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Display information for equation id:math.1199.233 on revision:1199

* Page found: Elektrodynamik Schöll (eq math.1199.233)

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\begin{align}
& {{\Delta }_{r\acute{\ }}}\Phi (\bar{r}\acute{\ })=\int_{V}^{{}}{{{d}^{3}}r{{\Delta }_{r\acute{\ }}}}G\left( \bar{r}-\bar{r}\acute{\ } \right)\rho \left( {\bar{r}} \right)+{{\varepsilon }_{0}}\sum\limits_{\alpha =1}^{n}{{{\Phi }_{\alpha }}\oint\limits_{S\alpha }{{}}d\bar{f}\cdot {{\nabla }_{r}}}{{\Delta }_{r\acute{\ }}}G\left( \bar{r}-\bar{r}\acute{\ } \right) \\
& {{\Delta }_{r\acute{\ }}}G\left( \bar{r}-\bar{r}\acute{\ } \right)=-\frac{1}{{{\varepsilon }_{0}}}\delta \left( \bar{r}-\bar{r}\acute{\ } \right) \\
& \delta \left( \bar{r}-\bar{r}\acute{\ } \right)=0,da\ \bar{r}\in S\alpha ,\bar{r}\acute{\ }\in V-\partial V \\
& \Rightarrow {{\varepsilon }_{0}}\sum\limits_{\alpha =1}^{n}{{{\Phi }_{\alpha }}\oint\limits_{S\alpha }{{}}d\bar{f}\cdot {{\nabla }_{r}}}{{\Delta }_{r\acute{\ }}}G\left( \bar{r}-\bar{r}\acute{\ } \right)=0 \\
& {{\Delta }_{r\acute{\ }}}\Phi (\bar{r}\acute{\ })=\int_{V}^{{}}{{{d}^{3}}r{{\Delta }_{r\acute{\ }}}}G\left( \bar{r}-\bar{r}\acute{\ } \right)\rho \left( {\bar{r}} \right)=-\int_{V}^{{}}{{{d}^{3}}r\frac{1}{{{\varepsilon }_{0}}}\delta \left( \bar{r}-\bar{r}\acute{\ } \right)}\rho \left( {\bar{r}} \right)=-\frac{\rho \left( \bar{r}\acute{\ } \right)}{{{\varepsilon }_{0}}} \\
\end{align}

TeX (checked): 

{\begin{aligned}&{{\Delta }_{r{\acute {\ }}}}\Phi ({\bar {r}}{\acute {\ }})=\int _{V}^{}{{{d}^{3}}r{{\Delta }_{r{\acute {\ }}}}}G\left({\bar {r}}-{\bar {r}}{\acute {\ }}\right)\rho \left({\bar {r}}\right)+{{\varepsilon }_{0}}\sum \limits _{\alpha =1}^{n}{{{\Phi }_{\alpha }}\oint \limits _{S\alpha }{}d{\bar {f}}\cdot {{\nabla }_{r}}}{{\Delta }_{r{\acute {\ }}}}G\left({\bar {r}}-{\bar {r}}{\acute {\ }}\right)\\&{{\Delta }_{r{\acute {\ }}}}G\left({\bar {r}}-{\bar {r}}{\acute {\ }}\right)=-{\frac {1}{{\varepsilon }_{0}}}\delta \left({\bar {r}}-{\bar {r}}{\acute {\ }}\right)\\&\delta \left({\bar {r}}-{\bar {r}}{\acute {\ }}\right)=0,da\ {\bar {r}}\in S\alpha ,{\bar {r}}{\acute {\ }}\in V-\partial V\\&\Rightarrow {{\varepsilon }_{0}}\sum \limits _{\alpha =1}^{n}{{{\Phi }_{\alpha }}\oint \limits _{S\alpha }{}d{\bar {f}}\cdot {{\nabla }_{r}}}{{\Delta }_{r{\acute {\ }}}}G\left({\bar {r}}-{\bar {r}}{\acute {\ }}\right)=0\\&{{\Delta }_{r{\acute {\ }}}}\Phi ({\bar {r}}{\acute {\ }})=\int _{V}^{}{{{d}^{3}}r{{\Delta }_{r{\acute {\ }}}}}G\left({\bar {r}}-{\bar {r}}{\acute {\ }}\right)\rho \left({\bar {r}}\right)=-\int _{V}^{}{{{d}^{3}}r{\frac {1}{{\varepsilon }_{0}}}\delta \left({\bar {r}}-{\bar {r}}{\acute {\ }}\right)}\rho \left({\bar {r}}\right)=-{\frac {\rho \left({\bar {r}}{\acute {\ }}\right)}{{\varepsilon }_{0}}}\\\end{aligned}}

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Δr´Φ(r¯´)=∫Vd3rΔr´G(r¯−r¯´)ρ(r¯)+ε0∑α=1nΦα∮Sαdf¯⋅∇rΔr´G(r¯−r¯´)Δr´G(r¯−r¯´)=−1ε0δ(r¯−r¯´)δ(r¯−r¯´)=0,dar¯∈Sα,r¯´∈V−∂V⇒ε0∑α=1nΦα∮Sαdf¯⋅∇rΔr´G(r¯−r¯´)=0Δr´Φ(r¯´)=∫Vd3rΔr´G(r¯−r¯´)ρ(r¯)=−∫Vd3r1ε0δ(r¯−r¯´)ρ(r¯)=−ρ(r¯´)ε0
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