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이 항목의 수학노트 원문주소==
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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%;">이 항목의 수학노트 원문주소 | + | <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%;">이 항목의 수학노트 원문주소== |
* [[리만 곡률 텐서]] | * [[리만 곡률 텐서]] | ||
| 7번째 줄: | 7번째 줄: | ||
| − | ==개요 | + | ==개요== |
* [[접속 (connection)]]<math>\nabla</math>이 정의되어 있다고 하자 | * [[접속 (connection)]]<math>\nabla</math>이 정의되어 있다고 하자 | ||
| 17번째 줄: | 17번째 줄: | ||
| − | ==리만 곡률 텐서의 성분 | + | ==리만 곡률 텐서의 성분== |
* <math>{R^\rho}_{\sigma\mu\nu} = dx^\rho(R(\partial_{\mu},\partial_{\nu})\partial_{\sigma})</math> | * <math>{R^\rho}_{\sigma\mu\nu} = dx^\rho(R(\partial_{\mu},\partial_{\nu})\partial_{\sigma})</math> | ||
| 26번째 줄: | 26번째 줄: | ||
| − | <h5 style="margin: 0px; line-height: 2em;">곡률 2형식 | + | <h5 style="margin: 0px; line-height: 2em;">곡률 2형식== |
* <math>R(X,Y)\partial_{j}=\Omega_{j}^{s}(X,Y)\partial_s</math><br> | * <math>R(X,Y)\partial_{j}=\Omega_{j}^{s}(X,Y)\partial_s</math><br> | ||
| 38번째 줄: | 38번째 줄: | ||
| − | <h5 style="margin: 0px; line-height: 2em;">곡면의 경우 | + | <h5 style="margin: 0px; line-height: 2em;">곡면의 경우== |
* 제1기본형식이 <math>E=e(u,v),F=0,G=g(u,v)</math> 로 주어진 경우, 리만 곡률 텐서는 다음과 같다 (이외의 <math> R_{jkl}^i</math>는 0이다)<br><math> R_{212}^1 = \frac{e(u,v) \left(e^{(0,1)}(u,v) g^{(0,1)}(u,v)+g^{(1,0)}(u,v)^2\right)+g(u,v) \left(e^{(1,0)}(u,v) g^{(1,0)}(u,v)-2 e(u,v) \left(e^{(0,2)}(u,v)+g^{(2,0)}(u,v)\right)+e^{(0,1)}(u,v)^2\right)}{4 e(u,v)^2 g(u,v)}</math><br><math>R_{112}^2 = -\frac{e(u,v) \left(e^{(0,1)}(u,v) g^{(0,1)}(u,v)+g^{(1,0)}(u,v)^2\right)+g(u,v) \left(e^{(1,0)}(u,v) g^{(1,0)}(u,v)-2 e(u,v) \left(e^{(0,2)}(u,v)+g^{(2,0)}(u,v)\right)+e^{(0,1)}(u,v)^2\right)}{4 e(u,v) g(u,v)^2}</math><br><math>R_{221}^1 = -\frac{e(u,v) \left(e^{(0,1)}(u,v) g^{(0,1)}(u,v)+g^{(1,0)}(u,v)^2\right)+g(u,v) \left(e^{(1,0)}(u,v) g^{(1,0)}(u,v)-2 e(u,v) \left(e^{(0,2)}(u,v)+g^{(2,0)}(u,v)\right)+e^{(0,1)}(u,v)^2\right)}{4 e(u,v)^2 g(u,v)}</math><br><math>R_{121}^2 = \frac{e(u,v) \left(e^{(0,1)}(u,v) g^{(0,1)}(u,v)+g^{(1,0)}(u,v)^2\right)+g(u,v) \left(e^{(1,0)}(u,v) g^{(1,0)}(u,v)-2 e(u,v) \left(e^{(0,2)}(u,v)+g^{(2,0)}(u,v)\right)+e^{(0,1)}(u,v)^2\right)}{4 