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Display information for equation id:math.2077.19 on revision:2077

* Page found: Poisson- Gleichung und Greensche Funktion (eq math.2077.19)

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TeX (original user input): 

\begin{align}
& \int_{V}^{{}}{{{d}^{3}}}r\Delta \Phi (\bar{r})=\frac{1}{4\pi {{\varepsilon }_{0}}}\int_{{{R}^{3}}}^{{}}{{{d}^{3}}r\acute{\ }}\rho \left( \bar{r}\acute{\ } \right)\oint\limits_{\partial V}{d\bar{f}\cdot {{\nabla }_{r}}}\frac{1}{|\bar{r}-\bar{r}\acute{\ }|} \\
& {{\nabla }_{r}}\frac{1}{|\bar{r}-\bar{r}\acute{\ }|}=-\frac{\left( \bar{r}-\bar{r}\acute{\ } \right)}{|\bar{r}-\bar{r}\acute{\ }{{|}^{3}}} \\
\end{align}

TeX (checked): 

{\begin{aligned}&\int _{V}^{}{{d}^{3}}r\Delta \Phi ({\bar {r}})={\frac {1}{4\pi {{\varepsilon }_{0}}}}\int _{{R}^{3}}^{}{{{d}^{3}}r{\acute {\ }}}\rho \left({\bar {r}}{\acute {\ }}\right)\oint \limits _{\partial V}{d{\bar {f}}\cdot {{\nabla }_{r}}}{\frac {1}{|{\bar {r}}-{\bar {r}}{\acute {\ }}|}}\\&{{\nabla }_{r}}{\frac {1}{|{\bar {r}}-{\bar {r}}{\acute {\ }}|}}=-{\frac {\left({\bar {r}}-{\bar {r}}{\acute {\ }}\right)}{|{\bar {r}}-{\bar {r}}{\acute {\ }}{{|}^{3}}}}\\\end{aligned}}

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MathML (38.344 KB / 4.326 KB) :

∫ V d 3 ⁢ r ⁢ Δ ⁢ Φ ⁢ ( r ¯ ) = 1 4 ⁢ π ⁢ ε 0 ⁢ ∫ R 3 d 3 ⁢ r ⁢ ´ ⁢ ρ ⁢ ( r ¯ ⁢ ´ ) ⁢ ∮ ∂ ⁡ V 𝑑 f ¯ ⋅ ∇ r ⁡ 1 | r ¯ - r ¯ ⁢ ´ | ∇ r ⁡ 1 | r ¯ - r ¯ ⁢ ´ | = - ( r ¯ - r ¯ ⁢ ´ ) | r ¯ - r ¯ ⁢ ´ | 3 missing-subexpression subscript 𝑉 superscript 𝑑 3 𝑟 Δ Φ ¯ 𝑟 1 4 𝜋 subscript 𝜀 0 subscript superscript 𝑅 3 superscript 𝑑 3 𝑟 ´ absent 𝜌 ¯ 𝑟 ´ absent subscript contour-integral 𝑉 ⋅ differential-d ¯ 𝑓 subscript ∇ 𝑟 1 ¯ 𝑟 ¯ 𝑟 ´ absent missing-subexpression subscript ∇ 𝑟 1 ¯ 𝑟 ¯ 𝑟 ´ absent ¯ 𝑟 ¯ 𝑟 ´ absent superscript ¯ 𝑟 ¯ 𝑟 ´ absent 3 {\displaystyle{\displaystyle\begin{aligned} &\displaystyle\int_{V}{{{d}^{3}}}r% \Delta\Phi(\bar{r})=\frac{1}{4\pi{{\varepsilon}_{0}}}\int_{{{R}^{3}}}{{{d}^{3}% }r\acute{\ }}\rho\left(\bar{r}\acute{\ }\right)\oint\limits_{\partial V}{d\bar% {f}\cdot{{\nabla}_{r}}}\frac{1}{|\bar{r}-\bar{r}\acute{\ }|}\\ &\displaystyle{{\nabla}_{r}}\frac{1}{|\bar{r}-\bar{r}\acute{\ }|}=-\frac{\left% (\bar{r}-\bar{r}\acute{\ }\right)}{|\bar{r}-\bar{r}\acute{\ }{{|}^{3}}}\\ \end{aligned}}}

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∫Vd3rΔΦ(r¯)=14πε0∫R3d3r´ρ(r¯´)∮∂Vdf¯⋅∇r1|r¯−r¯´|∇r1|r¯−r¯´|=−(r¯−r¯´)|r¯−r¯´|3

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Identifiers

  • V
  • r
  • Δ
  • Φ
  • r¯
  • π
  • ε0
  • R
  • r
  • ´
  • ρ
  • r¯
  • ´
  • V
  • d
  • f¯
  • r
  • r¯
  • r¯
  • ´
  • r
  • r¯
  • r¯
  • ´
  • r¯
  • r¯
  • ´
  • r¯
  • r¯
  • ´

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