"구면좌표계"의 두 판 사이의 차이

수학노트
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<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">이 항목의 스프링노트 원문주소</h5>
 
  
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==개요==
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<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">개요</h5>
 
  
 
* <math>\rho ,\phi ,\theta</math><br>
 
* <math>\rho ,\phi ,\theta</math><br>
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* <math>\rho>0</math>, <math>0<\phi<2\pi</math>, <math>0<\theta<\pi</math><br>
 
* <math>\rho>0</math>, <math>0<\phi<2\pi</math>, <math>0<\theta<\pi</math><br>
  
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<h5>메트릭 텐서</h5>
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==메트릭 텐서==
  
 
<math>\left( \begin{array}{ccc}  1 & 0 & 0 \\  0 & \rho ^2 \sin ^2(\theta ) & 0 \\  0 & 0 & \rho ^2 \end{array} \right)</math>
 
<math>\left( \begin{array}{ccc}  1 & 0 & 0 \\  0 & \rho ^2 \sin ^2(\theta ) & 0 \\  0 & 0 & \rho ^2 \end{array} \right)</math>
  
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<h5>라플라시안</h5>
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==라플라시안==
  
 
* [[라플라시안(Laplacian)|라플라시안]]<br><math>\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}</math><br>
 
* [[라플라시안(Laplacian)|라플라시안]]<br><math>\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}</math><br>
  
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<h5>크리스토펠 기호</h5>
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==크리스토펠 기호==
  
 
* [[크리스토펠 기호]] 항목 참조<math>\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}</math>
 
* [[크리스토펠 기호]] 항목 참조<math>\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}</math>
  
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<h5>리만 곡률 텐서</h5>
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==리만 곡률 텐서==
  
 
* [[리만 곡률 텐서]]<br><math>\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}</math><br>
 
* [[리만 곡률 텐서]]<br><math>\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}</math><br>
 
*  covariant tensor<br><math>\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}</math><br>
 
*  covariant tensor<br><math>\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}</math><br>
  
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<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">역사</h5>
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==역사==
  
 
* [[수학사연표 (역사)|수학사연표]]
 
* [[수학사연표 (역사)|수학사연표]]
  
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<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">메모</h5>
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==메모==
  
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<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">관련된 항목들</h5>
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==관련된 항목들==
  
 
* [[구면조화함수(spherical harmonics)]]<br>
 
* [[구면조화함수(spherical harmonics)]]<br>
70번째 줄: 67번째 줄:
 
* [[구면(sphere)]]<br>
 
* [[구면(sphere)]]<br>
  
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<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">수학용어번역</h5>
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==수학용어번역==
  
 
* http://www.google.com/dictionary?langpair=en|ko&q=
 
* http://www.google.com/dictionary?langpair=en|ko&q=
 
* [http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=&fstr= 대한수학회 수학 학술 용어집]<br>
 
* [http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=&fstr= 대한수학회 수학 학술 용어집]<br>
 
** http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr=
 
** http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr=
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* [http://kms.or.kr/home/kor/board/bulletin_list_subject.asp?bulletinid=%7BD6048897-56F9-43D7-8BB6-50B362D1243A%7D&boardname=%BC%F6%C7%D0%BF%EB%BE%EE%C5%E4%B7%D0%B9%E6&globalmenu=7&localmenu=4 대한수학회 수학용어한글화 게시판]
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* [http://kms.or.kr/home/kor/board/bulletin_list_subject.asp?bulletinid=%7BD6048897-56F9-43D7-8BB6-50B362D1243A%7D&boardname=%BC%F6%C7%D0%BF%EB%BE%EE%C5%E4%B7%D0%B9%E6&globalmenu=7&localmenu=4 대한수학회 수학용어한글화 게시판]
  
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<h5>매스매티카 파일 및 계산 리소스</h5>
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==매스매티카 파일 및 계산 리소스==
  
 
* https://docs.google.com/file/d/0B8XXo8Tve1cxZEc2Q0c5M3p3QlU/edit
 
* https://docs.google.com/file/d/0B8XXo8Tve1cxZEc2Q0c5M3p3QlU/edit
96번째 줄: 93번째 줄:
 
* [https://docs.google.com/open?id=0B8XXo8Tve1cxMWI0NzNjYWUtNmIwZi00YzhkLTkzNzQtMDMwYmVmYmIxNmIw 매스매티카 파일 목록]
 
* [https://docs.google.com/open?id=0B8XXo8Tve1cxMWI0NzNjYWUtNmIwZi00YzhkLTkzNzQtMDMwYmVmYmIxNmIw 매스매티카 파일 목록]
  
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<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">사전 형태의 자료</h5>
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==사전 형태의 자료==
  
 
* [http://ko.wikipedia.org/wiki/%EA%B5%AC%EB%A9%B4%EC%A2%8C%ED%91%9C%EA%B3%84 http://ko.wikipedia.org/wiki/구면좌표계]
 
* [http://ko.wikipedia.org/wiki/%EA%B5%AC%EB%A9%B4%EC%A2%8C%ED%91%9C%EA%B3%84 http://ko.wikipedia.org/wiki/구면좌표계]
110번째 줄: 107번째 줄:
 
* [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]<br>
 
* [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]<br>
 
** http://www.research.att.com/~njas/sequences/?q=
 
** http://www.research.att.com/~njas/sequences/?q=
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<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">관련논문</h5>
 
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* http://www.jstor.org/action/doBasicSearch?Query=
 
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* http://dx.doi.org/
 
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<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">관련도서 및 추천도서</h5>
 
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*  도서내검색<br>
 
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** http://books.google.com/books?q=
 
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** http://book.daum.net/search/contentSearch.do?query=
 
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*  도서검색<br>
 
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** http://books.google.com/books?q=
 
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** http://book.daum.net/search/mainSearch.do?query=
 
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** http://book.daum.net/search/mainSearch.do?query=
 
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<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">관련기사</h5>
 
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*  네이버 뉴스 검색 (키워드 수정)<br>
 
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** http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=
 
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** http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=
 
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** http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=
 
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<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">블로그</h5>
 
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*  구글 블로그 검색<br>
 
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** http://blogsearch.google.com/blogsearch?q=
 
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* [http://navercast.naver.com/science/list 네이버 오늘의과학]
 
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* [http://math.dongascience.com/ 수학동아]
 
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* [http://www.ams.org/mathmoments/ Mathematical Moments from the AMS]
 
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* [http://betterexplained.com/ BetterExplained]
 

2012년 10월 20일 (토) 16:46 판


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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사전 형태의 자료

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