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Display information for equation id:math.2133.58 on revision:2133

* Page found: Multipolstrahlung (eq math.2133.58)

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\begin{align}
& \left( \bar{r}\bar{r}\acute{\ } \right)\bar{j}\left( \bar{r}\acute{\ },\tau  \right)=\frac{1}{2}\left( \bar{r}\acute{\ }\times \bar{j} \right)\times \bar{r}+\frac{1}{2}\left[ \left( \bar{r}\bar{r}\acute{\ } \right)\bar{j}+\left( \bar{r}\bar{j} \right)\bar{r}\acute{\ } \right] \\
& und \\
& {{\nabla }_{r\acute{\ }}}\left[ {{x}_{k}}\acute{\ }\left( \bar{r}\bar{r}\acute{\ } \right)\bar{j} \right]=\left[ \left( \bar{r}\bar{r}\acute{\ } \right){{j}_{k}}+{{x}_{k}}\acute{\ }\left( \bar{r}\bar{j} \right)+{{x}_{k\acute{\ }}}\left( \bar{r}\bar{r}\acute{\ } \right){{\nabla }_{r\acute{\ }}}\cdot \bar{j} \right] \\
& {{\nabla }_{r\acute{\ }}}\cdot \bar{j}=-\frac{\partial }{\partial \tau }\rho \left( \bar{r}\acute{\ },\tau  \right) \\
\end{align}

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( r ¯ ⁢ r ¯ ⁢ ´ ) ⁢ j ¯ ⁢ ( r ¯ ⁢ ´ , τ ) = 1 2 ⁢ ( r ¯ ⁢ ´ × j ¯ ) × r ¯ + 1 2 ⁢ [ ( r ¯ ⁢ r ¯ ⁢ ´ ) ⁢ j ¯ + ( r ¯ ⁢ j ¯ ) ⁢ r ¯ ⁢ ´ ] u ⁢ n ⁢ d ∇ r ⁢ ´ ⁡ [ x k ⁢ ´ ⁢ ( r ¯ ⁢ r ¯ ⁢ ´ ) ⁢ j ¯ ] = [ ( r ¯ ⁢ r ¯ ⁢ ´ ) ⁢ j k + x k ⁢ ´ ⁢ ( r ¯ ⁢ j ¯ ) + x k ⁢ ´ ⁢ ( r ¯ ⁢ r ¯ ⁢ ´ ) ⁢ ∇ r ⁢ ´ ⋅ j ¯ ] ∇ r ⁢ ´ ⋅ j ¯ = - ∂ ∂ ⁡ τ ⁢ ρ ⁢ ( r ¯ ⁢ ´ , τ ) missing-subexpression ¯ 𝑟 ¯ 𝑟 ´ absent ¯ 𝑗 ¯ 𝑟 ´ absent 𝜏 1 2 ¯ 𝑟 ´ absent ¯ 𝑗 ¯ 𝑟 1 2 delimited-[] ¯ 𝑟 ¯ 𝑟 ´ absent ¯ 𝑗 ¯ 𝑟 ¯ 𝑗 ¯ 𝑟 ´ absent missing-subexpression 𝑢 𝑛 𝑑 missing-subexpression subscript ∇ 𝑟 ´ absent subscript 𝑥 𝑘 ´ absent ¯ 𝑟 ¯ 𝑟 ´ absent ¯ 𝑗 delimited-[] ¯ 𝑟 ¯ 𝑟 ´ absent subscript 𝑗 𝑘 subscript 𝑥 𝑘 ´ absent ¯ 𝑟 ¯ 𝑗 ⋅ subscript 𝑥 𝑘 ´ absent ¯ 𝑟 ¯ 𝑟 ´ absent subscript ∇ 𝑟 ´ absent ¯ 𝑗 missing-subexpression ⋅ subscript ∇ 𝑟 ´ absent ¯ 𝑗 𝜏 𝜌 ¯ 𝑟 ´ absent 𝜏 {\displaystyle{\displaystyle\begin{aligned} &\displaystyle\left(\bar{r}\bar{r}% \acute{\ }\right)\bar{j}\left(\bar{r}\acute{\ },\tau\right)=\frac{1}{2}\left(% \bar{r}\acute{\ }\times\bar{j}\right)\times\bar{r}+\frac{1}{2}\left[\left(\bar% {r}\bar{r}\acute{\ }\right)\bar{j}+\left(\bar{r}\bar{j}\right)\bar{r}\acute{\ % }\right]\\ &\displaystyle und\\ &\displaystyle{{\nabla}_{r\acute{\ }}}\left[{{x}_{k}}\acute{\ }\left(\bar{r}% \bar{r}\acute{\ }\right)\bar{j}\right]=\left[\left(\bar{r}\bar{r}\acute{\ }% \right){{j}_{k}}+{{x}_{k}}\acute{\ }\left(\bar{r}\bar{j}\right)+{{x}_{k\acute{% \ }}}\left(\bar{r}\bar{r}\acute{\ }\right){{\nabla}_{r\acute{\ }}}\cdot\bar{j}% \right]\\ &\displaystyle{{\nabla}_{r\acute{\ }}}\cdot\bar{j}=-\frac{\partial}{\partial% \tau}\rho\left(\bar{r}\acute{\ },\tau\right)\\ \end{aligned}}}

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(r¯r¯´)j¯(r¯´,τ)=12(r¯´×j¯)×r¯+12[(r¯r¯´)j¯+(r¯j¯)r¯´]und∇r´[xk´(r¯r¯´)j¯]=[(r¯r¯´)jk+xk´(r¯j¯)+xk´(r¯r¯´)∇r´⋅j¯]∇r´⋅j¯=−∂∂τρ(r¯´,τ)

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