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Display information for equation id:math.1199.327 on revision:1199

* Page found: Elektrodynamik Schöll (eq math.1199.327)

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TeX (original user input): 

\begin{align}
& \nabla \times \bar{B}(\bar{r})=\nabla \times \left( \nabla \times \bar{A}(\bar{r}) \right)=\nabla \left( \nabla \cdot \bar{A}(\bar{r}) \right)-\Delta \bar{A}(\bar{r}) \\
& \nabla \cdot \bar{A}(\bar{r})=\nabla \cdot \frac{{{\mu }_{0}}}{4\pi }\int_{{{R}^{3}}}^{{}}{{}}{{d}^{3}}r\acute{\ }\frac{\bar{j}(\bar{r}\acute{\ })}{\left| \bar{r}-\bar{r}\acute{\ } \right|}=\frac{{{\mu }_{0}}}{4\pi }\int_{{{R}^{3}}}^{{}}{{}}{{d}^{3}}r\acute{\ }{{\nabla }_{r}}\cdot \left( \frac{\bar{j}(\bar{r}\acute{\ })}{\left| \bar{r}-\bar{r}\acute{\ } \right|} \right)=\frac{{{\mu }_{0}}}{4\pi }\int_{{{R}^{3}}}^{{}}{{}}{{d}^{3}}r\acute{\ }\bar{j}(\bar{r}\acute{\ }){{\nabla }_{r}}\cdot \frac{1}{\left| \bar{r}-\bar{r}\acute{\ } \right|} \\
& {{\nabla }_{r}}\cdot \frac{1}{\left| \bar{r}-\bar{r}\acute{\ } \right|}=-{{\nabla }_{r\acute{\ }}}\cdot \frac{1}{\left| \bar{r}-\bar{r}\acute{\ } \right|} \\
& \Rightarrow \nabla \cdot \bar{A}(\bar{r})=-\frac{{{\mu }_{0}}}{4\pi }\int_{{{R}^{3}}}^{{}}{{}}{{d}^{3}}r\acute{\ }\left[ {{\nabla }_{r\acute{\ }}}\cdot \left( \frac{\bar{j}(\bar{r}\acute{\ })}{\left| \bar{r}-\bar{r}\acute{\ } \right|} \right)+\frac{1}{\left| \bar{r}-\bar{r}\acute{\ } \right|}{{\nabla }_{r\acute{\ }}}\cdot \bar{j}(\bar{r}\acute{\ }) \right] \\
& {{\nabla }_{r\acute{\ }}}\cdot \bar{j}(\bar{r}\acute{\ })=-\frac{\partial }{\partial t}\rho =0 \\
& \Rightarrow \nabla \cdot \bar{A}(\bar{r})=-\frac{{{\mu }_{0}}}{4\pi }\int_{{{R}^{3}}}^{{}}{{}}{{d}^{3}}r\acute{\ }{{\nabla }_{r\acute{\ }}}\cdot \left( \frac{\bar{j}(\bar{r}\acute{\ })}{\left| \bar{r}-\bar{r}\acute{\ } \right|} \right) \\
\end{align}

TeX (checked): 

{\begin{aligned}&\nabla \times {\bar {B}}({\bar {r}})=\nabla \times \left(\nabla \times {\bar {A}}({\bar {r}})\right)=\nabla \left(\nabla \cdot {\bar {A}}({\bar {r}})\right)-\Delta {\bar {A}}({\bar {r}})\\&\nabla \cdot {\bar {A}}({\bar {r}})=\nabla \cdot {\frac {{\mu }_{0}}{4\pi }}\int _{{R}^{3}}^{}{}{{d}^{3}}r{\acute {\ }}{\frac {{\bar {j}}({\bar {r}}{\acute {\ }})}{\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}}={\frac {{\mu }_{0}}{4\pi }}\int _{{R}^{3}}^{}{}{{d}^{3}}r{\acute {\ }}{{\nabla }_{r}}\cdot \left({\frac {{\bar {j}}({\bar {r}}{\acute {\ }})}{\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}}\right)={\frac {{\mu }_{0}}{4\pi }}\int _{{R}^{3}}^{}{}{{d}^{3}}r{\acute {\ }}{\bar {j}}({\bar {r}}{\acute {\ }}){{\nabla }_{r}}\cdot {\frac {1}{\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}}\\&{{\nabla }_{r}}\cdot {\frac {1}{\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}}=-{{\nabla }_{r{\acute {\ }}}}\cdot {\frac {1}{\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}}\\&\Rightarrow \nabla \cdot {\bar {A}}({\bar {r}})=-{\frac {{\mu }_{0}}{4\pi }}\int _{{R}^{3}}^{}{}{{d}^{3}}r{\acute {\ }}\left[{{\nabla }_{r{\acute {\ }}}}\cdot \left({\frac {{\bar {j}}({\bar {r}}{\acute {\ }})}{\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}}\right)+{\frac {1}{\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}}{{\nabla }_{r{\acute {\ }}}}\cdot {\bar {j}}({\bar {r}}{\acute {\ }})\right]\\&{{\nabla }_{r{\acute {\ }}}}\cdot {\bar {j}}({\bar {r}}{\acute {\ }})=-{\frac {\partial }{\partial t}}\rho =0\\&\Rightarrow \nabla \cdot {\bar {A}}({\bar {r}})=-{\frac {{\mu }_{0}}{4\pi }}\int _{{R}^{3}}^{}{}{{d}^{3}}r{\acute {\ }}{{\nabla }_{r{\acute {\ }}}}\cdot \left({\frac {{\bar {j}}({\bar {r}}{\acute {\ }})}{\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}}\right)\\\end{aligned}}

