N차원 구면

수학노트
(N차원 구면의 매개화에서 넘어옴)
둘러보기로 이동 검색으로 이동

개요

  • 반지름 r인 n-차원 구면(n-sphere)
    • (n+1)-차원 유클리드 공간에서 다음 을 만족시키는 점들의 집합 <math>x_1^2+\cdots+x_{n+1}^2= r^2</math>


매개화

1차원 구면 <math>S^1</math>

<math>\begin{array}{ccc} x_1 & = & r \cos (\theta ) \\ x_2 & = & r \sin (\theta ) \end{array}</math> <math>0\leq \theta \leq 2\pi</math>

야코비안 <math>r</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> 야코비안 <math>r^2 \sin \left(\phi _1\right)</math> <math>0\leq \theta \leq 2\pi</math>, <math>0\leq \phi_1 \leq \pi</math>


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> 야코비안 <math>r^3 \sin \left(\phi _1\right) \sin ^2\left(\phi _2\right)</math> <math>0\leq \theta \leq 2\pi</math>, <math>0\leq \phi_1,\phi_2 \leq \pi</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> 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>


단위구면의 부피에의 응용

  • 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>


메모

  • 야코비안 행렬 : 7차원의 경우
<math>

\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) </math>


  • 역행렬
<math>

\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)</math>


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.
원본 주소 "https://wiki.mathnt.net/index.php?title=N차원_구면&oldid=47838"