e(u,v) g(u,v)^2}</math><br> | * 제1기본형식이 <math>E=e(u,v),F=0,G=g(u,v)</math> 로 주어진 경우, 리만 곡률 텐서는 다음과 같다 (이외의 <math> R_{jkl}^i</math>는 0이다)<br><math> R_{212}^1 = \frac{e(u,v) \left(e^{(0,1)}(u,v) g^{(0,1)}(u,v)+g^{(1,0)}(u,v)^2\right)+g(u,v) \left(e^{(1,0)}(u,v) g^{(1,0)}(u,v)-2 e(u,v) \left(e^{(0,2)}(u,v)+g^{(2,0)}(u,v)\right)+e^{(0,1)}(u,v)^2\right)}{4 e(u,v)^2 g(u,v)}</math><br><math>R_{112}^2 = -\frac{e(u,v) \left(e^{(0,1)}(u,v) g^{(0,1)}(u,v)+g^{(1,0)}(u,v)^2\right)+g(u,v) \left(e^{(1,0)}(u,v) g^{(1,0)}(u,v)-2 e(u,v) \left(e^{(0,2)}(u,v)+g^{(2,0)}(u,v)\right)+e^{(0,1)}(u,v)^2\right)}{4 e(u,v) g(u,v)^2}</math><br><math>R_{221}^1 = -\frac{e(u,v) \left(e^{(0,1)}(u,v) g^{(0,1)}(u,v)+g^{(1,0)}(u,v)^2\right)+g(u,v) \left(e^{(1,0)}(u,v) g^{(1,0)}(u,v)-2 e(u,v) \left(e^{(0,2)}(u,v)+g^{(2,0)}(u,v)\right)+e^{(0,1)}(u,v)^2\right)}{4 e(u,v)^2 g(u,v)}</math><br><math>R_{121}^2 = \frac{e(u,v) \left(e^{(0,1)}(u,v) g^{(0,1)}(u,v)+g^{(1,0)}(u,v)^2\right)+g(u,v) \left(e^{(1,0)}(u,v) g^{(1,0)}(u,v)-2 e(u,v) \left(e^{(0,2)}(u,v)+g^{(2,0)}(u,v)\right)+e^{(0,1)}(u,v)^2\right)}{4 e(u,v) g(u,v)^2}</math><br> | ||
| 46번째 줄: | 46번째 줄: | ||
| − | ==역사 | + | ==역사== |
| 57번째 줄: | 57번째 줄: | ||
| − | ==메모 | + | ==메모== |
* [http://www.math.sunysb.edu/%7Ebrweber/401s09/coursefiles/Lecture24.pdf http://www.math.sunysb.edu/~brweber/401s09/coursefiles/Lecture24.pdf] | * [http://www.math.sunysb.edu/%7Ebrweber/401s09/coursefiles/Lecture24.pdf http://www.math.sunysb.edu/~brweber/401s09/coursefiles/Lecture24.pdf] | ||
| 68번째 줄: | 68번째 줄: | ||
| − | ==관련된 항목들 | + | ==관련된 항목들== |
* [[접속 (connection)]] | * [[접속 (connection)]] | ||
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| − | ==매스매티카 파일 및 계산 리소스 | + | ==매스매티카 파일 및 계산 리소스== |
* https://docs.google.com/leaf?id=0B8XXo8Tve1cxN2ZmMGViMGQtMmI4Ny00MmI3LWE4ZTYtYmQyNjZiYWVhMTc5&sort=name&layout=list&num=50 | * https://docs.google.com/leaf?id=0B8XXo8Tve1cxN2ZmMGViMGQtMmI4Ny00MmI3LWE4ZTYtYmQyNjZiYWVhMTc5&sort=name&layout=list&num=50 | ||
| 93번째 줄: | 93번째 줄: | ||
| − | <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%;">수학용어번역 | + | <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%;">수학용어번역== |
* 단어사전 http://www.google.com/dictionary?langpair=en|ko&q= | * 단어사전 http://www.google.com/dictionary?langpair=en|ko&q= | ||
