구면좌표계

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개요

  • \(\rho ,\phi ,\theta\)
  • \(x=\rho \cos\phi \, \sin\theta\)
  • \(y=\rho \sin\phi \, \sin\theta\)
  • \(z=\rho \cos\theta\)
  • \(\rho>0\), \(0<\phi<2\pi\), \(0<\theta<\pi\)



메트릭 텐서

\(\left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & \rho ^2 \sin ^2(\theta ) & 0 \\ 0 & 0 & \rho ^2 \end{array} \right)\)



라플라시안

  • 라플라시안\[\Delta f = {1 \over r^2} {\partial \over \partial r} \left( r^2 {\partial f \over \partial r} \right) + {1 \over r^2 \sin \theta} {\partial \over \partial \theta} \left( \sin \theta {\partial f \over \partial \theta} \right) + {1 \over r^2 \sin^2 \theta} {\partial^2 f \over \partial \phi^2}\]



크리스토펠 기호

  • 크리스토펠 기호 항목 참조\[\begin{array}{ll} \Gamma _{11}^1 & 0 \\ \Gamma _{12}^1 & 0 \\ \Gamma _{13}^1 & 0 \\ \Gamma _{21}^1 & 0 \\ \Gamma _{22}^1 & -\rho \sin ^2(\theta ) \\ \Gamma _{23}^1 & 0 \\ \Gamma _{31}^1 & 0 \\ \Gamma _{32}^1 & 0 \\ \Gamma _{33}^1 & -\rho \\ \Gamma _{11}^2 & 0 \\ \Gamma _{12}^2 & \frac{1}{\rho } \\ \Gamma _{13}^2 & 0 \\ \Gamma _{21}^2 & \frac{1}{\rho } \\ \Gamma _{22}^2 & 0 \\ \Gamma _{23}^2 & \cot (\theta ) \\ \Gamma _{31}^2 & 0 \\ \Gamma _{32}^2 & \cot (\theta ) \\ \Gamma _{33}^2 & 0 \\ \Gamma _{11}^3 & 0 \\ \Gamma _{12}^3 & 0 \\ \Gamma _{13}^3 & \frac{1}{\rho } \\ \Gamma _{21}^3 & 0 \\ \Gamma _{22}^3 & \sin (\theta ) (-\cos (\theta )) \\ \Gamma _{23}^3 & 0 \\ \Gamma _{31}^3 & \frac{1}{\rho } \\ \Gamma _{32}^3 & 0 \\ \Gamma _{33}^3 & 0 \end{array}\]

