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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 Weak derivative ： ウィキペディア英語版 Weak derivative In mathematics, a weak derivative is a generalization of the concept of the derivative of a function (''strong derivative'') for functions not assumed differentiable, but only integrable, i.e. to lie in the L''p'' space $\backslash mathrm^1(())$. See distributions for an even more general definition. == Definition == Let $u$ be a function in the Lebesgue space $L^1(())$. We say that $v$ in $L^1(())$ is a ''weak derivative'' of $u$ if, :$\backslash int\_a^b\; u(t)\backslash varphi\text{'}(t)dt=-\backslash int\_a^b\; v(t)\backslash varphi(t)dt$ for all infinitely differentiable functions $\backslash varphi$ with $\backslash varphi(a)=\backslash varphi(b)=0$. This definition is motivated by the integration technique of Integration by parts. Generalizing to $n$ dimensions, if $u$ and $v$ are in the space $L\_^1(U)$ of locally integrable functions for some open set $U\; \backslash subset\; \backslash mathbb^n$, and if $\backslash alpha$ is a multi-index, we say that $v$ is the $\backslash alpha^$-weak derivative of $u$ if :$\backslash int\_U\; u\; D^\; \backslash varphi=(-1)^\; \backslash int\_U\; v\backslash varphi$ for all $\backslash varphi\; \backslash in\; C^\_c\; (U)$, that is, for all infinitely differentiable functions $\backslash varphi$ with compact support in $U$. If $u$ has a weak derivative, it is often written $D^u$ since weak derivatives are unique (at least, up to a set of measure zero, see below). 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「Weak derivative」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.