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* Page found: Elektrodynamik Schöll (eq math.1199.154)

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\begin{align}
& {{Q}_{2}}=\frac{1}{2}\int_{-\infty }^{\infty }{{}}{{d}^{3}}r\acute{\ }\rho (\bar{r}\acute{\ })r{{\acute{\ }}^{2}}\left( 3{{\cos }^{2}}\vartheta -1 \right)=\frac{1}{2}\int_{-\infty }^{\infty }{{}}{{d}^{3}}r\acute{\ }\rho (\bar{r}\acute{\ })\left( 3\frac{\bar{r}\acute{\ }\cdot \bar{r}}{r}\frac{\bar{r}\acute{\ }\cdot \bar{r}}{r}-\bar{r}{{\acute{\ }}^{2}} \right) \\
& \frac{\bar{r}\acute{\ }\cdot \bar{r}}{r}\frac{\bar{r}\acute{\ }\cdot \bar{r}}{r}=\frac{{{x}_{k}}\acute{\ }{{x}_{k}}{{x}_{l}}\acute{\ }{{x}_{l}}}{{{r}^{2}}} \\
& \Rightarrow {{Q}_{2}}=\frac{1}{2{{r}^{2}}}\int_{-\infty }^{\infty }{{}}{{d}^{3}}r\acute{\ }\rho (\bar{r}\acute{\ })\left( 3{{x}_{k}}\acute{\ }{{x}_{l}}\acute{\ }-\bar{r}{{\acute{\ }}^{2}}{{\delta }_{kl}} \right) \\
\end{align}

TeX (checked): 

{\begin{aligned}&{{Q}_{2}}={\frac {1}{2}}\int _{-\infty }^{\infty }{}{{d}^{3}}r{\acute {\ }}\rho ({\bar {r}}{\acute {\ }})r{{\acute {\ }}^{2}}\left(3{{\cos }^{2}}\vartheta -1\right)={\frac {1}{2}}\int _{-\infty }^{\infty }{}{{d}^{3}}r{\acute {\ }}\rho ({\bar {r}}{\acute {\ }})\left(3{\frac {{\bar {r}}{\acute {\ }}\cdot {\bar {r}}}{r}}{\frac {{\bar {r}}{\acute {\ }}\cdot {\bar {r}}}{r}}-{\bar {r}}{{\acute {\ }}^{2}}\right)\\&{\frac {{\bar {r}}{\acute {\ }}\cdot {\bar {r}}}{r}}{\frac {{\bar {r}}{\acute {\ }}\cdot {\bar {r}}}{r}}={\frac {{{x}_{k}}{\acute {\ }}{{x}_{k}}{{x}_{l}}{\acute {\ }}{{x}_{l}}}{{r}^{2}}}\\&\Rightarrow {{Q}_{2}}={\frac {1}{2{{r}^{2}}}}\int _{-\infty }^{\infty }{}{{d}^{3}}r{\acute {\ }}\rho ({\bar {r}}{\acute {\ }})\left(3{{x}_{k}}{\acute {\ }}{{x}_{l}}{\acute {\ }}-{\bar {r}}{{\acute {\ }}^{2}}{{\delta }_{kl}}\right)\\\end{aligned}}

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Q2=12∫−∞∞d3r´ρ(r¯´)r´2(3cos2ϑ−1)=12∫−∞∞d3r´ρ(r¯´)(3r¯´⋅r¯rr¯´⋅r¯r−r¯´2)r¯´⋅r¯rr¯´⋅r¯r=xk´xkxl´xlr2⇒Q2=12r2∫−∞∞d3r´ρ(r¯´)(3xk´xl´−r¯´2δkl)
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Identifiers

  • Q2
  • r
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  • ρ
  • r¯
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  • ϑ
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  • ρ
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  • r¯
  • r
  • r¯
  • ´
  • r¯
  • r
  • r¯
  • ´
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  • r¯
  • r
  • r¯
  • ´
  • r¯
  • r
  • xk
  • ´
  • xk
  • xl
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  • Q2
  • r
  • r
  • ´
  • ρ
  • r¯
  • ´
  • xk
  • ´
  • xl
  • ´
  • r¯
  • ´
  • δkl

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