| 108번째 줄: | 108번째 줄: | ||
| − | ==사전 형태의 자료 | + | ==사전 형태의 자료== |
* http://ko.wikipedia.org/wiki/ | * http://ko.wikipedia.org/wiki/ | ||
| 118번째 줄: | 118번째 줄: | ||
| − | ==리뷰논문, 에세이, 강의노트 | + | ==리뷰논문, 에세이, 강의노트== |
* http://www.math.csusb.edu/faculty/dunn/lecture1.pdf | * http://www.math.csusb.edu/faculty/dunn/lecture1.pdf | ||
| 126번째 줄: | 126번째 줄: | ||
| − | ==관련논문 | + | ==관련논문== |
* http://www.jstor.org/action/doBasicSearch?Query= | * http://www.jstor.org/action/doBasicSearch?Query= | ||
| 136번째 줄: | 136번째 줄: | ||
| − | ==관련도서 | + | ==관련도서== |
* 도서내검색<br> | * 도서내검색<br> | ||
| 146번째 줄: | 146번째 줄: | ||
| − | ==링크 | + | ==링크== |
* [http://www.ams.org/news/math-in-the-media/mathdigest-index Summaries of Media Coverage of Math] | * [http://www.ams.org/news/math-in-the-media/mathdigest-index Summaries of Media Coverage of Math] | ||
* 구글 블로그 검색<br> | * 구글 블로그 검색<br> | ||
** http://blogsearch.google.com/blogsearch?q= | ** http://blogsearch.google.com/blogsearch?q= | ||
2012년 11월 1일 (목) 13:48 판
이 항목의 수학노트 원문주소==
개요
- 접속 (connection)\(\nabla\)이 정의되어 있다고 하자
- 세 개의 벡터장 X,Y,Z 가 주어지면, 새로운 벡터장 R(X,Y)Z 를 얻는다
\(R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z\)
- covariant tensor
리만 곡률 텐서의 성분
- \({R^\rho}_{\sigma\mu\nu} = dx^\rho(R(\partial_{\mu},\partial_{\nu})\partial_{\sigma})\)
- 크리스토펠 기호 를 이용한 성분의 계산
\({R^\rho}_{\sigma\mu\nu} = \partial_\mu\Gamma^\rho_{\nu\sigma} - \partial_\nu\Gamma^\rho_{\mu\sigma} + \Gamma^\rho_{\mu\lambda}\Gamma^\lambda_{\nu\sigma} - \Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\sigma}\)
\({R^l}_{kij} = \partial_i\Gamma^l_{jk} - \partial_j\Gamma^l_{ik} + \Gamma^l_{is}\Gamma^s_{jk} - \Gamma^l_{js}\Gamma^s_{ik}\)
\(R_{\rho\sigma\mu\nu} = g_{\rho \zeta} {R^\zeta}_{\sigma\mu\nu} .\)
곡률 2형식==
- \(R(X,Y)\partial_{j}=\Omega_{j}^{s}(X,Y)\partial_s\)
- \(\,\Omega=d\omega +\frac{1}{2}[\omega,\omega]=d\omega +\omega\wedge \omega\)
- \(\Omega^i_{j}=d\omega^i_{j} +\sum_k \omega^i_{k}\wedge\omega^k_{j}\)
곡면의 경우==
- 제1기본형식이 \(E=e(u,v),F=0,G=g(u,v)\) 로 주어진 경우, 리만 곡률 텐서는 다음과 같다 (이외의 \( R_{jkl}^i\)는 0이다)
\( R_{212}^1 = \frac{e(u,v) \left(e^{(0,1)}(u,v) g^{(0,1)}(u,v)+g^{(1,0)}(u,v)^2\right)+g(u,v) \left(e^{(1,0)}(u,v) g^{(1,0)}(u,v)-2 e(u,v) \left(e^{(0,2)}(u,v)+g^{(2,0)}(u,v)\right)+e^{(0,1)}(u,v)^2\right)}{4 e(u,v)^2 g(u,v)}\)
