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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 Geometric calculus ： ウィキペディア英語版 Geometric calculus In mathematics, geometric calculus extends the geometric algebra to include differentiation and integration. The formalism is powerful and can be shown to encompass other mathematical theories including differential geometry and differential forms.〔David Hestenes, Garrett Sobczyk: Clifford Algebra to Geometric Calculus, a Unified Language for mathematics and Physics (Dordrecht/Boston:G.Reidel Publ.Co., 1984, ISBN 90-277-2561-6〕 ==Differentiation== With a geometric algebra given, let ''a'' and ''b'' be vectors and let ''F(a)'' be a multivector-valued function. The directional derivative of ''F(a)'' along ''b'' is defined as :$\backslash nabla\_b\; F(a)\; =\; \backslash lim\_\}$ provided that the limit exists, where the limit is taken for scalar ''ε''. This is similar to the usual definition of a directional derivative but extends it to functions that are not necessarily scalar-valued. Next, choose a set of basis vectors $\backslash$ and consider the operators, noted $(\backslash partial\_i)$, that perform directional derivatives in the directions of $(e\_i)$: :$\backslash partial\_i\; :\; F\; \backslash mapsto\; (x\backslash mapsto\; \backslash nabla\_\; F(x))$ Then, using the Einstein summation notation, consider the operator : :$e^i\backslash partial\_i$ which means: :$F\; \backslash mapsto\; e^i\backslash partial\_i\; F$ or, more verbosely: :$F\; \backslash mapsto\; (x\backslash mapsto\; e^i\backslash nabla\_\; F(x))$ It can be shown that this operator is independent of the choice of frame, and can thus be used to define the ''geometric derivative'': :$\backslash nabla\; =\; e^i\backslash partial\_i$ This is similar to the usual definition of the gradient, but it, too, extends to functions that are not necessarily scalar-valued. It can be shown that the directional derivative is linear regarding its direction, that is: :$\backslash nabla\_\; =\; \backslash alpha\backslash nabla\_a\; +\; \backslash beta\backslash nabla\_b$ From this follows that the directional derivative is the inner product of its direction by the geometric derivative. All needs to be observed is that the direction $a$ can be written $a\; =\; (a\backslash cdot\; e^i)\; e\_i$, so that: :$\backslash nabla\_a\; =\; \backslash nabla\_\; =\; (a\backslash cdot\; e^i)\backslash nabla\_\; =\; a\backslash cdot(e^i\backslash nabla\_)\; =\; a\backslash cdot\; \backslash nabla$ For this reason, $\backslash nabla\_a\; F(x)$ is often noted $a\backslash cdot\; \backslash nabla\; F(x)$. The standard order of operations for the geometric derivative is that it acts only on the function closest to its immediate right. Given two functions ''F'' and ''G'', then for example we have :$\backslash nabla\; FG\; =\; (\backslash nabla\; F)G.$ 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「Geometric calculus」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.