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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 Invariants of tensors ： ウィキペディア英語版 Invariants of tensors In mathematics, in the fields of multilinear algebra and representation theory, invariants of tensors are coefficients of the characteristic polynomial of the tensor ''A'':〔SPENCER, A. J. M. Continuum Mechanics. Longman, 1980.〕 :$\backslash \; p(\backslash lambda):=\backslash det\; (\backslash mathbf-\backslash lambda\; \backslash mathbf)$, where $\backslash mathbf\; E$ is the identity tensor and $\backslash lambda\backslash in\backslash mathbb\; C$ is the polynomial's indeterminate (it is important to bear in mind that a polynomial's indeterminate $\backslash lambda$ may also be a non-scalar as long as power, scaling and adding make sense for it, e.g., $p(\backslash mathbf\; A)$ is legitimate, and in fact, quite useful). The first invariant of an ''n''×''n'' tensor A ($I\_A$) is the coefficient for $\backslash lambda^$ (because the coefficient for $\backslash lambda^n$ is always 1), the second invariant ($II\_A$) is the coefficient for $\backslash lambda^$, etc., the ''n''th invariant is the free term. The definition of the ''invariants of tensors'' and specific notations used throughout the article were introduced into the field of rheology by Ronald Rivlin and became extremely popular there. In fact even the trace of a tensor $A$ is usually denoted as $I\_A$ in the textbooks on rheology. ==Properties== The first invariant (trace) is always the sum of the diagonal components: :$\backslash \; I\_A=A\_+A\_+\; \backslash cdots\; +\; A\_=\backslash mathrm(\backslash mathbf)\; \backslash ,$ The ''n''th invariant is just $\backslash pm\; \backslash det\; \backslash mathbf$, the determinant of $\backslash mathbf$ (up to sign). The invariants do not change with rotation of the coordinate system (they are objective). Obviously, any function of the invariants only is also objective. 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「Invariants of tensors」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.