\(R_{112}^2 = -\frac{e(u,v) \left(e^{(0,1)}(u,v) g^{(0,1)}(u,v)+g^{(1,0)}(u,v)^2\right)+g(u,v) \left(e^{(1,0)}(u,v) g^{(1,0)}(u,v)-2 e(u,v) \left(e^{(0,2)}(u,v)+g^{(2,0)}(u,v)\right)+e^{(0,1)}(u,v)^2\right)}{4 e(u,v) g(u,v)^2}\)
\(R_{221}^1 = -\frac{e(u,v) \left(e^{(0,1)}(u,v) g^{(0,1)}(u,v)+g^{(1,0)}(u,v)^2\right)+g(u,v) \left(e^{(1,0)}(u,v) g^{(1,0)}(u,v)-2 e(u,v) \left(e^{(0,2)}(u,v)+g^{(2,0)}(u,v)\right)+e^{(0,1)}(u,v)^2\right)}{4 e(u,v)^2 g(u,v)}\)
\(R_{121}^2 = \frac{e(u,v) \left(e^{(0,1)}(u,v) g^{(0,1)}(u,v)+g^{(1,0)}(u,v)^2\right)+g(u,v) \left(e^{(1,0)}(u,v) g^{(1,0)}(u,v)-2 e(u,v) \left(e^{(0,2)}(u,v)+g^{(2,0)}(u,v)\right)+e^{(0,1)}(u,v)^2\right)}{4 e(u,v) g(u,v)^2}\)
역사
메모
- http://www.math.sunysb.edu/~brweber/401s09/coursefiles/Lecture24.pdf
- http://users-phys.au.dk/fedorov/nucltheo/GTR/09/note6.pdf
관련된 항목들
매스매티카 파일 및 계산 리소스
- https://docs.google.com/leaf?id=0B8XXo8Tve1cxN2ZmMGViMGQtMmI4Ny00MmI3LWE4ZTYtYmQyNjZiYWVhMTc5&sort=name&layout=list&num=50
- http://www.wolframalpha.com/input/?i=
- 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
수학용어번역==
- 단어사전 http://www.google.com/dictionary?langpair=en%7Cko&q=
- 발음사전 http://www.forvo.com/search/
- 대한수학회 수학 학술 용어집
- 남·북한수학용어비교
- 대한수학회 수학용어한글화 게시판
사전 형태의 자료
- http://ko.wikipedia.org/wiki/
- http://en.wikipedia.org/wiki/Riemann_curvature_tensor
- The Online Encyclopaedia of Mathematics[1]
리뷰논문, 에세이, 강의노트
관련논문
관련도서
링크
\(R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z\)
\({R^\rho}_{\sigma\mu\nu} = \partial_\mu\Gamma^\rho_{\nu\sigma} - \partial_\nu\Gamma^\rho_{\mu\sigma} + \Gamma^\rho_{\mu\lambda}\Gamma^\lambda_{\nu\sigma} - \Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\sigma}\)
\({R^l}_{kij} = \partial_i\Gamma^l_{jk} - \partial_j\Gamma^l_{ik} + \Gamma^l_{is}\Gamma^s_{jk} - \Gamma^l_{js}\Gamma^s_{ik}\)
\(R_{\rho\sigma\mu\nu} = g_{\rho \zeta} {R^\zeta}_{\sigma\mu\nu} .\)
- \(R(X,Y)\partial_{j}=\Omega_{j}^{s}(X,Y)\partial_s\)
- \(\,\Omega=d\omega +\frac{1}{2}[\omega,\omega]=d\omega +\omega\wedge \omega\)
- \(\Omega^i_{j}=d\omega^i_{j} +\sum_k \omega^i_{k}\wedge\omega^k_{j}\)
곡면의 경우==
- 제1기본형식이 \(E=e(u,v),F=0,G=g(u,v)\) 로 주어진 경우, 리만 곡률 텐서는 다음과 같다 (이외의 \( R_{jkl}^i\)는 0이다)
