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* Page found: Magnetische Multipole (eq math.2100.42)

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\begin{align}
& \bar{F}=\int_{{}}^{{}}{{}}{{d}^{3}}r\acute{\ }\bar{j}(\bar{r}\acute{\ })\times \left[ \left( \bar{r}\acute{\ } \right){{\nabla }_{r}} \right]\bar{B}(\bar{r})-\int_{{}}^{{}}{{}}{{d}^{3}}r\acute{\ }\bar{j}(\bar{r}\acute{\ })\times \left[ \left( {\bar{r}} \right){{\nabla }_{r}} \right]\bar{B}(\bar{r}) \\
& \int_{{}}^{{}}{{}}{{d}^{3}}r\acute{\ }\bar{j}(\bar{r}\acute{\ })\times \left[ \left( {\bar{r}} \right){{\nabla }_{r}} \right]\bar{B}(\bar{r})=0,da\int_{{}}^{{}}{{}}{{d}^{3}}r\acute{\ }\bar{j}(\bar{r}\acute{\ })=0 \\
& \Rightarrow \bar{F}=\int_{{}}^{{}}{{}}{{d}^{3}}r\acute{\ }\bar{j}(\bar{r}\acute{\ })\times \left[ \left( \bar{r}\acute{\ } \right){{\nabla }_{r}} \right]\bar{B}(\bar{r}) \\
& \left[ \left( \bar{r}\acute{\ } \right){{\nabla }_{r}} \right]\bar{B}(\bar{r})={{\nabla }_{r}}\left[ \left( \bar{r}\acute{\ } \right)\cdot \bar{B}(\bar{r}) \right]-\bar{r}\acute{\ }\times \left[ {{\nabla }_{r}}\times \bar{B}(\bar{r}) \right] \\
\end{align}

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F ¯ = ∫ d 3 ⁢ r ⁢ ´ ⁢ j ¯ ⁢ ( r ¯ ⁢ ´ ) × [ ( r ¯ ⁢ ´ ) ⁢ ∇ r ] ⁢ B ¯ ⁢ ( r ¯ ) - ∫ d 3 ⁢ r ⁢ ´ ⁢ j ¯ ⁢ ( r ¯ ⁢ ´ ) × [ ( r ¯ ) ⁢ ∇ r ] ⁢ B ¯ ⁢ ( r ¯ ) ∫ d 3 ⁢ r ⁢ ´ ⁢ j ¯ ⁢ ( r ¯ ⁢ ´ ) × [ ( r ¯ ) ⁢ ∇ r ] ⁢ B ¯ ⁢ ( r ¯ ) = 0 , d ⁢ a ⁢ ∫ d 3 ⁢ r ⁢ ´ ⁢ j ¯ ⁢ ( r ¯ ⁢ ´ ) = 0 ⇒ F ¯ = ∫ d 3 ⁢ r ⁢ ´ ⁢ j ¯ ⁢ ( r ¯ ⁢ ´ ) × [ ( r ¯ ⁢ ´ ) ⁢ ∇ r ] ⁢ B ¯ ⁢ ( r ¯ ) [ ( r ¯ ⁢ ´ ) ⁢ ∇ r ] ⁢ B ¯ ⁢ ( r ¯ ) = ∇ r ⁡ [ ( r ¯ ⁢ ´ ) ⋅ B ¯ ⁢ ( r ¯ ) ] - r ¯ ⁢ ´ × [ ∇ r × B ¯ ⁢ ( r ¯ ) ] missing-subexpression ¯ 𝐹 superscript 𝑑 3 𝑟 ´ absent ¯ 𝑗 ¯ 𝑟 ´ absent delimited-[] ¯ 𝑟 ´ absent subscript ∇ 𝑟 ¯ 𝐵 ¯ 𝑟 superscript 𝑑 3 𝑟 ´ absent ¯ 𝑗 ¯ 𝑟 ´ absent delimited-[] ¯ 𝑟 subscript ∇ 𝑟 ¯ 𝐵 ¯ 𝑟 missing-subexpression formulae-sequence superscript 𝑑 3 𝑟 ´ absent ¯ 𝑗 ¯ 𝑟 ´ absent delimited-[] ¯ 𝑟 subscript ∇ 𝑟 ¯ 𝐵 ¯ 𝑟 0 𝑑 𝑎 superscript 𝑑 3 𝑟 ´ absent ¯ 𝑗 ¯ 𝑟 ´ absent 0 missing-subexpression ⇒ absent ¯ 𝐹 superscript 𝑑 3 𝑟 ´ absent ¯ 𝑗 ¯ 𝑟 ´ absent delimited-[] ¯ 𝑟 ´ absent subscript ∇ 𝑟 ¯ 𝐵 ¯ 𝑟 missing-subexpression delimited-[] ¯ 𝑟 ´ absent subscript ∇ 𝑟 ¯ 𝐵 ¯ 𝑟 subscript ∇ 𝑟 ⋅ ¯ 𝑟 ´ absent ¯ 𝐵 ¯ 𝑟 ¯ 𝑟 ´ absent delimited-[] subscript ∇ 𝑟 ¯ 𝐵 ¯ 𝑟 {\displaystyle{\displaystyle\begin{aligned} &\displaystyle\bar{F}=\int{{}}{{d}% ^{3}}r\acute{\ }\bar{j}(\bar{r}\acute{\ })\times\left[\left(\bar{r}\acute{\ }% \right){{\nabla}_{r}}\right]\bar{B}(\bar{r})-\int{{}}{{d}^{3}}r\acute{\ }\bar{% j}(\bar{r}\acute{\ })\times\left[\left({\bar{r}}\right){{\nabla}_{r}}\right]% \bar{B}(\bar{r})\\ &\displaystyle\int{{}}{{d}^{3}}r\acute{\ }\bar{j}(\bar{r}\acute{\ })\times% \left[\left({\bar{r}}\right){{\nabla}_{r}}\right]\bar{B}(\bar{r})=0,da\int{{}}% {{d}^{3}}r\acute{\ }\bar{j}(\bar{r}\acute{\ })=0\\ &\displaystyle\Rightarrow\bar{F}=\int{{}}{{d}^{3}}r\acute{\ }\bar{j}(\bar{r}% \acute{\ })\times\left[\left(\bar{r}\acute{\ }\right){{\nabla}_{r}}\right]\bar% {B}(\bar{r})\\ &\displaystyle\left[\left(\bar{r}\acute{\ }\right){{\nabla}_{r}}\right]\bar{B}% (\bar{r})={{\nabla}_{r}}\left[\left(\bar{r}\acute{\ }\right)\cdot\bar{B}(\bar{% r})\right]-\bar{r}\acute{\ }\times\left[{{\nabla}_{r}}\times\bar{B}(\bar{r})% \right]\\ \end{aligned}}}

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F¯=∫d3r´j¯(r¯´)×[(r¯´)∇r]B¯(r¯)−∫d3r´j¯(r¯´)×[(r¯)∇r]B¯(r¯)∫d3r´j¯(r¯´)×[(r¯)∇r]B¯(r¯)=0,da∫d3r´j¯(r¯´)=0⇒F¯=∫d3r´j¯(r¯´)×[(r¯´)∇r]B¯(r¯)[(r¯´)∇r]B¯(r¯)=∇r[(r¯´)⋅B¯(r¯)]−r¯´×[∇r×B¯(r¯)]

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