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Display information for equation id:math.1199.387 on revision:1199

* Page found: Elektrodynamik Schöll (eq math.1199.387)

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TeX (original user input): 

\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}

TeX (checked): 

{\begin{aligned}&{\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{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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