"N차원 구면"의 두 판 사이의 차이
Pythagoras0 (토론 | 기여) |
Pythagoras0 (토론 | 기여) |
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| (같은 사용자의 중간 판 하나는 보이지 않습니다) | |||
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** (n+1)-차원 유클리드 공간에서 다음 을 만족시키는 점들의 집합 <math>x_1^2+\cdots+x_{n+1}^2= r^2</math> | ** (n+1)-차원 유클리드 공간에서 다음 을 만족시키는 점들의 집합 <math>x_1^2+\cdots+x_{n+1}^2= r^2</math> | ||
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==매개화== | ==매개화== | ||
===1차원 구면 <math>S^1</math>=== | ===1차원 구면 <math>S^1</math>=== | ||
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* [[원의 매개화와 삼각함수의 탄생]] 참조 | * [[원의 매개화와 삼각함수의 탄생]] 참조 | ||
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===2차원 구면 <math>S^2</math>=== | ===2차원 구면 <math>S^2</math>=== | ||
<math>\begin{array}{ccc} x_1 & = & r \cos (\theta ) \sin \left(\phi _1\right) \\ x_2 & = & r \sin (\theta ) \sin \left(\phi _1\right) \\ x_3 & = & r \cos \left(\phi _1\right) \end{array}</math> | <math>\begin{array}{ccc} x_1 & = & r \cos (\theta ) \sin \left(\phi _1\right) \\ x_2 & = & r \sin (\theta ) \sin \left(\phi _1\right) \\ x_3 & = & r \cos \left(\phi _1\right) \end{array}</math> | ||
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<math>0\leq \theta \leq 2\pi</math>, <math>0\leq \phi_1 \leq \pi</math> | <math>0\leq \theta \leq 2\pi</math>, <math>0\leq \phi_1 \leq \pi</math> | ||
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===3차원 구면 <math>S^3</math>=== | ===3차원 구면 <math>S^3</math>=== | ||
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<math>0\leq \theta \leq 2\pi</math>, <math>0\leq \phi_1,\phi_2 \leq \pi</math> | <math>0\leq \theta \leq 2\pi</math>, <math>0\leq \phi_1,\phi_2 \leq \pi</math> | ||
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===4차원 구면 <math>S^4</math>=== | ===4차원 구면 <math>S^4</math>=== | ||
<math>\begin{array}{ccc} x_1 & = & r \cos (\theta ) \sin \left(\phi _1\right) \sin \left(\phi _2\right) \sin \left(\phi _3\right) \\ x_2 & = & r \sin (\theta ) \sin \left(\phi _1\right) \sin \left(\phi _2\right) \sin \left(\phi _3\right) \\ x_3 & = & r \sin \left(\phi _2\right) \sin \left(\phi _3\right) \cos \left(\phi _1\right) \\ x_4 & = & r \sin \left(\phi _3\right) \cos \left(\phi _2\right) \\ x_5 & = & r \cos \left(\phi _3\right) \end{array}</math> | <math>\begin{array}{ccc} x_1 & = & r \cos (\theta ) \sin \left(\phi _1\right) \sin \left(\phi _2\right) \sin \left(\phi _3\right) \\ x_2 & = & r \sin (\theta ) \sin \left(\phi _1\right) \sin \left(\phi _2\right) \sin \left(\phi _3\right) \\ x_3 & = & r \sin \left(\phi _2\right) \sin \left(\phi _3\right) \cos \left(\phi _1\right) \\ x_4 & = & r \sin \left(\phi _3\right) \cos \left(\phi _2\right) \\ x_5 & = & r \cos \left(\phi _3\right) \end{array}</math> | ||
야코비안 <math>r^4 \sin \left(\phi _1\right) \sin ^2\left(\phi _2\right) \sin ^3\left(\phi _3\right)</math> | 야코비안 <math>r^4 \sin \left(\phi _1\right) \sin ^2\left(\phi _2\right) \sin ^3\left(\phi _3\right)</math> | ||
<math>0\leq \theta \leq 2\pi</math>, <math>0\leq \phi_1,\phi_2,\phi_3 \leq \pi</math> | <math>0\leq \theta \leq 2\pi</math>, <math>0\leq \phi_1,\phi_2,\phi_3 \leq \pi</math> | ||
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==메모== | ==메모== | ||
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\end{array} | \end{array} | ||
