"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%;">개요</h5>
 
<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>
  
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* <math>\rho ,\phi ,\theta</math><br>
 
* <math>x=\rho \cos\phi \, \sin\theta</math><br>
 
* <math>x=\rho \cos\phi \, \sin\theta</math><br>
 
* <math>y=\rho \sin\phi \, \sin\theta</math><br>
 
* <math>y=\rho \sin\phi \, \sin\theta</math><br>
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<h5>메트릭 텐서</h5>
 
<h5>메트릭 텐서</h5>
  
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<math>\left( \begin{array}{ccc}  1 & 0 & 0 \\  0 & \rho ^2 \text{Sin}[\theta ]^2 & 0 \\  0 & 0 & \rho ^2 \end{array} \right)</math>
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<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(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]] 항목 참조
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<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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2012년 8월 15일 (수) 18:18 판

이 항목의 스프링노트 원문주소

 

 

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

 

 

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

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

 

 

 

 

역사

 

 

메모

 

 

관련된 항목들

 

 

수학용어번역

 

 

사전 형태의 자료

 

 

관련논문

 

관련도서 및 추천도서

 

 

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