리만 곡률 텐서

  • 리만 곡률 텐서\[\begin{array}{lll} \begin{array}{ll} R_{111}^1 & 0 \\ R_{112}^1 & 0 \\ R_{113}^1 & 0 \end{array} & \begin{array}{ll} R_{121}^1 & 0 \\ R_{122}^1 & 0 \\ R_{123}^1 & 0 \end{array} & \begin{array}{ll} R_{131}^1 & 0 \\ R_{132}^1 & 0 \\ R_{133}^1 & 0 \end{array} \\ \begin{array}{ll} R_{211}^1 & 0 \\ R_{212}^1 & 0 \\ R_{213}^1 & 0 \end{array} & \begin{array}{ll} R_{221}^1 & 0 \\ R_{222}^1 & 0 \\ R_{223}^1 & 0 \end{array} & \begin{array}{ll} R_{231}^1 & 0 \\ R_{232}^1 & 0 \\ R_{233}^1 & 0 \end{array} \\ \begin{array}{ll} R_{311}^1 & 0 \\ R_{312}^1 & 0 \\ R_{313}^1 & 0 \end{array} & \begin{array}{ll} R_{321}^1 & 0 \\ R_{322}^1 & 0 \\ R_{323}^1 & 0 \end{array} & \begin{array}{ll} R_{331}^1 & 0 \\ R_{332}^1 & 0 \\ R_{333}^1 & 0 \end{array} \\ \begin{array}{ll} R_{111}^2 & 0 \\ R_{112}^2 & 0 \\ R_{113}^2 & 0 \end{array} & \begin{array}{ll} R_{121}^2 & 0 \\ R_{122}^2 & 0 \\ R_{123}^2 & 0 \end{array} & \begin{array}{ll} R_{131}^2 & 0 \\ R_{132}^2 & 0 \\ R_{133}^2 & 0 \end{array} \\ \begin{array}{ll} R_{211}^2 & 0 \\ R_{212}^2 & 0 \\ R_{213}^2 & 0 \end{array} & \begin{array}{ll} R_{221}^2 & 0 \\ R_{222}^2 & 0 \\ R_{223}^2 & 0 \end{array} & \begin{array}{ll} R_{231}^2 & 0 \\ R_{232}^2 & 0 \\ R_{233}^2 & 0 \end{array} \\ \begin{array}{ll} R_{311}^2 & 0 \\ R_{312}^2 & 0 \\ R_{313}^2 & 0 \end{array} & \begin{array}{ll} R_{321}^2 & 0 \\ R_{322}^2 & 0 \\ R_{323}^2 & 0 \end{array} & \begin{array}{ll} R_{331}^2 & 0 \\ R_{332}^2 & 0 \\ R_{333}^2 & 0 \end{array} \\ \begin{array}{ll} R_{111}^3 & 0 \\ R_{112}^3 & 0 \\ R_{113}^3 & 0 \end{array} & \begin{array}{ll} R_{121}^3 & 0 \\ R_{122}^3 & 0 \\ R_{123}^3 & 0 \end{array} & \begin{array}{ll} R_{131}^3 & 0 \\ R_{132}^3 & 0 \\ R_{133}^3 & 0 \end{array} \\ \begin{array}{ll} R_{211}^3 & 0 \\ R_{212}^3 & 0 \\ R_{213}^3 & 0 \end{array} & \begin{array}{ll} R_{221}^3 & 0 \\ R_{222}^3 & 0 \\ R_{223}^3 & 0 \end{array} & \begin{array}{ll} R_{231}^3 & 0 \\ R_{232}^3 & 0 \\ R_{233}^3 & 0 \end{array} \\ \begin{array}{ll} R_{311}^3 & 0 \\ R_{312}^3 & 0 \\ R_{313}^3 & 0 \end{array} & \begin{array}{ll} R_{321}^3 & 0 \\ R_{322}^3 & 0 \\ R_{323}^3 & 0 \end{array} & \begin{array}{ll} R_{331}^3 & 0 \\ R_{332}^3 & 0 \\ R_{333}^3 & 0 \end{array} \end{array}\]
  • covariant tensor\[\begin{array}{lll} \begin{array}{ll} R_{1111} & 0 \\ R_{1112} & 0 \\ R_{1113} & 0 \end{array} & \begin{array}{ll} R_{1121} & 0 \\ R_{1122} & 0 \\ R_{1123} & 0 \end{array} & \begin{array}{ll} R_{1131} & 0 \\ R_{1132} & 0 \\ R_{1133} & 0 \end{array} \\ \begin{array}{ll} R_{1211} & 0 \\ R_{1212} & 0 \\ R_{1213} & 0 \end{array} & \begin{array}{ll} R_{1221} & 0 \\ R_{1222} & 0 \\ R_{1223} & 0 \end{array} & \begin{array}{ll} R_{1231} & 0 \\ R_{1232} & 0 \\ R_{1233} & 0 \end{array} \\ \begin{array}{ll} R_{1311} & 0 \\ R_{1312} & 0 \\ R_{1313} & 0 \end{array} & \begin{array}{ll} R_{1321} & 0 \\ R_{1322} & 0 \\ R_{1323} & 0 \end{array} & \begin{array}{ll} R_{1331} & 0 \\ R_{1332} & 0 \\ R_{1333} & 0 \end{array} \\ \begin{array}{ll} R_{2111} & 0 \\ R_{2112} & 0 \\ R_{2113} & 0 \end{array} & \begin{array}{ll} R_{2121} & 0 \\ R_{2122} & 0 \\ R_{2123} & 0 \end{array} & \begin{array}{ll} R_{2131} & 0 \\ R_{2132} & 0 \\ R_{2133} & 0 \end{array} \\ \begin{array}{ll} R_{2211} & 0 \\ R_{2212} & 0 \\ R_{2213} & 0 \end{array} & \begin{array}{ll} R_{2221} & 0 \\ R_{2222} & 0 \\ R_{2223} & 0 \end{array} & \begin{array}{ll} R_{2231} & 0 \\ R_{2232} & 0 \\ R_{2233} & 0 \end{array} \\ \begin{array}{ll} R_{2311} & 0 \\ R_{2312} & 0 \\ R_{2313} & 0 \end{array} & \begin{array}{ll} R_{2321} & 0 \\ R_{2322} & 0 \\ R_{2323} & 0 \end{array} & \begin{array}{ll} R_{2331} & 0 \\ R_{2332} & 0 \\ R_{2333} & 0 \end{array} \\ \begin{array}{ll} R_{3111} & 0 \\ R_{3112} & 0 \\ R_{3113} & 0 \end{array} & \begin{array}{ll} R_{3121} & 0 \\ R_{3122} & 0 \\ R_{3123} & 0 \end{array} & \begin{array}{ll} R_{3131} & 0 \\ R_{3132} & 0 \\ R_{3133} & 0 \end{array} \\ \begin{array}{ll} R_{3211} & 0 \\ R_{3212} & 0 \\ R_{3213} & 0 \end{array} & \begin{array}{ll} R_{3221} & 0 \\ R_{3222} & 0 \\ R_{3223} & 0 \end{array} & \begin{array}{ll} R_{3231} & 0 \\ R_{3232} & 0 \\ R_{3233} & 0 \end{array} \\ \begin{array}{ll} R_{3311} & 0 \\ R_{3312} & 0 \\ R_{3313} & 0 \end{array} & \begin{array}{ll} R_{3321} & 0 \\ R_{3322} & 0 \\ R_{3323} & 0 \end{array} & \begin{array}{ll} R_{3331} & 0 \\ R_{3332} & 0 \\ R_{3333} & 0 \end{array} \end{array}\]



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