Delta V}^{{}}{{}}{{d}^{3}}s\int_{{{R}^{3}}}^{{}}{{}}{{d}^{3}}r\acute{\ }{{\nabla }_{r}}\left\{ \frac{1}{\left| \bar{r}+\bar{s}-\bar{r}\acute{\ } \right|}{{{\bar{P}}}_{m}}\left( \bar{r}\acute{\ },t-\frac{\left| \bar{r}+\bar{s}-\bar{r}\acute{\ } \right|}{c} \right) \right\} \\
& =-\frac{1}{4\pi {{\varepsilon }_{0}}}\int_{{{R}^{3}}}^{{}}{{}}{{d}^{3}}r\acute{\ }\acute{\ }{{\nabla }_{r}}\left\{ \frac{1}{\left| \bar{r}-\bar{r}\acute{\ }\acute{\ } \right|}\bar{P}\left( \bar{r}\acute{\ }\acute{\ },t-\frac{\left| \bar{r}-\bar{r}\acute{\ }\acute{\ } \right|}{c} \right) \right\} \\
\end{align}

TeX (checked): 

{\begin{aligned}&\Phi \left({\bar {r}},t\right)={\frac {1}{\Delta V}}\int _{\Delta V}^{}{}{{d}^{3}}s{{\Phi }_{m}}\left({\bar {r}}+{\bar {s}},t\right)\\&=-{\frac {1}{4\pi {{\varepsilon }_{0}}}}{\frac {1}{\Delta V}}\int _{\Delta V}^{}{}{{d}^{3}}s\int _{{R}^{3}}^{}{}{{d}^{3}}r{\acute {\ }}{{\nabla }_{r}}\left\{{\frac {1}{\left|{\bar {r}}+{\bar {s}}-{\bar {r}}{\acute {\ }}\right|}}{{\bar {P}}_{m}}\left({\bar {r}}{\acute {\ }},t-{\frac {\left|{\bar {r}}+{\bar {s}}-{\bar {r}}{\acute {\ }}\right|}{c}}\right)\right\}\\&=-{\frac {1}{4\pi {{\varepsilon }_{0}}}}\int _{{R}^{3}}^{}{}{{d}^{3}}r{\acute {\ }}{\acute {\ }}{{\nabla }_{r}}\left\{{\frac {1}{\left|{\bar {r}}-{\bar {r}}{\acute {\ }}{\acute {\ }}\right|}}{\bar {P}}\left({\bar {r}}{\acute {\ }}{\acute {\ }},t-{\frac {\left|{\bar {r}}-{\bar {r}}{\acute {\ }}{\acute {\ }}\right|}{c}}\right)\right\}\\\end{aligned}}

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Φ(r¯,t)=1ΔV∫ΔVd3sΦm(r¯+s¯,t)=−14πε01ΔV∫ΔVd3s∫R3d3r´∇r{1|r¯+s¯−r¯´|P¯m(r¯´,t−|r¯+s¯−r¯´|c)}=−14πε0∫R3d3r´´∇r{1|r¯−r¯´´|P¯(r¯´´,t−|r¯−r¯´´|c)}
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Calculated based on the variables occurring on the entire Elektrodynamik Schöll page

Identifiers

  • Φ
  • r¯
  • t
  • Δ
  • V
  • Δ
  • V
  • s
  • Φm
  • r¯
  • s¯
  • t
  • π
  • ε0
  • Δ
  • V
  • Δ
  • V
  • s
  • R
  • r
  • ´
  • r
  • r¯
  • s¯
  • r¯
  • ´
  • P¯m
  • r¯
  • ´
  • t
  • r¯
  • s¯
  • r¯
  • ´
  • c
  • π
  • ε0
  • R
  • r
  • ´
  • ´
  • r
  • r¯
  • r¯
  • ´
  • ´
  • P¯
  • r¯
  • ´
  • ´
  • t
  • r¯
  • r¯
  • ´
  • ´
  • c

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