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* Page found: Elektrodynamik Schöll (eq math.1199.320)

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\begin{align}
& rot\bar{A}(\bar{r})=\nabla \times \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}}\frac{1}{\left| \bar{r}-\bar{r}\acute{\ } \right|}\times \bar{j}(\bar{r}\acute{\ }) \\
& {{\nabla }_{r}}\frac{1}{\left| \bar{r}-\bar{r}\acute{\ } \right|}=-\frac{\bar{r}-\bar{r}\acute{\ }}{{{\left| \bar{r}-\bar{r}\acute{\ } \right|}^{3}}} \\
& \Rightarrow rot\bar{A}(\bar{r})=\frac{{{\mu }_{0}}}{4\pi }\int_{{{R}^{3}}}^{{}}{{}}{{d}^{3}}r\acute{\ }\bar{j}(\bar{r}\acute{\ })\times \frac{\bar{r}-\bar{r}\acute{\ }}{{{\left| \bar{r}-\bar{r}\acute{\ } \right|}^{3}}}=\bar{B}(\bar{r}) \\
\end{align}

TeX (checked): 

{\begin{aligned}&rot{\bar {A}}({\bar {r}})=\nabla \times {\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}}{\frac {1}{\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}}\times {\bar {j}}({\bar {r}}{\acute {\ }})\\&{{\nabla }_{r}}{\frac {1}{\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}}=-{\frac {{\bar {r}}-{\bar {r}}{\acute {\ }}}{{\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}^{3}}}\\&\Rightarrow rot{\bar {A}}({\bar {r}})={\frac {{\mu }_{0}}{4\pi }}\int _{{R}^{3}}^{}{}{{d}^{3}}r{\acute {\ }}{\bar {j}}({\bar {r}}{\acute {\ }})\times {\frac {{\bar {r}}-{\bar {r}}{\acute {\ }}}{{\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}^{3}}}={\bar {B}}({\bar {r}})\\\end{aligned}}

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rotA¯(r¯)=∇×μ04π∫R3d3r´j¯(r¯´)|r¯−r¯´|=μ04π∫R3d3r´∇r1|r¯−r¯´|×j¯(r¯´)∇r1|r¯−r¯´|=−r¯−r¯´|r¯−r¯´|3⇒rotA¯(r¯)=μ04π∫R3d3r´j¯(r¯´)×r¯−r¯´|r¯−r¯´|3=B¯(r¯)
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Calculated based on the variables occurring on the entire Elektrodynamik Schöll page

Identifiers

  • r
  • o
  • t
  • A¯
  • r¯
  • μ0
  • π
  • R
  • r
  • ´
  • j¯
  • r¯
  • ´
  • r¯
  • r¯
  • ´
  • μ0
  • π
  • R
  • r
  • ´
  • r
  • r¯
  • r¯
  • ´
  • j¯
  • r¯
  • ´
  • r
  • r¯
  • r¯
  • ´
  • r¯
  • r¯
  • ´
  • r¯
  • r¯
  • ´
  • r
  • o
  • t
  • A¯
  • r¯
  • μ0
  • π
  • R
  • r
  • ´
  • j¯
  • r¯
  • ´
  • r¯
  • r¯
  • ´
  • r¯
  • r¯
  • ´
  • B¯
  • r¯

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