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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Vector-valued differential form ： ウィキペディア英語版 Vector-valued differential form In mathematics, a vector-valued differential form on a manifold ''M'' is a differential form on ''M'' with values in a vector space ''V''. More generally, it is a differential form with values in some vector bundle ''E'' over ''M''. Ordinary differential forms can be viewed as R-valued differential forms. An important case of vector-valued differential forms are Lie algebra-valued forms. (A connection form is an example of such a form.) ==Formal definition== Let ''M'' be a smooth manifold and ''E'' → ''M'' be a smooth vector bundle over ''M''. We denote the space of smooth sections of a bundle ''E'' by Γ(''E''). An ''E''-valued differential form of degree ''p'' is a smooth section of the tensor product bundle of ''E'' with Λ''p''(''T'' *''M''), the ''p''-th exterior power of the cotangent bundle of ''M''. The space of such forms is denoted by :$\backslash Omega^p(M,E)\; =\; \backslash Gamma(E\backslash otimes\backslash Lambda^pT^$ *M). Because Γ is a monoidal functor, this can also be interpreted as :$\backslash Gamma(E\backslash otimes\backslash Lambda^pT^$ *M) = \Gamma(E) \otimes_ \Gamma(\Lambda^pT^ *M) = \Gamma(E) \otimes_ \Omega^p(M), where the latter two tensor products are the tensor product of modules over the ring Ω0(''M'') of smooth R-valued functions on ''M'' (see the sixth example here). By convention, an ''E''-valued 0-form is just a section of the bundle ''E''. That is, :$\backslash Omega^0(M,E)\; =\; \backslash Gamma(E).\backslash ,$ Equivalently, a ''E''-valued differential form can be defined as a bundle morphism :$TM\backslash otimes\backslash cdots\backslash otimes\; TM\; \backslash to\; E$ which is totally skew-symmetric. Let ''V'' be a fixed vector space. A ''V''-valued differential form of degree ''p'' is a differential form of degree ''p'' with values in the trivial bundle ''M'' × ''V''. The space of such forms is denoted Ω''p''(''M'', ''V''). When ''V'' = R one recovers the definition of an ordinary differential form. If ''V'' is finite-dimensional, then one can show that the natural homomorphism :$\backslash Omega^p(M)\; \backslash otimes\_\backslash mathbb\; V\; \backslash to\; \backslash Omega^p(M,V),$ where the first tensor product is of vector spaces over R, is an isomorphism.〔Proof: One can verify this for ''p''=0 by turning a basis for ''V'' into a set of constant functions to ''V'', which allows the construction of an inverse to the above homomorphism. The general case can be proved by noting that :$\backslash Omega^p(M,\; V)\; =\; \backslash Omega^0(M,\; V)\; \backslash otimes\_\; \backslash Omega^p(M),$ and that because $\backslash mathbb$ is a sub-ring of Ω0(''M'') via the constant functions, :$\backslash Omega^0(M,\; V)\; \backslash otimes\_\; \backslash Omega^p(M)\; =\; (V\; \backslash otimes\_\backslash mathbb\; \backslash Omega^0(M))\; \backslash otimes\_\; \backslash Omega^p(M)\; =\; V\; \backslash otimes\_\backslash mathbb\; (\backslash Omega^0(M)\; \backslash otimes\_\; \backslash Omega^p(M))\; =\; V\; \backslash otimes\_\backslash mathbb\; \backslash Omega^p(M).$〕 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「Vector-valued differential form」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.