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$\displaystyle \mathop{\mathop{\emph ¢¦}\nolimits \cdot}\nolimits \emph a$ $\displaystyle =$ $\displaystyle \partial_ia^i
=\partial_i\left(\sum_\mu\alpha^i_\mu a^\mu\right)$  
  $\displaystyle =$    

$\displaystyle \D{g_{\mu\nu}}{x^\lambda}=\sum_i\alpha^i_\mu\D{\alpha^i_\nu}{x^\l...
...a^i_\mu}{x^\lambda}\alpha^i_\nu =2\sum_i\alpha^i_\mu\D{\alpha^i_\nu}{x_\lambda}$ (C.66)

$\displaystyle \D{\alpha^i_\nu}{x^\lambda}=\frac12\sum_i\alpha^i_\mu\D{g_{\mu\nu}}{x^\lambda}$ (C.67)

$\displaystyle \emph A=\sum_\mu A^\mu\emph e_\mu =\sum_{i\mu}A^\mu\frac{\alpha^i_\mu}{\sqrt{g_{\mu\mu}}}\emph e_i$ (C.68)

$\displaystyle A^i=\sum_\mu\alpha^i_\mu\frac{A^\mu}{\sqrt{g_{\mu\mu}}}$ (C.69)


$\displaystyle (\mathop{\mathop{\emph ¢¦}\nolimits \cdot}\nolimits \emph A)$ $\displaystyle =$ $\displaystyle (\mathop{\mathop{\emph ¢¦}\nolimits \cdot}\nolimits \emph A)_0=\s...
...=\sum_{ijk}\alpha^i_j\D{}{x^j}\left(\alpha^i_k\frac{A^k}
{\sqrt{g_{kk}}}\right)$  
  $\displaystyle =$ $\displaystyle \sum_{ijk}\alpha^i_j\alpha^i_k\D{}{x^j}\left(\frac{A^k}
{\sqrt{g_...
...ght)
=\sum_{jk}g_{jj}\delta{jk}\D{}{x^j}\left(\frac{A^k}
{\sqrt{g_{kk}}}\right)$  
  $\displaystyle =$ $\displaystyle \sum_jg_{jj}\D{}{x^j}\left(\frac{A^j}{\sqrt{g_{jj}}}\right)$ (C.70)

¶ËºÂɸ¤Î¾ì¹ç

$\displaystyle x=r\sin\theta\cos\phi \hspace{2em} y=r\sin\theta\sin\phi \hspace{2em} z=r\cos\theta$ (C.71)


$\displaystyle dx$ $\displaystyle =$ $\displaystyle \sin\theta\cos\phi\,dr+r\cos\theta\cos\phi\,d\theta
-r\sin\theta\sin\phi\,d\phi$  
$\displaystyle dy$ $\displaystyle =$ $\displaystyle \sin\theta\sin\phi\,dr+r\cos\theta\sin\phi\,d\theta
+r\sin\theta\cos\phi\,d\phi$  
$\displaystyle dz$ $\displaystyle =$ $\displaystyle \cos\theta\,dr-r\sin\theta\,d\theta$ (C.72)


$\displaystyle \D fr$ $\displaystyle =$ $\displaystyle \D fx\sin\theta\cos\phi
+\D fy\sin\theta\sin\phi+\D fz\cos\theta$  
$\displaystyle \D f\theta$ $\displaystyle =$ $\displaystyle -\D fxr\cos\theta\cos\phi
+\D fyr\sin\theta\cos\phi$  
$\displaystyle \D f\phi$ $\displaystyle =$ $\displaystyle -\D fx\sin\theta\sin\phi
+\D fy\sin\theta\sin\phi$ (C.73)

$\displaystyle g_{rr}=1 \hspace{4em} g_{\theta\theta}=r \hspace{4em} g_{\phi\phi}=r\sin\theta$ (C.74)


$\displaystyle \emph e_r$ $\displaystyle =$ $\displaystyle \sin\theta\cos\phi\,\emph e_x
+\sin\theta\sin\phi\,\emph e_y
+\cos\theta\,\emph e_z$  
$\displaystyle \emph e_\theta$ $\displaystyle =$ $\displaystyle \cos\theta\cos\phi\,\emph e_x
+\cos\theta\sin\phi\,\emph e_y
-\sin\theta\,\emph e_z$  
$\displaystyle \emph e_\phi$ $\displaystyle =$ $\displaystyle -\sin\phi\,\emph e_x+\cos\phi\,\emph e_y$ (C.75)

