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Magnetisierung Eq: math.2144.18

Hash: 4acf8c512dc45d9f75af4f48e3257919

TeX (original user input):

\begin{align}
& \bar{A}\left( \bar{r},t \right)=\frac{{{\mu }_{0}}}{4\pi }\int_{{}}^{{}}{{}}{{d}^{3}}r\acute{\ }\left[ \frac{1}{\left| \bar{r}-\bar{r}\acute{\ } \right|}\dot{\bar{P}}\left( \bar{r}\acute{\ },t-\frac{\left| \bar{r}-\bar{r}\acute{\ } \right|}{c} \right)+{{\nabla }_{r}}\times \left( \frac{1}{\left| \bar{r}-\bar{r}\acute{\ } \right|}\bar{M}\left( \bar{r}\acute{\ },t-\frac{\left| \bar{r}-\bar{r}\acute{\ } \right|}{c} \right) \right) \right] \\
& \int_{{}}^{{}}{{}}{{d}^{3}}r\acute{\ }{{\nabla }_{r}}\times \left( \frac{1}{\left| \bar{r}-\bar{r}\acute{\ } \right|}\bar{M}\left( \bar{r}\acute{\ },t-\frac{\left| \bar{r}-\bar{r}\acute{\ } \right|}{c} \right) \right)= \\
& =-\int_{{}}^{{}}{{}}{{d}^{3}}r\acute{\ }{{\nabla }_{r\acute{\ }}}\times \left( \frac{1}{\left| \bar{r}-\bar{r}\acute{\ } \right|}\bar{M}\left( \bar{r}\acute{\ },t-\frac{\left| \bar{r}-\bar{r}\acute{\ } \right|}{c} \right) \right)+\int_{{}}^{{}}{{}}{{d}^{3}}r\acute{\ }\frac{1}{\left| \bar{r}-\bar{r}\acute{\ } \right|}{{\nabla }_{r\acute{\ }}}\times \bar{M}\left( \bar{r}\acute{\ },t\acute{\ } \right) \\
& t\acute{\ }=t-\frac{\left| \bar{r}-\bar{r}\acute{\ } \right|}{c} \\
& \int_{{}}^{{}}{{}}{{d}^{3}}r\acute{\ }{{\nabla }_{r\acute{\ }}}\times \left( \frac{1}{\left| \bar{r}-\bar{r}\acute{\ } \right|}\bar{M}\left( \bar{r}\acute{\ },t-\frac{\left| \bar{r}-\bar{r}\acute{\ } \right|}{c} \right) \right)=0 \\
& \Rightarrow \bar{A}\left( \bar{r},t \right)=\frac{{{\mu }_{0}}}{4\pi }\int_{{}}^{{}}{{}}{{d}^{3}}r\acute{\ }\left[ \frac{1}{\left| \bar{r}-\bar{r}\acute{\ } \right|}\dot{\bar{P}}\left( \bar{r}\acute{\ },t-\frac{\left| \bar{r}-\bar{r}\acute{\ } \right|}{c} \right)+\frac{1}{\left| \bar{r}-\bar{r}\acute{\ } \right|}{{\nabla }_{r\acute{\ }}}\times \bar{M}\left( \bar{r}\acute{\ },t\acute{\ } \right) \right] \\
\end{align}

TeX (checked):

