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

$$=$$

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

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

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

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

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

$$= \sum_{ijk}\alpha^i_j\alpha^i_k\D{}{x^j}\left(\frac{A^k} {\sqrt{g_... ...ght) =\sum_{jk}g_{jj}\delta{jk}\D{}{x^j}\left(\frac{A^k} {\sqrt{g_{kk}}}\right)$$

$$= \sum_jg_{jj}\D{}{x^j}\left(\frac{A^j}{\sqrt{g_{jj}}}\right)$$ (C.70)

¶ËºÂɸ¤Î¾ì¹ç

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

$$dx = \sin\theta\cos\phi\,dr+r\cos\theta\cos\phi\,d\theta -r\sin\theta\sin\phi\,d\phi$$

$$dy = \sin\theta\sin\phi\,dr+r\cos\theta\sin\phi\,d\theta +r\sin\theta\cos\phi\,d\phi$$

$$dz = \cos\theta\,dr-r\sin\theta\,d\theta$$ (C.72)

$$\D fr = \D fx\sin\theta\cos\phi +\D fy\sin\theta\sin\phi+\D fz\cos\theta$$

$$\D f\theta = -\D fxr\cos\theta\cos\phi +\D fyr\sin\theta\cos\phi$$

$$\D f\phi = -\D fx\sin\theta\sin\phi +\D fy\sin\theta\sin\phi$$ (C.73)

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

$$\emph e_r = \sin\theta\cos\phi\,\emph e_x +\sin\theta\sin\phi\,\emph e_y +\cos\theta\,\emph e_z$$

$$\emph e_\theta = \cos\theta\cos\phi\,\emph e_x +\cos\theta\sin\phi\,\emph e_y -\sin\theta\,\emph e_z$$

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

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

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

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

$$(\mathop{\mathop{\emph ¢¦}\nolimits \times}\nolimits \emph A)_r = \frac1{r\sin\theta}\left\{\D{}\theta (A_\phi\sin\theta)-\D{A_\theta}\phi\right\}$$

$$(\mathop{\mathop{\emph ¢¦}\nolimits \times}\nolimits \emph A)_\theta = \frac1{r\sin\theta}\left\{\D{A_r}\phi -\sin\theta\D{}r(rA_\phi)\right\}$$

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

±ßÃìºÂɸ¤Î¾ì¹ç

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

$$dx = \cos\theta\,dr-r\sin\theta\,d\theta$$

$$dy = \sin\theta\,dr+r\cos\theta\,d\theta$$

$$dz = dz$$ (C.81)

$$\D fr =$$

$$\D f\theta =$$

$$\D fz =$$ (C.82)

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

$$\emph e_r =$$

$$\emph e_\theta =$$

$$\emph e_z =$$ (C.84)

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

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

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

$$(\mathop{\mathop{\emph ¢¦}\nolimits \times}\nolimits \emph A)_r = \frac1r\D{A_z}\theta-\D{A_\theta}z$$

$$(\mathop{\mathop{\emph ¢¦}\nolimits \times}\nolimits \emph A)_\theta = \D{A_r}z-\D{A_z}r$$

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