$\displaystyle \mathop{\emph ¢¦}\nolimits ^2f=\frac1{r^2}\D{}r\left(r^2\D fr\rig...
...\theta\left(\sin\theta\D f\theta\right) +\frac1{r^2\sin^2\theta}\D{^2f}{\phi^2}$ (C.76)

$\displaystyle (\mathop{\mathop{\emph ¢¦}\nolimits }\nolimits f)_r=\D fr \hspace...
...athop{\mathop{\emph ¢¦}\nolimits }\nolimits f)_\phi=\frac1{r\sin\theta}\D f\phi$ (C.77)

$\displaystyle \mathop{\mathop{\emph ¢¦}\nolimits \cdot}\nolimits \emph A=\frac1...
...ac1{r\sin\theta} \left\{\D{}\theta(\sin\theta A_\theta) +\D{A_\phi}\phi\right\}$ (C.78)


$\displaystyle (\mathop{\mathop{\emph ¢¦}\nolimits \times}\nolimits \emph A)_r$ $\displaystyle =$ $\displaystyle \frac1{r\sin\theta}\left\{\D{}\theta
(A_\phi\sin\theta)-\D{A_\theta}\phi\right\}$  
$\displaystyle (\mathop{\mathop{\emph ¢¦}\nolimits \times}\nolimits \emph A)_\theta$ $\displaystyle =$ $\displaystyle \frac1{r\sin\theta}\left\{\D{A_r}\phi
-\sin\theta\D{}r(rA_\phi)\right\}$  
$\displaystyle (\mathop{\mathop{\emph ¢¦}\nolimits \times}\nolimits \emph A)_\phi$ $\displaystyle =$ $\displaystyle \frac1r\left\{
\D{}r(rA_\theta)-\D{A_r}\theta\right\}$ (C.79)

±ßÃìºÂɸ¤Î¾ì¹ç

$\displaystyle x=r\cos\theta \hspace{4em} y=r\sin\theta \hspace{4em} z=z$ (C.80)


$\displaystyle dx$ $\displaystyle =$ $\displaystyle \cos\theta\,dr-r\sin\theta\,d\theta$  
$\displaystyle dy$ $\displaystyle =$ $\displaystyle \sin\theta\,dr+r\cos\theta\,d\theta$  
$\displaystyle dz$ $\displaystyle =$ $\displaystyle dz$ (C.81)


$\displaystyle \D fr$ $\displaystyle =$    
$\displaystyle \D f\theta$ $\displaystyle =$    
$\displaystyle \D fz$ $\displaystyle =$   (C.82)

$\displaystyle g_{rr}= \hspace{4em} g_{\theta\theta}= \hspace{4em} g_{\phi\phi}=$ (C.83)


$\displaystyle \emph e_r$ $\displaystyle =$    
$\displaystyle \emph e_\theta$ $\displaystyle =$    
$\displaystyle \emph e_z$ $\displaystyle =$   (C.84)

$\displaystyle \mathop{\emph ¢¦}\nolimits ^2f=\D{^2f}{r^2}+\frac1r\D fr +\frac1{r^2}\D{^2f}{\theta^2}+\D{^2f}{z^2}$ (C.85)

$\displaystyle (\mathop{\mathop{\emph ¢¦}\nolimits }\nolimits f)_r=\D fr \hspace...
... f\theta \hspace{2em} (\mathop{\mathop{\emph ¢¦}\nolimits }\nolimits f)_z=\D fz$ (C.86)

$\displaystyle \mathop{\mathop{\emph ¢¦}\nolimits \cdot}\nolimits \emph A=\D{A_r}r+\frac{A_r}r+\frac1r \D{A_\theta}\theta+\D{A_z}z$ (C.87)


$\displaystyle (\mathop{\mathop{\emph ¢¦}\nolimits \times}\nolimits \emph A)_r$ $\displaystyle =$ $\displaystyle \frac1r\D{A_z}\theta-\D{A_\theta}z$  
$\displaystyle (\mathop{\mathop{\emph ¢¦}\nolimits \times}\nolimits \emph A)_\theta$ $\displaystyle =$ $\displaystyle \D{A_r}z-\D{A_z}r$  
$\displaystyle (\mathop{\mathop{\emph ¢¦}\nolimits \times}\nolimits \emph A)_z$ $\displaystyle =$ $\displaystyle \frac1r\left\{
\D{}r(rA_\theta)-\D{A_r}\theta\right\}$ (C.88)


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Next: »Í¸µ¥Ù¥¯¥È¥ë Up: ºÂɸÊÑ´¹ Previous: ¸ûÇÛ   Contents   Index
Yoichi OKABE 2008-03-29