"N차원 구면"의 두 판 사이의 차이

수학노트
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잔글 (찾아 바꾸기 – “<h5>” 문자열을 “==” 문자열로)
 
(같은 사용자의 중간 판 18개는 보이지 않습니다)
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==이 항목의 수학노트 원문주소</h5>
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==개요==
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*  반지름 r인 n-차원 구면(n-sphere)
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==개요</h5>
 
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*  반지름 r인 n-차원 구면(n-sphere)<br>
 
 
** (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==
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==1차원 구면 <math>S^1</math></h5>
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===1차원 구면 <math>S^1</math>===
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<math>\begin{array}{ccc}  x_1 & = & r \cos (\theta ) \\  x_2 & = & r \sin (\theta ) \end{array}</math>
 
<math>\begin{array}{ccc}  x_1 & = & r \cos (\theta ) \\  x_2 & = & r \sin (\theta ) \end{array}</math>
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<math>0\leq \theta \leq 2\pi</math>
 
<math>0\leq \theta \leq 2\pi</math>
  
24번째 줄: 13번째 줄:
 
* [[원의 매개화와 삼각함수의 탄생]] 참조
 
* [[원의 매개화와 삼각함수의 탄생]] 참조
  
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===2차원 구면 <math>S^2</math>===
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==2차원 구면 <math>S^2</math></h5>
 
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<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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야코비안 <math>r^2 \sin \left(\phi _1\right)</math>
 
야코비안 <math>r^2 \sin \left(\phi _1\right)</math>
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<math>0\leq \theta \leq 2\pi</math>, <math>0\leq \phi_1 \leq \pi</math>
  
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<math>0\leq \theta \leq 2\pi</math>, <math>0\leq \phi_1 \leq 2\pi</math>
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==3차원 구면 <math>S^3</math></h5>
 
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===3차원 구면 <math>S^3</math>===
 
<math>\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}</math>
 
<math>\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}</math>
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야코비안 <math>r^3 \sin \left(\phi _1\right) \sin ^2\left(\phi _2\right)</math>
 
야코비안 <math>r^3 \sin \left(\phi _1\right) \sin ^2\left(\phi _2\right)</math>
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<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>===
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<math>0\leq \theta \leq 2\pi</math>, <math>0\leq \phi_1,\phi_2 \leq 2\pi</math>
 
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==4차원 구면 <math>S^4</math></h5>
 
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<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>
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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>
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<math>0\leq \theta \leq 2\pi</math>, <math>0\leq \phi_1,\phi_2,\phi_3  \leq \pi</math>
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<math>0\leq \theta \leq 2\pi</math>, <math>0\leq \phi_1,\phi_2,\phi_3  \leq 2\pi</math>
 
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==단위구면의 부피에의 응용</h5>
 
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* [[n차원 구면의 매개화|n차원 구면의 매개화]]<br> 다음의 점화식을 얻을 수 있다<br><math> \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)}</math><br><math>\omega_1=2\pi </math><br>
 
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==역사</h5>
 
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* http://www.google.com/search?hl=en&tbs=tl:1&q=
 
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* [[수학사연표 (역사)|수학사연표]]
 
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==메모</h5>
 
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* Math Overflow http://mathoverflow.net/search?q=
 
  
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===단위구면의 부피에의 응용===
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* [[n차원 구면의 매개화|n차원 구면의 매개화]] 다음의 점화식을 얻을 수 있다:<math> \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)}</math>:<math>\omega_1=2\pi </math>
  
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==관련된 항목들</h5>
 
  
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==메모==
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* 야코비안 행렬 : 7차원의 경우
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:<math>
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\left(
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\begin{array}{cccccccc}
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\cos \left(\theta _1\right) & -r \sin \left(\theta _1\right) & 0 & 0 & 0 & 0 & 0 & 0 \\
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\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 \\
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\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 \\
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\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 \\
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\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 \\
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\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 \\
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\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) \\
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\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) \\
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\end{array}
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\right)
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</math>
  
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==수학용어번역</h5>
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* 역행렬
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:<math>
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\left(
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\begin{array}{cccccccc}
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\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) \\
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-\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} \\
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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} \\
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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)</math>
 +
  
−
*  단어사전<br>
+
==관련된 항목들==
−
** http://translate.google.com/#en|ko|
+
* [[N차원 구면의 부피(면적)]]
−
** http://ko.wiktionary.org/wiki/
+
* [[자코비안]]
−
* 발음사전 http://www.forvo.com/search/
 
−
* [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://www.kss.or.kr/pds/sec/dic.aspx 한국통계학회 통계학 용어 온라인 대조표]
 
−
* [http://cgi.postech.ac.kr/cgi-bin/cgiwrap/sand/terms/terms.cgi 한국물리학회 물리학 용어집 검색기]
 
−
* [http://www.nktech.net/science/term/term_l.jsp?l_mode=cate&s_code_cd=MA 남·북한수학용어비교]
 
−
* [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]
 
  
−
 
 
  
−
 
+
  
−
==매스매티카 파일 및 계산 리소스</h5>
+
==매스매티카 파일 및 계산 리소스==
  
 
* https://docs.google.com/file/d/0B8XXo8Tve1cxakJjVmNBWHh2VTg/edit
 
* https://docs.google.com/file/d/0B8XXo8Tve1cxakJjVmNBWHh2VTg/edit
−
* http://www.wolframalpha.com/input/?i=
 
−
* http://functions.wolfram.com/
 
−
* [http://dlmf.nist.gov/ NIST Digital Library of Mathematical Functions]
 
−
* [http://people.math.sfu.ca/%7Ecbm/aands/toc.htm Abramowitz and Stegun Handbook of mathematical functions]
 
−
* [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]
 
−
* [http://numbers.computation.free.fr/Constants/constants.html Numbers, constants and computation]
 
−
* [https://docs.google.com/open?id=0B8XXo8Tve1cxMWI0NzNjYWUtNmIwZi00YzhkLTkzNzQtMDMwYmVmYmIxNmIw 매스매티카 파일 목록]
 
−
 
−
 
 
−
 
−
 
 
−
 
−
==사전 형태의 자료</h5>
 
−
 
−
* http://ko.wikipedia.org/wiki/
 
−
* http://en.wikipedia.org/wiki/
 
−
* [http://www.encyclopediaofmath.org/index.php/Main_Page Encyclopaedia of Mathematics]
 
−
* [http://dlmf.nist.gov NIST Digital Library of Mathematical Functions]
 
−
* [http://eqworld.ipmnet.ru/ The World of Mathematical Equations]
 
−
 
−
 
 
−
 
−
 
 
−
 
−
==리뷰논문, 에세이, 강의노트</h5>
 
−
 
−
 
 
−
 
−
 
 
−
 
−
 
 
−
 
−
==관련논문</h5>
 
−
 
−
* http://www.jstor.org/action/doBasicSearch?Query=
 
−
* http://www.ams.org/mathscinet
 
−
* http://dx.doi.org/
 
  
−
 
+
 +
  
−
 
+
==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.
  
−
==관련도서</h5>
 
  
−
*  도서내검색<br>
+
[[분류:미적분학]]
−
** http://books.google.com/books?q=
 
−
** http://book.daum.net/search/contentSearch.do?query=
 

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.
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