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∇×B¯(r¯)=∇×(∇×A¯(r¯))=∇(∇⋅A¯(r¯))−ΔA¯(r¯)∇⋅A¯(r¯)=∇⋅μ04π∫R3d3r´j¯(r¯´)|r¯−r¯´|=μ04π∫R3d3r´∇r⋅(j¯(r¯´)|r¯−r¯´|)=μ04π∫R3d3r´j¯(r¯´)∇r⋅1|r¯−r¯´|∇r⋅1|r¯−r¯´|=−∇r´⋅1|r¯−r¯´|⇒∇⋅A¯(r¯)=−μ04π∫R3d3r´[∇r´⋅(j¯(r¯´)|r¯−r¯´|)+1|r¯−r¯´|∇r´⋅j¯(r¯´)]∇r´⋅j¯(r¯´)=−∂∂tρ=0⇒∇⋅A¯(r¯)=−μ04π∫R3d3r´∇r´⋅(j¯(r¯´)|r¯−r¯´|)
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data-mjx-texclass="ORD"><mover><mi>j</mi><mo>¯</mo></mover></mrow></mrow><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mspace width="0.5em"/><mo data-mjx-pseudoscript="true">´</mo></mover></mrow></mrow><mo stretchy="false">)</mo><mo>=</mo><mo>&#x2212;</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>&#x2202;</mi></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>&#x2202;</mi><mi>t</mi></mrow></mrow></mfrac></mrow><mi>&#x03C1;</mi><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd></mtd><mtd><mo>&#x21D2;</mo><mi mathvariant="normal">&#x2207;</mi><mo>&#x22C5;</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>A</mi><mo>¯</mo></mover></mrow></mrow><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mo stretchy="false">)</mo><mo>=</mo><mo>&#x2212;</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><msub><mi>&#x03BC;</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>4</mn><mi>&#x03C0;</mi></mrow></mrow></mfrac></mrow><mstyle displaystyle="true" scriptlevel="0"><munderover><mo texclass="OP">&#x222B;</mo><mrow data-mjx-texclass="ORD"><msup><mi>R</mi><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msup></mrow><mrow data-mjx-texclass="ORD"></mrow></munderover></mstyle><msup><mi>d</mi><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msup><mi>r</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mspace width="0.5em"/><mo data-mjx-pseudoscript="true">´</mo></mover></mrow></mrow><msub><mi mathvariant="normal">&#x2207;</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>r</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mspace width="0.5em"/><mo data-mjx-pseudoscript="true">´</mo></mover></mrow></mrow></mrow></mrow></msub><mo>&#x22C5;</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>j</mi><mo>¯</mo></mover></mrow></mrow><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mspace width="0.5em"/><mo data-mjx-pseudoscript="true">´</mo></mover></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mo>&#x2212;</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mspace width="0.5em"/><mo data-mjx-pseudoscript="true">´</mo></mover></mrow></mrow><mo data-mjx-texclass="CLOSE">|</mo></mrow></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd></mtr><mtr><mtd></mtd></mtr></mtable></mrow></mstyle></mrow></math>

Translations to Computer Algebra Systems

Translation to Maple

In Maple:

Translation to Mathematica

In Mathematica:

Similar pages

Calculated based on the variables occurring on the entire Elektrodynamik Schöll page

Identifiers

  • B¯
  • r¯
  • A¯
  • r¯
  • A¯
  • r¯
  • Δ
  • A¯
  • r¯
  • A¯
  • r¯
  • μ0
  • π
  • R
  • r
  • ´
  • j¯
  • r¯
  • ´
  • r¯
  • r¯
  • ´
  • μ0
  • π
  • R
  • r
  • ´
  • r
  • j¯
  • r¯
  • ´
  • r¯
  • r¯
  • ´
  • μ0
  • π
  • R
  • r
  • ´
  • j¯
  • r¯
  • ´
  • r
  • r¯
  • r¯
  • ´
  • r
  • r¯
  • r¯
  • ´
  • r
  • ´
  • r¯
  • r¯
  • ´
  • A¯
  • r¯
  • μ0
  • π
  • R
  • r
  • ´
  • r
  • ´
  • j¯
  • r¯
  • ´
  • r¯
  • r¯
  • ´
  • r¯
  • r¯
  • ´
  • r
  • ´
  • j¯
  • r¯
  • ´
  • r
  • ´
  • j¯
  • r¯
  • ´
  • t
  • ρ
  • A¯
  • r¯
  • μ0
  • π
  • R
  • r
  • ´
  • r
  • ´
  • j¯
  • r¯
  • ´
  • r¯
  • r¯
  • ´

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