{\begin{aligned}&{\bar {A}}\left({\bar {r}},t\right)={\frac {{\mu }_{0}}{4\pi }}\int _{}^{}{}{{d}^{3}}r{\acute {\ }}\left[{\frac {1}{\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}}{\dot {\bar {P}}}\left({\bar {r}}{\acute {\ }},t-{\frac {\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}{c}}\right)+{{\nabla }_{r}}\times \left({\frac {1}{\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}}{\bar {M}}\left({\bar {r}}{\acute {\ }},t-{\frac {\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}{c}}\right)\right)\right]\\&\int _{}^{}{}{{d}^{3}}r{\acute {\ }}{{\nabla }_{r}}\times \left({\frac {1}{\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}}{\bar {M}}\left({\bar {r}}{\acute {\ }},t-{\frac {\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}{c}}\right)\right)=\\&=-\int _{}^{}{}{{d}^{3}}r{\acute {\ }}{{\nabla }_{r{\acute {\ }}}}\times \left({\frac {1}{\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}}{\bar {M}}\left({\bar {r}}{\acute {\ }},t-{\frac {\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}{c}}\right)\right)+\int _{}^{}{}{{d}^{3}}r{\acute {\ }}{\frac {1}{\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}}{{\nabla }_{r{\acute {\ }}}}\times {\bar {M}}\left({\bar {r}}{\acute {\ }},t{\acute {\ }}\right)\\&t{\acute {\ }}=t-{\frac {\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}{c}}\\&\int _{}^{}{}{{d}^{3}}r{\acute {\ }}{{\nabla }_{r{\acute {\ }}}}\times \left({\frac {1}{\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}}{\bar {M}}\left({\bar {r}}{\acute {\ }},t-{\frac {\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}{c}}\right)\right)=0\\&\Rightarrow {\bar {A}}\left({\bar {r}},t\right)={\frac {{\mu }_{0}}{4\pi }}\int _{}^{}{}{{d}^{3}}r{\acute {\ }}\left[{\frac {1}{\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}}{\dot {\bar {P}}}\left({\bar {r}}{\acute {\ }},t-{\frac {\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}{c}}\right)+{\frac {1}{\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}}{{\nabla }_{r{\acute {\ }}}}\times {\bar {M}}\left({\bar {r}}{\acute {\ }},t{\acute {\ }}\right)\right]\\\end{aligned}}

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\begin{matrix} \bar{A} (\bar{r}, t) = \frac{μ_{0}}{4 π} \int_{}^{} d^{3} r \overset{´}{} [\frac{1}{| \bar{r} - \bar{r} \overset{´}{} |} \dot{\bar{P}} (\bar{r} \overset{´}{}, t - \frac{| \bar{r} - \bar{r} \overset{´}{} |}{c}) + \nabla_{r} \times (\frac{1}{| \bar{r} - \bar{r} \overset{´}{} |} \bar{M} (\bar{r} \overset{´}{}, t - \frac{| \bar{r} - \bar{r} \overset{´}{} |}{c}))] \\ \int_{}^{} d^{3} r \overset{´}{} \nabla_{r} \times (\frac{1}{| \bar{r} - \bar{r} \overset{´}{} |} \bar{M} (\bar{r} \overset{´}{}, t - \frac{| \bar{r} - \bar{r} \overset{´}{} |}{c})) = \\ = - \int_{}^{} d^{3} r \overset{´}{} \nabla_{r \overset{´}{}} \times (\frac{1}{| \bar{r} - \bar{r} \overset{´}{} |} \bar{M} (\bar{r} \overset{´}{}, t - \frac{| \bar{r} - \bar{r} \overset{´}{} |}{c})) + \int_{}^{} d^{3} r \overset{´}{} \frac{1}{| \bar{r} - \bar{r} \overset{´}{} |} \nabla_{r \overset{´}{}} \times \bar{M} (\bar{r} \overset{´}{}, t \overset{´}{}) \\ t \overset{´}{} = t - \frac{| \bar{r} - \bar{r} \overset{´}{} |}{c} \\ \int_{}^{} d^{3} r \overset{´}{} \nabla_{r \overset{´}{}} \times (\frac{1}{| \bar{r} - \bar{r} \overset{´}{} |} \bar{M} (\bar{r} \overset{´}{}, t - \frac{| \bar{r} - \bar{r} \overset{´}{} |}{c})) = 0 \\ \Rightarrow \bar{A} (\bar{r}, t) = \frac{μ_{0}}{4 π} \int_{}^{} d^{3} r \overset{´}{} [\frac{1}{| \bar{r} - \bar{r} \overset{´}{} |} \dot{\bar{P}} (\bar{r} \overset{´}{}, t - \frac{| \bar{r} - \bar{r} \overset{´}{} |}{c}) + \frac{1}{| \bar{r} - \bar{r} \overset{´}{} |} \nabla_{r \overset{´}{}} \times \bar{M} (\bar{r} \overset{´}{}, t \overset{´}{})] \end{matrix}

