구면(sphere)

이 항목의 스프링노트 원문주소

• 구면(sphere)

 

 

개요

• 구면기하학의 모델

 

 

==매개화

• 3차원상의 반지름이 R인 구면 \( x^2+y^2+z^2 = R^2\)
• 매개화
  \(X(u,v)=R(\cos u \sin v, \sin u \sin v, \cos v)\)
  \(0<u<2\pi\), \(0<v<\pi\)
  
• \(X_u=R(- \sin u \sin v , \cos u \sin v ,0)\)
  \(X_v=R( \cos u \cos v , \sin u \cos v ,-\sin v)\)
  \(N=(-\cos u \sin v, -\sin u \sin v, -\cos v)\)
  \(X_{uu}=R(-\cos u \sin v , -\sin u \sin v ,0)\)
  \(X_{uv}=R(-\cos v \sin u , \cos u \cos v , 0)\)
  \(X_{vv}=R(- \cos u \sin v , - \sin u \sin v , - \cos v )\)
  

 

 

==제1기본형식 (메트릭 텐서)

• \(E=R^2\sin^2 v\)
• \(F=0\)
• \(G=R^2\)

 

 

==크리스토펠 기호

• 크리스토펠 기호 항목 참조
  \(\Gamma^1_{11}=0\)
  \(\Gamma^1_{12}=\cot v\)
  \(\Gamma^1_{21}=\cot v\)
  \(\Gamma^1_{22}=0\)
  \(\Gamma^2_{11}=-\sin v \cos v\)
  \(\Gamma^2_{12}=0\)
  \(\Gamma^2_{21}=0\)
  \(\Gamma^2_{22}=0\)
  

 

 

==리만 곡률 텐서

• 리만 곡률 텐서
  \(\begin{array}{ll} \begin{array}{ll} R_{111}^1 & 0 \\ R_{112}^1 & 0 \end{array} & \begin{array}{ll} R_{121}^1 & 0 \\ R_{122}^1 & 0 \end{array} \\ \begin{array}{ll} R_{211}^1 & 0 \\ R_{212}^1 & 1 \end{array} & \begin{array}{ll} R_{221}^1 & -1 \\ R_{222}^1 & 0 \end{array} \\ \begin{array}{ll} R_{111}^2 & 0 \\ R_{112}^2 & -\sin ^2(v) \end{array} & \begin{array}{ll} R_{121}^2 & \sin ^2(v) \\ R_{122}^2 & 0 \end{array} \\ \begin{array}{ll} R_{211}^2 & 0 \\ R_{212}^2 & 0 \end{array} & \begin{array}{ll} R_{221}^2 & 0 \\ R_{222}^2 & 0 \end{array} \end{array}\)
  
• covariant tensor
  \(\begin{array}{ll} \begin{array}{ll} R_{1111} & 0 \\ R_{1112} & 0 \end{array} & \begin{array}{ll} R_{1121} & 0 \\ R_{1122} & 0 \end{array} \\ \begin{array}{ll} R_{1211} & 0 \\ R_{1212} & R^2 \sin ^2(v) \end{array} & \begin{array}{ll} R_{1221} & -R^2 \sin ^2(v) \\ R_{1222} & 0 \end{array} \\ \begin{array}{ll} R_{2111} & 0 \\ R_{2112} & -R^2 \sin ^2(v) \end{array} & \begin{array}{ll} R_{2121} & R^2 \sin ^2(v) \\ R_{2122} & 0 \end{array} \\ \begin{array}{ll} R_{2211} & 0 \\ R_{2212} & 0 \end{array} & \begin{array}{ll} R_{2221} & 0 \\ R_{2222} & 0 \end{array} \end{array}\)
  

 

 

==측지선

• 측지선 이 만족시키는 미분방정식
  \(\frac{d^2\alpha_k }{dt^2} + \Gamma^{k}_{~i j }\frac{d\alpha_i }{dt}\frac{d\alpha_j }{dt} = 0\)
  
• 풀어쓰면, 
  \(\frac{d^2 u}{dt^2} + 2\Gamma^{1}_{~1 2 }\frac{du }{dt}\frac{dv }{dt} = 0\)
  \(\frac{d^2 v}{dt^2} + \Gamma^{2}_{~1 1 }\frac{du }{dt}\frac{du }{dt} = 0\)
  

 

 

==가우스곡률

• 가우스곡률 항목 참조
  \(K = -\frac{1}{2\sqrt{EG}}\left(\frac{\partial}{\partial u}\frac{G_u}{\sqrt{EG}} + \frac{\partial}{\partial v}\frac{E_v}{\sqrt{EG}}\right)\)
  
• 반지름 R인 구면의 가우스곡률
  \(K=\frac{1}{R^2}\)
  

 

 

==라플라시안

• 위의 좌표계에서 \(u=\phi,v=\theta\) 로 생각하자.
  
• 라플라시안
  \(\Delta f = {1 \over r^2 \sin \theta} {\partial \over \partial \theta} \left( \sin \theta {\partial f \over \partial \theta} \right) + {1 \over r^2 \sin^2 \theta} {\partial^2 f \over \partial \phi^2}={1 \over r^2 }({\partial^2 f \over \partial \theta^2} +\cot\theta {\partial f \over \partial \theta} + \frac{1}{ \sin^2 \theta} {\partial^2 f \over \partial \phi^2})\)
  

 

 

==역사

 

• http://www.google.com/search?hl=en&tbs=tl:1&q=
• 수학사연표
•  

 

 

==메모

• http://users-phys.au.dk/fedorov/nucltheo/GTR/09/note6.pdf

 

 

==관련된 항목들

• 미분기하학
• 지도와 수학
• n차원 공의 부피
• 구면좌표계

 

 

==매스매티카 파일 및 계산 리소스

• https://docs.google.com/leaf?id=0B8XXo8Tve1cxYWZjZjFkNWEtNTVlNC00OTVjLWJhMDMtODc1NjgyMGQ1OTA5&sort=name&layout=list&num=50
• http://www.wolframalpha.com/input/?i=sphere
• http://functions.wolfram.com/
• NIST Digital Library of Mathematical Functions
• Abramowitz and Stegun Handbook of mathematical functions
• The On-Line Encyclopedia of Integer Sequences
• Numbers, constants and computation
• 매스매티카 파일 목록

 

 

==사전 형태의 자료

• http://ko.wikipedia.org/wiki/
• http://en.wikipedia.org/wiki/sphere
• http://mathworld.wolfram.com/GreatCircle.html

 

 

==관련논문

• http://www.jstor.org/action/doBasicSearch?Query=
• http://dx.doi.org/