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Display information for equation id:math.1199.255 on revision:1199
* Page found: Elektrodynamik Schöll (eq math.1199.255)
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\begin{align}
& {{Q}_{\alpha }}=-{{\varepsilon }_{0}}\oint\limits_{S\alpha }{{}}d\bar{f}\cdot {{\nabla }_{r}}\int_{V}^{{}}{{{d}^{3}}r}G\left( \bar{r}-\bar{r}\acute{\ } \right)\rho \left( \bar{r}\acute{\ } \right)=-{{\varepsilon }_{0}}^{2}\oint\limits_{S\alpha }{{}}d\bar{f}\cdot {{\nabla }_{r}}\sum\limits_{\beta =1}^{n}{{{\Phi }_{\beta }}\oint\limits_{{{S}_{\beta }}}{{}}d\bar{f}\acute{\ }\cdot {{\nabla }_{r\acute{\ }}}}G\left( \bar{r}-\bar{r}\acute{\ } \right) \\
& {{Q}_{\alpha }}=-{{\varepsilon }_{0}}\int_{L\alpha }^{{}}{{}}{{d}^{3}}r\int_{V}^{{}}{{}}{{d}^{3}}r\acute{\ }{{\Delta }_{r}}G\left( \bar{r}-\bar{r}\acute{\ } \right)\rho \left( \bar{r}\acute{\ } \right)-\sum\limits_{\beta =1}^{n}{{{\Phi }_{\beta }}{{\varepsilon }_{0}}^{2}\oint\limits_{{{S}_{\alpha }}}{{}}d\bar{f}\cdot {{\nabla }_{r}}}\oint\limits_{{{S}_{\beta }}}{{}}d\bar{f}\acute{\ }\cdot {{\nabla }_{r\acute{\ }}}G\left( \bar{r}-\bar{r}\acute{\ } \right) \\
& {{\Delta }_{r}}G\left( \bar{r}-\bar{r}\acute{\ } \right)=-\frac{1}{{{\varepsilon }_{0}}}\delta \left( \bar{r}-\bar{r}\acute{\ } \right)=0\quad f\ddot{u}r\quad \bar{r}\in {{L}_{\alpha }},\bar{r}\acute{\ }\in V, \\
& {{\varepsilon }_{0}}^{2}\oint\limits_{{{S}_{\alpha }}}{{}}d\bar{f}\cdot {{\nabla }_{r}}\oint\limits_{{{S}_{\beta }}}{{}}d\bar{f}\acute{\ }\cdot {{\nabla }_{r\acute{\ }}}G\left( \bar{r}-\bar{r}\acute{\ } \right)=:-{{C}_{\alpha \beta }} \\
& \Rightarrow {{Q}_{\alpha }}=-\sum\limits_{\beta =1}^{n}{{{\Phi }_{\beta }}{{\varepsilon }_{0}}^{2}\oint\limits_{{{S}_{\alpha }}}{{}}d\bar{f}\cdot {{\nabla }_{r}}}\oint\limits_{{{S}_{\beta }}}{{}}d\bar{f}\acute{\ }\cdot {{\nabla }_{r\acute{\ }}}G\left( \bar{r}-\bar{r}\acute{\ } \right)=\sum\limits_{\beta =1}^{n}{{{C}_{\alpha \beta }}{{\Phi }_{\beta }}} \\
\end{align}
TeX (checked):
{\begin{aligned}&{{Q}_{\alpha }}=-{{\varepsilon }_{0}}\oint \limits _{S\alpha }{}d{\bar {f}}\cdot {{\nabla }_{r}}\int _{V}^{}{{{d}^{3}}r}G\left({\bar {r}}-{\bar {r}}{\acute {\ }}\right)\rho \left({\bar {r}}{\acute {\ }}\right)=-{{\varepsilon }_{0}}^{2}\oint \limits _{S\alpha }{}d{\bar {f}}\cdot {{\nabla }_{r}}\sum \limits _{\beta =1}^{n}{{{\Phi }_{\beta }}\oint \limits _{{S}_{\beta }}{}d{\bar {f}}{\acute {\ }}\cdot {{\nabla }_{r{\acute {\ }}}}}G\left({\bar {r}}-{\bar {r}}{\acute {\ }}\right)\\&{{Q}_{\alpha }}=-{{\varepsilon }_{0}}\int _{L\alpha }^{}{}{{d}^{3}}r\int _{V}^{}{}{{d}^{3}}r{\acute {\ }}{{\Delta }_{r}}G\left({\bar {r}}-{\bar {r}}{\acute {\ }}\right)\rho \left({\bar {r}}{\acute {\ }}\right)-\sum \limits _{\beta =1}^{n}{{{\Phi }_{\beta }}{{\varepsilon }_{0}}^{2}\oint \limits _{{S}_{\alpha }}{}d{\bar {f}}\cdot {{\nabla }_{r}}}\oint \limits _{{S}_{\beta }}{}d{\bar {f}}{\acute {\ }}\cdot {{\nabla }_{r{\acute {\ }}}}G\left({\bar {r}}-{\bar {r}}{\acute {\ }}\right)\\&{{\Delta }_{r}}G\left({\bar {r}}-{\bar {r}}{\acute {\ }}\right)=-{\frac {1}{{\varepsilon }_{0}}}\delta \left({\bar {r}}-{\bar {r}}{\acute {\ }}\right)=0\quad f{\ddot {u}}r\quad {\bar {r}}\in {{L}_{\alpha }},{\bar {r}}{\acute {\ }}\in V,\\&{{\varepsilon }_{0}}^{2}\oint \limits _{{S}_{\alpha }}{}d{\bar {f}}\cdot {{\nabla }_{r}}\oint \limits _{{S}_{\beta }}{}d{\bar {f}}{\acute {\ }}\cdot {{\nabla }_{r{\acute {\ }}}}G\left({\bar {r}}-{\bar {r}}{\acute {\ }}\right)=:-{{C}_{\alpha \beta }}\\&\Rightarrow {{Q}_{\alpha }}=-\sum \limits _{\beta =1}^{n}{{{\Phi }_{\beta }}{{\varepsilon }_{0}}^{2}\oint \limits _{{S}_{\alpha }}{}d{\bar {f}}\cdot {{\nabla }_{r}}}\oint \limits _{{S}_{\beta }}{}d{\bar {f}}{\acute {\ }}\cdot {{\nabla }_{r{\acute {\ }}}}G\left({\bar {r}}-{\bar {r}}{\acute {\ }}\right)=\sum \limits _{\beta =1}^{n}{{{C}_{\alpha \beta }}{{\Phi }_{\beta }}}\\\end{aligned}}
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data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><munder><mstyle displaystyle="true"><mo>∮</mo></mstyle><mrow data-mjx-texclass="ORD"><msub><mi>S</mi><mrow data-mjx-texclass="ORD"><mi>α</mi></mrow></msub></mrow></munder><mi>d</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>f</mi><mo>¯</mo></mover></mrow></mrow><mo>⋅</mo><msub><mi mathvariant="normal">∇</mi><mrow data-mjx-texclass="ORD"><mi>r</mi></mrow></msub></mrow><munder><mstyle displaystyle="true"><mo>∮</mo></mstyle><mrow data-mjx-texclass="ORD"><msub><mi>S</mi><mrow data-mjx-texclass="ORD"><mi>β</mi></mrow></msub></mrow></munder><mi>d</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>f</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mspace width="0.5em"/><mo data-mjx-pseudoscript="true">´</mo></mover></mrow></mrow><mo>⋅</mo><msub><mi mathvariant="normal">∇</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>r</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mspace width="0.5em"/><mo data-mjx-pseudoscript="true">´</mo></mover></mrow></mrow></mrow></mrow></msub><mi>G</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mo>−</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mspace width="0.5em"/><mo data-mjx-pseudoscript="true">´</mo></mover></mrow></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>=</mo><munderover><mo form="prefix" texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>β</mi><mo>=</mo><mn>1</mn></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></munderover><mrow data-mjx-texclass="ORD"><msub><mi>C</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>α</mi><mi>β</mi></mrow></mrow></msub><msub><mi mathvariant="normal">Φ</mi><mrow data-mjx-texclass="ORD"><mi>β</mi></mrow></msub></mrow></mtd></mtr><mtr><mtd></mtd></mtr></mtable></mrow></mstyle></mrow></math>
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