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* Page found: Stationäre Ströme und Magnetfeld (eq math.1432.104)

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TeX (original user input): 

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

TeX (checked): 

{\begin{aligned}&{\bar {F}}=\int _{}^{}{}{{d}^{3}}r{\acute {\ }}{\bar {j}}({\bar {r}}{\acute {\ }})\times {{\nabla }_{r}}\left[\left({\bar {r}}{\acute {\ }}\right)\cdot {\bar {B}}({\bar {r}})\right]\\&{\bar {j}}({\bar {r}}{\acute {\ }})\times {{\nabla }_{r}}\left[\left({\bar {r}}{\acute {\ }}\right)\cdot {\bar {B}}({\bar {r}})\right]=-{{\nabla }_{r}}\times \left[\left(\left({\bar {r}}{\acute {\ }}\right)\cdot {\bar {B}}({\bar {r}})\right){\bar {j}}({\bar {r}}{\acute {\ }})\right]+\left[\left({\bar {r}}{\acute {\ }}\right)\cdot {\bar {B}}({\bar {r}})\right]{{\nabla }_{r}}\times {\bar {j}}({\bar {r}}{\acute {\ }})\\&{{\nabla }_{r}}\times {\bar {j}}({\bar {r}}{\acute {\ }})=0\\&\Rightarrow {\bar {F}}=-\int _{}^{}{}{{d}^{3}}r{\acute {\ }}{{\nabla }_{r}}\times \left[\left(\left({\bar {r}}{\acute {\ }}\right)\cdot {\bar {B}}({\bar {r}})\right){\bar {j}}({\bar {r}}{\acute {\ }})\right]=-{{\nabla }_{r}}\times \left({\bar {m}}\times {\bar {B}}({\bar {r}})\right)\\&{\bar {F}}=-{{\nabla }_{r}}\times \left({\bar {m}}\times {\bar {B}}({\bar {r}})\right)=\left({\bar {m}}\cdot {{\nabla }_{r}}\right){\bar {B}}({\bar {r}})=-{{\nabla }_{r}}\left(-{\bar {m}}\cdot {\bar {B}}({\bar {r}})\right)\\\end{aligned}}

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F¯=∫d3r ´j¯(r¯ ´)×∇r[(r¯ ´)⋅B¯(r¯)]j¯(r¯ ´)×∇r[(r¯ ´)⋅B¯(r¯)]=−∇r×[((r¯ ´)⋅B¯(r¯))j¯(r¯ ´)]+[(r¯ ´)⋅B¯(r¯)]∇r×j¯(r¯ ´)∇r×j¯(r¯ ´)=0⇒F¯=−∫d3r ´∇r×[((r¯ ´)⋅B¯(r¯))j¯(r¯ ´)]=−∇r×(m¯×B¯(r¯))F¯=−∇r×(m¯×B¯(r¯))=(m¯⋅∇r)B¯(r¯)=−∇r(−m¯⋅B¯(r¯))
<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>&#160;</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>&#160;</mtext><mo data-mjx-pseudoscript="true">´</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>&#160;</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></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mover><mi>j</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mover><mi>r</mi><mo>¯</mo></mover><mover><mtext>&#160;</mtext><mo data-mjx-pseudoscript="true">´</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>&#160;</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><mo stretchy="false">−</mo><msub><mi mathvariant="normal">∇</mi><mrow data-mjx-texclass="ORD"><mi>r</mi></mrow></msub><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><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mover><mi>r</mi><mo>¯</mo></mover><mover><mtext>&#160;</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><mover><mi>j</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mover><mi>r</mi><mo>¯</mo></mover><mover><mtext>&#160;</mtext><mo data-mjx-pseudoscript="true">´</mo></mover><mo stretchy="false">)</mo><mo data-mjx-texclass="CLOSE">]</mo></mrow><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>&#160;</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><msub><mi mathvariant="normal">∇</mi><mrow data-mjx-texclass="ORD"><mi>r</mi></mrow></msub><mo stretchy="false">×</mo><mover><mi>j</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mover><mi>r</mi><mo>¯</mo></mover><mover><mtext>&#160;</mtext><mo data-mjx-pseudoscript="true">´</mo></mover><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><msub><mi mathvariant="normal">∇</mi><mrow data-mjx-texclass="ORD"><mi>r</mi></mrow></msub><mo stretchy="false">×</mo><mover><mi>j</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mover><mi>r</mi><mo>¯</mo></mover><mover><mtext>&#160;</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><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>&#160;</mtext><mo data-mjx-pseudoscript="true">´</mo></mover><msub><mi mathvariant="normal">∇</mi><mrow data-mjx-texclass="ORD"><mi>r</mi></mrow></msub><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><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mover><mi>r</mi><mo>¯</mo></mover><mover><mtext>&#160;</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><mover><mi>j</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mover><mi>r</mi><mo>¯</mo></mover><mover><mtext>&#160;</mtext><mo data-mjx-pseudoscript="true">´</mo></mover><mo stretchy="false">)</mo><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo stretchy="false">=</mo><mo stretchy="false">−</mo><msub><mi mathvariant="normal">∇</mi><mrow data-mjx-texclass="ORD"><mi>r</mi></mrow></msub><mo stretchy="false">×</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mover><mi>m</mi><mo>¯</mo></mover><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><mtd class="mwe-math-columnalign-l"><mover><mi>F</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><mo stretchy="false">×</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mover><mi>m</mi><mo>¯</mo></mover><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><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mover><mi>m</mi><mo>¯</mo></mover><mo stretchy="false">⋅</mo><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><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><mo stretchy="false">−</mo><mover><mi>m</mi><mo>¯</mo></mover><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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