\right)</math> | \right)</math> | ||
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==관련된 항목들== | ==관련된 항목들== | ||
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==매스매티카 파일 및 계산 리소스== | ==매스매티카 파일 및 계산 리소스== | ||
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* https://docs.google.com/file/d/0B8XXo8Tve1cxakJjVmNBWHh2VTg/edit | * https://docs.google.com/file/d/0B8XXo8Tve1cxakJjVmNBWHh2VTg/edit | ||
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==관련논문== | ==관련논문== | ||
2020년 12월 28일 (월) 02:57 기준 최신판
개요
- 반지름 r인 n-차원 구면(n-sphere)
- (n+1)-차원 유클리드 공간에서 다음 을 만족시키는 점들의 집합 \(x_1^2+\cdots+x_{n+1}^2= r^2\)
매개화
1차원 구면 \(S^1\)
\(\begin{array}{ccc} x_1 & = & r \cos (\theta ) \\ x_2 & = & r \sin (\theta ) \end{array}\) \(0\leq \theta \leq 2\pi\)
야코비안 \(r\)
2차원 구면 \(S^2\)
\(\begin{array}{ccc} x_1 & = & r \cos (\theta ) \sin \left(\phi _1\right) \\ x_2 & = & r \sin (\theta ) \sin \left(\phi _1\right) \\ x_3 & = & r \cos \left(\phi _1\right) \end{array}\) 야코비안 \(r^2 \sin \left(\phi _1\right)\) \(0\leq \theta \leq 2\pi\), \(0\leq \phi_1 \leq \pi\)
3차원 구면 \(S^3\)
\(\begin{array}{ccc} x_1 & = & r \cos (\theta ) \sin \left(\phi _1\right) \sin \left(\phi _2\right) \\ x_2 & = & r \sin (\theta ) \sin \left(\phi _1\right) \sin \left(\phi _2\right) \\ x_3 & = & r \sin \left(\phi _2\right) \cos \left(\phi _1\right) \\ x_4 & = & r \cos \left(\phi _2\right) \end{array}\) 야코비안 \(r^3 \sin \left(\phi _1\right) \sin ^2\left(\phi _2\right)\) \(0\leq \theta \leq 2\pi\), \(0\leq \phi_1,\phi_2 \leq \pi\)
4차원 구면 \(S^4\)
\(\begin{array}{ccc} x_1 & = & r \cos (\theta ) \sin \left(\phi _1\right) \sin \left(\phi _2\right) \sin \left(\phi _3\right) \\ x_2 & = & r \sin (\theta ) \sin \left(\phi _1\right) \sin \left(\phi _2\right) \sin \left(\phi _3\right) \\ x_3 & = & r \sin \left(\phi _2\right) \sin \left(\phi _3\right) \cos \left(\phi _1\right) \\ x_4 & = & r \sin \left(\phi _3\right) \cos \left(\phi _2\right) \\ x_5 & = & r \cos \left(\phi _3\right) \end{array}\) 야코비안 \(r^4 \sin \left(\phi _1\right) \sin ^2\left(\phi _2\right) \sin ^3\left(\phi _3\right)\) \(0\leq \theta \leq 2\pi\), \(0\leq \phi_1,\phi_2,\phi_3 \leq \pi\)
단위구면의 부피에의 응용
- n차원 구면의 매개화 다음의 점화식을 얻을 수 있다\[ \omega_{n}=\omega_{n-1}\left(\int_0^{\pi }\sin ^{n-1} \phi \, d\phi\right)=\omega_{n-1}\frac{\sqrt{\pi } \Gamma \left(\frac{n}{2}\right)}{\Gamma \left(\frac{n+1}{2}\right)}\]\[\omega_1=2\pi \]
메모
- 야코비안 행렬 : 7차원의 경우
\[ \left( \begin{array}{cccccccc} \cos \left(\theta _1\right) & -r \sin \left(\theta _1\right) & 0 & 0 & 0 & 0 & 0 & 0 \\ \sin \left(\theta _1\right) \cos \left(\theta _2\right) & r \cos \left(\theta _1\right) \cos \left(\theta _2\right) & -r \sin \left(\theta _1\right) \sin \left(\theta _2\right) & 0 & 0 & 0 & 0 & 0 \\ \sin \left(\theta _1\right) \sin \left(\theta _2\right) \cos \left(\theta _3\right) & r \sin \left(\theta _2\right) \cos \left(\theta _1\right) \cos \left(\theta _3\right) & r \sin \left(\theta _1\right) \cos \left(\theta _2\right) \cos \left(\theta _3\right) & -r \sin \left(\theta _1\right) \sin \left(\theta _2\right) \sin \left(\theta _3\right) & 0 & 0 & 0 & 0 \\ \sin \left(\theta _1\right) \sin \left(\theta _2\right) \sin \left(\theta _3\right) \cos \left(\theta _4\right) & r \sin \left(\theta _2\right) \sin \left(\theta _3\right) \cos \left(\theta _1\right) \cos \left(\theta _4\right) & r \sin \left(\theta _1\right) \sin \left(\theta _3\right) \cos \left(\theta _2\right) \cos \left(\theta _4\right) & r \sin \left(\theta _1\right) \sin \left(\theta _2\right) \cos \left(\theta _3\right) \cos \left(\theta _4\right) & -r \sin \left(\theta _1\right) \sin \left(\theta _2\right) \sin \left(\theta _3\right) \sin \left(\theta _4\right) & 0 & 0 & 0 \\ \sin \left(\theta _1\right) \sin \left(\theta _2\right) \sin \left(\theta _3\right) \sin \left(\theta _4\right) \cos \left(\theta _5\right) & r \sin \left(\theta _2\right) \sin \left(\theta _3\right) \sin \left(\theta _4\right) \cos \left(\theta _1\right) \cos \left(\theta _5\right) & r \sin \left(\theta _1\right) \sin \left(\theta _3\right) \sin \left(\theta _4\right) \cos \left(\theta _2\right) \cos \left(\theta _5\right) & r \sin \left(\theta _1\right) \sin \left(\theta _2\right) \sin \left(\theta _4\right) \cos \left(\theta _3\right) \cos \left(\theta _5\right) & r \sin \left(\theta _1\right) \sin \left(\theta _2\right) \sin \left(\theta _3\right) \cos \left(\theta _4\right) \cos \left(\theta _5\right) & -r \sin \left(\theta _1\right) \sin \left(\theta _2\right) \sin \left(\theta _3\right) \sin \left(\theta _4\right) \sin \left(\theta _5\right) & 0 & 0 \\ \sin \left(\theta _1\right) \sin \left(\theta _2\right) \sin \left(\theta _3\right) \sin \left(\theta _4\right) \sin \left(\theta _5\right) \cos \left(\theta _6\right) & r \sin \left(\theta _2\right) \sin \left(\theta _3\right) \sin \left(\theta _4\right) \sin \left(\theta _5\right) \cos \left(\theta _1\right) \cos \left(\theta _6\right) & r \sin \left(\theta _1\right) \sin \left(\theta _3\right) \sin \left(\theta _4\right) \sin \left(\theta _5\right) \cos \left(\theta _2\right) \cos \left(\theta _6\right) & r \sin \left(\theta _1\right) \sin \left(\theta _2\right) \sin \left(\theta _4\right) \sin \left(\theta _5\right) \cos \left(\theta _3\right) \cos \left(\theta _6\right) & r \sin \left(\theta _1\right) \sin \left(\theta _2\right) \sin \left(\theta _3\right) \sin \left(\theta _5\right) \cos \left(\theta _4\right) \cos \left(\theta _6\right) & r \sin \left(\theta _1\right) \sin \left(\theta _2\right) \sin \left(\theta _3\right) \sin \left(\theta _4\right) \cos \left(\theta _5\right) \cos \left(\theta _6\right) & -r \sin \left(\theta _1\right) \sin \left(\theta _2\right) \sin \left(\theta _3\right) \sin \left(\theta _4\right) \sin \left(\theta _5\right) \sin \left(\theta _6\right) & 0 \\ \sin \left(\theta _1\right) \sin \left(\theta _2\right) \sin \left(\theta _3\right) \sin \left(\theta _4\right) \sin \left(\theta _5\right) \sin \left(\theta _6\right) \cos \left(\theta _7\right) & r \sin \left(\theta _2\right) \sin \left(\theta _3\right) \sin \left(\theta _4\right) \sin \left(\theta _5\right) \sin \left(\theta _6\right) \cos \left(\theta _1\right) \cos \left(\theta _7\right) & r \sin \left(\theta _1\right) \sin \left(\theta _3\right) \sin \left(\theta _4\right) \sin \left(\theta _5\right) \sin \left(\theta _6\right) \cos \left(\theta _2\right) \cos \left(\theta _7\right) & r \sin \left(\theta _1\right) \sin \left(\theta _2\right) \sin \left(\theta _4\right) \sin \left(\theta _5\right) \sin \left(\theta _6\right) \cos \left(\theta _3\right) \cos \left(\theta _7\right) & r \sin \left(\theta _1\right) \sin \left(\theta _2\right) \sin \left(\theta _3\right) \sin \left(\theta _5\right) \sin \left(\theta _6\right) \cos \left(\theta _4\right) \cos \left(\theta _7\right) & r \sin \left(\theta _1\right) \sin \left(\theta _2\right) \sin \left(\theta _3\right) \sin \left(\theta _4\right) \sin \left(\theta _6\right) \cos \left(\theta _5\right) \cos \left(\theta _7\right) & r \sin \left(\theta _1\right) \sin \left(\theta _2\right) \sin \left(\theta _3\right) \sin \left(\theta _4\right) \sin \left(\theta _5\right) \cos \left(\theta _6\right) \cos \left(\theta _7\right) & -r \sin \left(\theta _1\right) \sin \left(\theta _2\right) \sin \left(\theta _3\right) \sin \left(\theta _4\right) \sin \left(\theta _5\right) \sin \left(\theta _6\right) \sin \left(\theta _7\right) \\ \sin \left(\theta _1\right) \sin \left(\theta _2\right) \sin \left(\theta _3\right) \sin \left(\theta _4\right) \sin \left(\theta _5\right) \sin \left(\theta _6\right) \sin \left(\theta _7\right) & r \sin \left(\theta _2\right) \sin \left(\theta _3\right) \sin \left(\theta _4\right) \sin \left(\theta _5\right) \sin \left(\theta _6\right) \sin \left(\theta _7\right) \cos \left(\theta _1\right) & r \sin \left(\theta _1\right) \sin \left(\theta _3\right) \sin \left(\theta _4\right) \sin \left(\theta _5\right) \sin \left(\theta _6\right) \sin \left(\theta _7\right) \cos \left(\theta _2\right) & r \sin \left(\theta _1\right) \sin \left(\theta _2\right) \sin \left(\theta _4\right) \sin \left(\theta _5\right) \sin \left(\theta _6\right) \sin \left(\theta _7\right) \cos \left(\theta _3\right) & r \sin \left(\theta _1\right) \sin \left(\theta _2\right) \sin \left(\theta _3\right) \sin \left(\theta _5\right) \sin \left(\theta _6\right) \sin \left(\theta _7\right) \cos \left(\theta _4\right) & r \sin \left(\theta _1\right) \sin \left(\theta _2\right) \sin \left(\theta _3\right) \sin \left(\theta _4\right) \sin \left(\theta _6\right) \sin \left(\theta _7\right) \cos \left(\theta _5\right) & r \sin \left(\theta _1\right) \sin \left(\theta _2\right) \sin \left(\theta _3\right) \sin \left(\theta _4\right) \sin \left(\theta _5\right) \sin \left(\theta _7\right) \cos \left(\theta _6\right) & r \sin \left(\theta _1\right) \sin \left(\theta _2\right) \sin \left(\theta _3\right) \sin \left(\theta _4\right) \sin \left(\theta _5\right) \sin \left(\theta _6\right) \cos \left(\theta _7\right) \\ \end{array} \right) \]
- 역행렬
\[ \left( \begin{array}{cccccccc} \cos \theta _1 & \left(\sin \theta _1\right) \left(\cos \theta _2\right) & \left(\sin \theta _1\right) \left(\sin \theta _2\right) \left(\cos \theta _3\right) & \left(\sin \theta _1\right) \left(\sin \theta _2\right) \left(\sin \theta _3\right) \left(\cos \theta _4\right) & \left(\sin \theta _1\right) \left(\sin \theta _2\right) \left(\sin \theta _3\right) \left(\sin \theta _4\right) \left(\cos \theta _5\right) & \left(\sin \theta _1\right) \left(\sin \theta _2\right) \left(\sin \theta _3\right) \left(\sin \theta _4\right) \left(\sin \theta _5\right) \left(\cos \theta _6\right) & \left(\sin \theta _1\right) \left(\sin \theta _2\right) \left(\sin \theta _3\right) \left(\sin \theta _4\right) \left(\sin \theta _5\right) \left(\sin \theta _6\right) \left(\cos \theta _7\right) & \left(\sin \theta _1\right) \left(\sin \theta _2\right) \left(\sin \theta _3\right) \left(\sin \theta _4\right) \left(\sin \theta _5\right) \left(\sin \theta _6\right) \left(\sin \theta _7\right) \\ -\frac{\sin \theta _1}{r} & \frac{\left(\cos \theta _1\right) \left(\cos \theta _2\right)}{r} & \frac{\left(\sin \theta _2\right) \left(\cos \theta _1\right) \left(\cos \theta _3\right)}{r} & \frac{\left(\sin \theta _2\right) \left(\sin \theta _3\right) \left(\cos \theta _1\right) \left(\cos \theta _4\right)}{r} & \frac{\left(\sin \theta _2\right) \left(\sin \theta _3\right) \left(\sin \theta _4\right) \left(\cos \theta _1\right) \left(\cos \theta _5\right)}{r} & \frac{\left(\sin \theta _2\right) \left(\sin \theta _3\right) \left(\sin \theta _4\right) \left(\sin \theta _5\right) \left(\cos \theta _1\right) \left(\cos \theta _6\right)}{r} & \frac{\left(\sin \theta _2\right) \left(\sin \theta _3\right) \left(\sin \theta _4\right) \left(\sin \theta _5\right) \left(\sin \theta _6\right) \left(\cos \theta _1\right) \left(\cos \theta _7\right)}{r} & \frac{\left(\sin \theta _2\right) \left(\sin \theta _3\right) \left(\sin \theta _4\right) \left(\sin \theta _5\right) \left(\sin \theta _6\right) \left(\sin \theta _7\right) \left(\cos \theta _1\right)}{r} \\ 0 & -\frac{\left(\sin \theta _2\right) \left(\csc \theta _1\right)}{r} & \frac{\left(\cos \theta _2\right) \left(\cos \theta _3\right) \left(\csc \theta _1\right)}{r} & \frac{\left(\sin \theta _3\right) \left(\cos \theta _2\right) \left(\cos \theta _4\right) \left(\csc \theta _1\right)}{r} & \frac{\left(\sin \theta _3\right) \left(\sin \theta _4\right) \left(\cos \theta _2\right) \left(\cos \theta _5\right) \left(\csc \theta _1\right)}{r} & \frac{\left(\sin \theta _3\right) \left(\sin \theta _4\right) \left(\sin \theta _5\right) \left(\cos \theta _2\right) \left(\cos \theta _6\right) \left(\csc \theta _1\right)}{r} & \frac{\left(\sin \theta _3\right) \left(\sin \theta _4\right) \left(\sin \theta _5\right) \left(\sin \theta _6\right) \left(\cos \theta _2\right) \left(\cos \theta _7\right) \left(\csc \theta _1\right)}{r} & \frac{\left(\sin \theta _3\right) \left(\sin \theta _4\right) \left(\sin \theta _5\right) \left(\sin \theta _6\right) \left(\sin \theta _7\right) \left(\cos \theta _2\right) \left(\csc \theta _1\right)}{r} \\ 0 & 0 & -\frac{\left(\sin \theta _3\right) \left(\csc \theta _1\right) \left(\csc \theta _2\right)}{r} & \frac{\left(\cos \theta _3\right) \left(\cos \theta _4\right) \left(\csc \theta _1\right) \left(\csc \theta _2\right)}{r} & \frac{\left(\sin \theta _4\right) \left(\cos \theta _3\right) \left(\cos \theta _5\right) \left(\csc \theta _1\right) \left(\csc \theta _2\right)}{r} & \frac{\left(\sin \theta _4\right) \left(\sin \theta _5\right) \left(\cos \theta _3\right) \left(\cos \theta _6\right) \left(\csc \theta _1\right) \left(\csc \theta _2\right)}{r} & \frac{\left(\sin \theta _4\right) \left(\sin \theta _5\right) \left(\sin \theta _6\right) \left(\cos \theta _3\right) \left(\cos \theta _7\right) \left(\csc \theta _1\right) \left(\csc \theta _2\right)}{r} & \frac{\left(\sin \theta _4\right) \left(\sin \theta _5\right) \left(\sin \theta _6\right) \left(\sin \theta _7\right) \left(\cos \theta _3\right) \left(\csc \theta _1\right) \left(\csc \theta _2\right)}{r} \\ 0 & 0 & 0 & -\frac{\left(\sin \theta _4\right) \left(\csc \theta _1\right) \left(\csc \theta _2\right) \left(\csc \theta _3\right)}{r} & \frac{\left(\cos \theta _4\right) \left(\cos \theta _5\right) \left(\csc \theta _1\right) \left(\csc \theta _2\right) \left(\csc \theta _3\right)}{r} & \frac{\left(\sin \theta _5\right) \left(\cos \theta _4\right) \left(\cos \theta _6\right) \left(\csc \theta _1\right) \left(\csc \theta _2\right) \left(\csc \theta _3\right)}{r} & \frac{\left(\sin \theta _5\right) \left(\sin \theta _6\right) \left(\cos \theta _4\right) \left(\cos \theta _7\right) \left(\csc \theta _1\right) \left(\csc \theta _2\right) \left(\csc \theta _3\right)}{r} & \frac{\left(\sin \theta _5\right) \left(\sin \theta _6\right) \left(\sin \theta _7\right) \left(\cos \theta _4\right) \left(\csc \theta _1\right) \left(\csc \theta _2\right) \left(\csc \theta _3\right)}{r} \\ 0 & 0 & 0 & 0 & -\frac{\left(\sin \theta _5\right) \left(\csc \theta _1\right) \left(\csc \theta _2\right) \left(\csc \theta _3\right) \left(\csc \theta _4\right)}{r} & \frac{\left(\cos \theta _5\right) \left(\cos \theta _6\right) \left(\csc \theta _1\right) \left(\csc \theta _2\right) \left(\csc \theta _3\right) \left(\csc \theta _4\right)}{r} & \frac{\left(\sin \theta _6\right) \left(\cos \theta _5\right) \left(\cos \theta _7\right) \left(\csc \theta _1\right) \left(\csc \theta _2\right) \left(\csc \theta _3\right) \left(\csc \theta _4\right)}{r} & \frac{\left(\sin \theta _6\right) \left(\sin \theta _7\right) \left(\cos \theta _5\right) \left(\csc \theta _1\right) \left(\csc \theta _2\right) \left(\csc \theta _3\right) \left(\csc \theta _4\right)}{r} \\ 0 & 0 & 0 & 0 & 0 & -\frac{\left(\sin \theta _6\right) \left(\csc \theta _1\right) \left(\csc \theta _2\right) \left(\csc \theta _3\right) \left(\csc \theta _4\right) \left(\csc \theta _5\right)}{r} & \frac{\left(\cos \theta _6\right) \left(\cos \theta _7\right) \left(\csc \theta _1\right) \left(\csc \theta _2\right) \left(\csc \theta _3\right) \left(\csc \theta _4\right) \left(\csc \theta _5\right)}{r} & \frac{\left(\sin \theta _7\right) \left(\cos \theta _6\right) \left(\csc \theta _1\right) \left(\csc \theta _2\right) \left(\csc \theta _3\right) \left(\csc \theta _4\right) \left(\csc \theta _5\right)}{r} \\ 0 & 0 & 0 & 0 & 0 & 0 & -\frac{\left(\sin \theta _7\right) \left(\csc \theta _1\right) \left(\csc \theta _2\right) \left(\csc \theta _3\right) \left(\csc \theta _4\right) \left(\csc \theta _5\right) \left(\csc \theta _6\right)}{r} & \frac{\left(\cos \theta _7\right) \left(\csc \theta _1\right) \left(\csc \theta _2\right) \left(\csc \theta _3\right) \left(\csc \theta _4\right) \left(\csc \theta _5\right) \left(\csc \theta _6\right)}{r} \\ \end{array} \right)\]
관련된 항목들
매스매티카 파일 및 계산 리소스
관련논문
- Bruno P. Zimmermann, On topological actions of finite groups on S^3, arXiv:1606.07626 [math.GT], June 24 2016, http://arxiv.org/abs/1606.07626
- Chapling, Richard. “A Hypergeometric Integral with Applications to the Fundamental Solution of Laplace’s Equation on Hyperspheres.” arXiv:1508.06689 [math-Ph], August 26, 2015. http://arxiv.org/abs/1508.06689.