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Display information for equation id:math.1432.101 on revision:1432
* Page found: Stationäre Ströme und Magnetfeld (eq math.1432.101)
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Hash: 7bc615a0978e4fc2cbfb143851e74fd9
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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<math xmlns="http://www.w3.org/1998/Math/MathML" class="mwe-math-element mwe-math-element-inline"><mrow data-mjx-texclass="ORD"><mstyle displaystyle="true" scriptlevel="0"><mtable displaystyle="true"><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mover><mi>F</mi><mo>¯</mo></mover><mo stretchy="false">=</mo><msubsup><mo stretchy="false">∫</mo><mrow data-mjx-texclass="ORD"></mrow><mrow data-mjx-texclass="ORD"></mrow></msubsup><msup><mi>d</mi><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msup><mi>r</mi><mover><mtext> </mtext><mo data-mjx-pseudoscript="true">´</mo></mover><mover><mi>j</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mover><mi>r</mi><mo>¯</mo></mover><mover><mtext> </mtext><mo data-mjx-pseudoscript="true">´</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">×</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mover><mi>r</mi><mo>¯</mo></mover><mover><mtext> </mtext><mo data-mjx-pseudoscript="true">´</mo></mover><mo data-mjx-texclass="CLOSE">)</mo></mrow><msub><mi mathvariant="normal">∇</mi><mrow data-mjx-texclass="ORD"><mi>r</mi></mrow></msub><mo data-mjx-texclass="CLOSE">]</mo></mrow><mover><mi>B</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mover><mi>r</mi><mo>¯</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">−</mo><msubsup><mo stretchy="false">∫</mo><mrow data-mjx-texclass="ORD"></mrow><mrow data-mjx-texclass="ORD"></mrow></msubsup><msup><mi>d</mi><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msup><mi>r</mi><mover><mtext> </mtext><mo data-mjx-pseudoscript="true">´</mo></mover><mover><mi>j</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mover><mi>r</mi><mo>¯</mo></mover><mover><mtext> </mtext><mo data-mjx-pseudoscript="true">´</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">×</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mover><mi>r</mi><mo>¯</mo></mover><mo data-mjx-texclass="CLOSE">)</mo></mrow><msub><mi mathvariant="normal">∇</mi><mrow data-mjx-texclass="ORD"><mi>r</mi></mrow></msub><mo data-mjx-texclass="CLOSE">]</mo></mrow><mover><mi>B</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mover><mi>r</mi><mo>¯</mo></mover><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><msubsup><mo stretchy="false">∫</mo><mrow data-mjx-texclass="ORD"></mrow><mrow data-mjx-texclass="ORD"></mrow></msubsup><msup><mi>d</mi><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msup><mi>r</mi><mover><mtext> </mtext><mo data-mjx-pseudoscript="true">´</mo></mover><mover><mi>j</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mover><mi>r</mi><mo>¯</mo></mover><mover><mtext> </mtext><mo data-mjx-pseudoscript="true">´</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">×</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mover><mi>r</mi><mo>¯</mo></mover><mo data-mjx-texclass="CLOSE">)</mo></mrow><msub><mi mathvariant="normal">∇</mi><mrow data-mjx-texclass="ORD"><mi>r</mi></mrow></msub><mo data-mjx-texclass="CLOSE">]</mo></mrow><mover><mi>B</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mover><mi>r</mi><mo>¯</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mn>0</mn><mo>,</mo><mi>d</mi><mi>a</mi><msubsup><mo stretchy="false">∫</mo><mrow data-mjx-texclass="ORD"></mrow><mrow data-mjx-texclass="ORD"></mrow></msubsup><msup><mi>d</mi><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msup><mi>r</mi><mover><mtext> </mtext><mo data-mjx-pseudoscript="true">´</mo></mover><mover><mi>j</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mover><mi>r</mi><mo>¯</mo></mover><mover><mtext> </mtext><mo data-mjx-pseudoscript="true">´</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mn>0</mn></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mo stretchy="false">⇒</mo><mover><mi>F</mi><mo>¯</mo></mover><mo stretchy="false">=</mo><msubsup><mo stretchy="false">∫</mo><mrow data-mjx-texclass="ORD"></mrow><mrow data-mjx-texclass="ORD"></mrow></msubsup><msup><mi>d</mi><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msup><mi>r</mi><mover><mtext> </mtext><mo data-mjx-pseudoscript="true">´</mo></mover><mover><mi>j</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mover><mi>r</mi><mo>¯</mo></mover><mover><mtext> </mtext><mo data-mjx-pseudoscript="true">´</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">×</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mover><mi>r</mi><mo>¯</mo></mover><mover><mtext> </mtext><mo data-mjx-pseudoscript="true">´</mo></mover><mo data-mjx-texclass="CLOSE">)</mo></mrow><msub><mi mathvariant="normal">∇</mi><mrow data-mjx-texclass="ORD"><mi>r</mi></mrow></msub><mo data-mjx-texclass="CLOSE">]</mo></mrow><mover><mi>B</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mover><mi>r</mi><mo>¯</mo></mover><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mover><mi>r</mi><mo>¯</mo></mover><mover><mtext> </mtext><mo data-mjx-pseudoscript="true">´</mo></mover><mo data-mjx-texclass="CLOSE">)</mo></mrow><msub><mi mathvariant="normal">∇</mi><mrow data-mjx-texclass="ORD"><mi>r</mi></mrow></msub><mo data-mjx-texclass="CLOSE">]</mo></mrow><mover><mi>B</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mover><mi>r</mi><mo>¯</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">=</mo><msub><mi mathvariant="normal">∇</mi><mrow data-mjx-texclass="ORD"><mi>r</mi></mrow></msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mover><mi>r</mi><mo>¯</mo></mover><mover><mtext> </mtext><mo data-mjx-pseudoscript="true">´</mo></mover><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo stretchy="false">⋅</mo><mover><mi>B</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mover><mi>r</mi><mo>¯</mo></mover><mo stretchy="false">)</mo><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo stretchy="false">−</mo><mover><mi>r</mi><mo>¯</mo></mover><mover><mtext> </mtext><mo data-mjx-pseudoscript="true">´</mo></mover><mo stretchy="false">×</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><msub><mi mathvariant="normal">∇</mi><mrow data-mjx-texclass="ORD"><mi>r</mi></mrow></msub><mo stretchy="false">×</mo><mover><mi>B</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mover><mi>r</mi><mo>¯</mo></mover><mo stretchy="false">)</mo><mo data-mjx-texclass="CLOSE">]</mo></mrow></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd></mtr></mtable></mstyle></mrow></math>
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