翻訳と辞書 Words near each other ・ Tetragonoderus femoralis ・ Tetragonoderus figuratus ・ Tetragonoderus fimbriatus ・ Tetracycline ・ Tetracycline antibiotics ・ Tetracycline litigation ・ Tetracycline-controlled transcriptional activation ・ Tetracyclopropylmethane ・ Tetracynodon ・ Tetracystis ・ Tetrad ・ Tetrad (area) ・ Tetrad (astronomy) ・ Tetrad (genetics) ・ Tetrad (Greek philosophy) ・ Tetrad (index notation) ・ Tetrad (music) ・ Tetrad formalism ・ Tetrad Islands ・ Tetrad of media effects ・ Tetrad test ・ Tetradactylus ・ Tetradecagon ・ Tetradecahedron ・ Tetradecane ・ Tetradecathlon ・ Tetradecylthioacetic acid ・ Tetradenia ・ Tetradesmus ・ Tetradia lophoptera Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Tetrad (index notation) ： ウィキペディア英語版 Tetrad (index notation) In Riemannian geometry, we can introduce a coordinate system over the Riemannian manifold (at least, over a chart), giving ''n'' coordinates :$x\_\backslash ;\backslash text\backslash qquad\; i\; =\; 1,\; \backslash dots,\; n$ for an n-dimensional manifold. Locally, at least, this gives a basis for the 1-forms, dxi where d is the exterior derivative. The dual basis for the tangent space T is ei. Now, let's choose an orthonormal basis for the fibers of T. The rest is index manipulation. ==Example== Take a 3-sphere with the radius ''R'' and give it polar coordinates α, θ, φ. :e(eα)/R, :e(eθ)/R sin(α) and :e(eφ)/R sin(α) sin(θ) form an orthonormal basis of T. Call these e1, e2 and e3. Given the metric η, we can ignore the covariant and contravariant distinction for T. Then, the dreibein (triad), :$e\_1\; =\; R\backslash ,\; d\backslash alpha$ :$e\_2\; =\; R\backslash ,\; \backslash sin\; d\backslash theta$ :$e\_3\; =\; R\backslash ,\; \backslash sin\; \backslash sin\; d\backslash phi$. So, :$de\_1=0$ :$de\_2=R\; \backslash cos\; d\backslash alpha\; \backslash wedge\; d\backslash theta$ :$de\_3=R\; (\backslash cos\; \backslash sin\; d\backslash alpha\; \backslash wedge\; d\backslash phi\; +\; \backslash sin\; \backslash cos\; d\backslash theta\; \backslash wedge\; d\backslash phi)$. from the relation :$d\_\backslash mathbf\; e\; =\; de\; +\; A\; \backslash wedge\; e\; =\; 0$, we get :$A\_\; =\; -\backslash cos\; \backslash ,\; d\backslash theta$ :$A\_\; =\; -\backslash cos\; \backslash ,\; \backslash sin\; d\backslash phi$ :$A\_\; =\; -\backslash cos\; \backslash ,\; d\backslash phi$. (dAη=0 tells us A is antisymmetric) So, $\backslash mathbf\; =\; d\backslash mathbf\; +\; \backslash mathbf\; \backslash wedge\; \backslash mathbf$, :$F\_=\backslash sin\; d\backslash alpha\backslash wedge\; d\backslash theta$ :$F\_=\backslash sin\; \backslash sin\; d\backslash alpha\backslash wedge\; d\backslash phi$ :$F\_=\backslash sin^2\; \backslash sin\; d\backslash theta\backslash wedge\; d\backslash phi$ 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Tetrad (index notation)」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.