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data-mjx-texclass="OPEN">(</mo><mover><mi>r</mi><mo>¯</mo></mover><mover><mtext>&#160;</mtext><mo data-mjx-pseudoscript="true">´</mo></mover><mo>,</mo><mi>t</mi><mo stretchy="false">−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><mover><mi>r</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 data-mjx-texclass="CLOSE">|</mo></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>c</mi></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo stretchy="false">+</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><mover><mi>r</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 data-mjx-texclass="CLOSE">|</mo></mrow></mrow></mfrac></mrow><msub><mi mathvariant="normal">∇</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>r</mi><mover><mtext>&#160;</mtext><mo data-mjx-pseudoscript="true">´</mo></mover></mrow></mrow></msub><mo stretchy="false">×</mo><mover><mi>M</mi><mo>¯</mo></mover><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>,</mo><mi>t</mi><mover><mtext>&#160;</mtext><mo data-mjx-pseudoscript="true">´</mo></mover><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo data-mjx-texclass="CLOSE">]</mo></mrow></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd></mtr></mtable></mstyle></mrow></math>

Translations to Computer Algebra Systems

Translation to Maple

In Maple:

Translation to Mathematica

In Mathematica:

Identifiers

$\bar{A}$

$\bar{r}$

$t$

$μ_{0}$

$π$

$r$

$\overset{´}{}$

$\bar{r}$

$\bar{r}$

$\overset{´}{}$

$\dot{\bar{P}}$

$\bar{r}$

$\overset{´}{}$

$t$

$\bar{r}$

$\bar{r}$

$\overset{´}{}$

$c$

$r$

$\bar{r}$

$\bar{r}$

$\overset{´}{}$

$\bar{M}$

$\bar{r}$

$\overset{´}{}$

$t$

$\bar{r}$

$\bar{r}$

$\overset{´}{}$

$c$

$r$

$\overset{´}{}$

$r$

$\bar{r}$

$\bar{r}$

$\overset{´}{}$

$\bar{M}$

$\bar{r}$

$\overset{´}{}$

$t$

$\bar{r}$

$\bar{r}$

$\overset{´}{}$

$c$

$r$

$\overset{´}{}$

$r$

$\overset{´}{}$

$\bar{r}$

$\bar{r}$

$\overset{´}{}$

$\bar{M}$

$\bar{r}$

$\overset{´}{}$

$t$

$\bar{r}$

$\bar{r}$

$\overset{´}{}$

$c$

$r$

$\overset{´}{}$

$\bar{r}$

$\bar{r}$

$\overset{´}{}$

$r$

$\overset{´}{}$

$\bar{M}$

$\bar{r}$

$\overset{´}{}$

$t$

$\overset{´}{}$

$t$

$\overset{´}{}$

$t$

$\bar{r}$

$\bar{r}$

$\overset{´}{}$

$c$

$r$

$\overset{´}{}$

$r$

$\overset{´}{}$

$\bar{r}$

$\bar{r}$

$\overset{´}{}$

$\bar{M}$

$\bar{r}$

$\overset{´}{}$

$t$

$\bar{r}$

$\bar{r}$

$\overset{´}{}$

$c$

$\bar{A}$

$\bar{r}$

$t$

$μ_{0}$

$π$

$r$

$\overset{´}{}$

$\bar{r}$

$\bar{r}$

$\overset{´}{}$

$\dot{\bar{P}}$

$\bar{r}$

$\overset{´}{}$

$t$

$\bar{r}$

$\bar{r}$

$\overset{´}{}$

$c$

$\bar{r}$

$\bar{r}$

$\overset{´}{}$

$r$

$\overset{´}{}$

$\bar{M}$

$\bar{r}$

$\overset{´}{}$

$t$

$\overset{´}{}$

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General

LaTeXML (experimentell; verwendet MathML) rendering

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Translations to Computer Algebra Systems

Translation to Maple

Translation to Mathematica

Similar pages

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MathML observations

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