\( R_{212}^1 = \frac{e(u,v) \left(e^{(0,1)}(u,v) g^{(0,1)}(u,v)+g^{(1,0)}(u,v)^2\right)+g(u,v) \left(e^{(1,0)}(u,v) g^{(1,0)}(u,v)-2 e(u,v) \left(e^{(0,2)}(u,v)+g^{(2,0)}(u,v)\right)+e^{(0,1)}(u,v)^2\right)}{4 e(u,v)^2 g(u,v)}\)
\(R_{112}^2 = -\frac{e(u,v) \left(e^{(0,1)}(u,v) g^{(0,1)}(u,v)+g^{(1,0)}(u,v)^2\right)+g(u,v) \left(e^{(1,0)}(u,v) g^{(1,0)}(u,v)-2 e(u,v) \left(e^{(0,2)}(u,v)+g^{(2,0)}(u,v)\right)+e^{(0,1)}(u,v)^2\right)}{4 e(u,v) g(u,v)^2}\)
\(R_{221}^1 = -\frac{e(u,v) \left(e^{(0,1)}(u,v) g^{(0,1)}(u,v)+g^{(1,0)}(u,v)^2\right)+g(u,v) \left(e^{(1,0)}(u,v) g^{(1,0)}(u,v)-2 e(u,v) \left(e^{(0,2)}(u,v)+g^{(2,0)}(u,v)\right)+e^{(0,1)}(u,v)^2\right)}{4 e(u,v)^2 g(u,v)}\)
\(R_{121}^2 = \frac{e(u,v) \left(e^{(0,1)}(u,v) g^{(0,1)}(u,v)+g^{(1,0)}(u,v)^2\right)+g(u,v) \left(e^{(1,0)}(u,v) g^{(1,0)}(u,v)-2 e(u,v) \left(e^{(0,2)}(u,v)+g^{(2,0)}(u,v)\right)+e^{(0,1)}(u,v)^2\right)}{4 e(u,v) g(u,v)^2}\)
\( R_{212}^1 = \frac{e(u,v) \left(e^{(0,1)}(u,v) g^{(0,1)}(u,v)+g^{(1,0)}(u,v)^2\right)+g(u,v) \left(e^{(1,0)}(u,v) g^{(1,0)}(u,v)-2 e(u,v) \left(e^{(0,2)}(u,v)+g^{(2,0)}(u,v)\right)+e^{(0,1)}(u,v)^2\right)}{4 e(u,v)^2 g(u,v)}\)
\(R_{112}^2 = -\frac{e(u,v) \left(e^{(0,1)}(u,v) g^{(0,1)}(u,v)+g^{(1,0)}(u,v)^2\right)+g(u,v) \left(e^{(1,0)}(u,v) g^{(1,0)}(u,v)-2 e(u,v) \left(e^{(0,2)}(u,v)+g^{(2,0)}(u,v)\right)+e^{(0,1)}(u,v)^2\right)}{4 e(u,v) g(u,v)^2}\)
\(R_{221}^1 = -\frac{e(u,v) \left(e^{(0,1)}(u,v) g^{(0,1)}(u,v)+g^{(1,0)}(u,v)^2\right)+g(u,v) \left(e^{(1,0)}(u,v) g^{(1,0)}(u,v)-2 e(u,v) \left(e^{(0,2)}(u,v)+g^{(2,0)}(u,v)\right)+e^{(0,1)}(u,v)^2\right)}{4 e(u,v)^2 g(u,v)}\)
\(R_{121}^2 = \frac{e(u,v) \left(e^{(0,1)}(u,v) g^{(0,1)}(u,v)+g^{(1,0)}(u,v)^2\right)+g(u,v) \left(e^{(1,0)}(u,v) g^{(1,0)}(u,v)-2 e(u,v) \left(e^{(0,2)}(u,v)+g^{(2,0)}(u,v)\right)+e^{(0,1)}(u,v)^2\right)}{4 e(u,v) g(u,v)^2}\)
역사
메모
- http://www.math.sunysb.edu/~brweber/401s09/coursefiles/Lecture24.pdf
- http://users-phys.au.dk/fedorov/nucltheo/GTR/09/note6.pdf
관련된 항목들
매스매티카 파일 및 계산 리소스
- https://docs.google.com/leaf?id=0B8XXo8Tve1cxN2ZmMGViMGQtMmI4Ny00MmI3LWE4ZTYtYmQyNjZiYWVhMTc5&sort=name&layout=list&num=50
- http://www.wolframalpha.com/input/?i=
- 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